Finite Group Databasebeta

← #29742969

Subgroups — #29742969

Order 15616.

968 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840043C211yesmaximal, characteristic, complemented
2C2×C4×D488C211yesmaximal, characteristic
3#21449669C211yesmaximal, characteristic, complemented
4C4⋊C4×D244C211yesmaximal, characteristic, complemented
5#21447346C211yesmaximal, characteristic, complemented
6#21447292C211yesmaximal, characteristic, complemented
7#21447206C211yesmaximal, characteristic, complemented
8#21447148C211yesmaximal, characteristic, complemented
9#21446552C211yesmaximal, characteristic, complemented
10#21446166C211yesmaximal, characteristic
11#21446166C211yesmaximal, characteristic
12#21446069C211yesmaximal, characteristic, complemented
13#21446069C211yesmaximal, characteristic, complemented
14#21446069C211yesmaximal, characteristic, complemented
15C61×#1916C211yesmaximal, characteristic, complemented
16C22×D488C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519755—21no—
19C2×C4×D244C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21C2×C4×D244C2211yescharacteristic
22C2×C4×D244C2211yescharacteristic
23C2×C4⋊C4×D61—21no—
24#16840043—24no—
25C4×D488—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519711C2211yescharacteristic
28C2×#14519711C2211yescharacteristic
29C2×#14519711C2211yescharacteristic
30C2×#14519711C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32C2×#14519699C2211yescharacteristic
33C2×#14519699C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37#16839684—21no—
38#16839675—21no—
39#16839664C2211yescharacteristic
40#16839664C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839660C2211yescharacteristic
44#16839660C2211yescharacteristic
45#16839636—21no—
46#16839636—21no—
47#16839636—21no—
48#16839632C2211yescharacteristic
49#16839619—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C4×C8×C61C2211yescharacteristic
56C4×C4⋊C4×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×D244C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68C2×D488—22no—
69C2×D488—22no—
70C2×D488Q814yescomplemented
71#14519777—22no—
72#14519777—41no—
73#14519755—44no—
74C4×D244—22no—
75C4×D244—22no—
76C4×D244—22no—
77C4×D244—22no—
78C4×D244—22no—
79C4×D244—41no—
80C4×D244—41no—
81C4×D244—41no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C4⋊C4×D61—44no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104#14519711—22no—
105#14519711—22no—
106#14519711—22no—
107#14519711—22no—
108#14519711—41no—
109#14519699—22no—
110#14519699—22no—
111#14519699—22no—
112#14519699—22no—
113#14519699—41no—
114#14519699D414yes—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C4⋊C4×C61D411yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128C4⋊C8×C61—22no—
129C4×C8×C61—22no—
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C23×D61—21no—
133C23×D61—21no—
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244C2×D412yes—
156C2×D244C2×Q812yes—
157C2×D244C2×Q812yes—
158C2×D244#4112yes—
159C2×C4×D61—24no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61—81no—
169C2×C4×D61—81no—
170C2×C4×D61—81no—
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122—81no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61C2×D412yes—
185C2×C8×C61C2×Q812yes—
186C4⋊C4×C61—41no—
187C4⋊C4×C61D814yescomplemented
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194#11758198—24no—
195#11758198—42no—
196#11758198—42no—
197#11758198—42no—
198#11758198—42no—
199#11758198—42no—
200#11758198—42no—
201#11758198—81no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—42no—
213C4⋊Dic61—42no—
214C4⋊Dic61—81no—
215C4⋊Dic61C2×D412yes—
216C4⋊Dic61#4112yes—
217#11758196—42no—
218#11758196—42no—
219#11758196—42no—
220#11758196—42no—
221D488—28no—
222C23×C61C22×D411yescharacteristic
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—22no—
226C22×D61—22no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61C2×D812yes—
253C2×C4×C61C2×D812yes—
254C2×C4×C61C2×D812yes—
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61C22×Q811yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279D244—22no—
280D244—22no—
281D244—22no—
282D244—22no—
283D244—22no—
284D244—41no—
285D244—41no—
286D244C22⋊Q814yescomplemented
287C4×D61—48no—
288C4×D61—84no—
289C4×D61—84no—
290C4×D61—84no—
291Dic122—84no—
292C8×C61—22no—
293C8×C61—22no—
294C8×C61—41no—
295C8×C61—41no—
296#26368—611nomaximal, Sylow
297C22×C61#52511yescharacteristic
298C22×C61D4211yescharacteristic
299C22×C61D4×Q811yescharacteristic
300C22×C61C22×D811yescharacteristic
301C22×C61#57511yescharacteristic
302C22×C61#57611yescharacteristic
303C22×C61#57811yescharacteristic
304C2×D61—24no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—42no—
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—42no—
313C2×D61—42no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C4×C61—22no—
319C4×C61—22no—
320C4×C61—22no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61C2×C22⋊Q811yescharacteristic
325C4×C61#53911yescharacteristic
326C4×C61#54711yescharacteristic
327C4×C61#55411yescharacteristic
328Dic61—42no—
329Dic61—42no—
330Dic61—81no—
331Dic61—81no—
332Dic61—81no—
333Dic61—81no—
334C2×#474—611no—
335C2×C4×D8—611no—
336#2452—611no—
337D4×C4⋊C4—611no—
338#2093—611no—
339#2083—611no—
340#2067—611no—
341#2057—611no—
342#1942—611no—
343#1916D6111yescharacteristic, complemented
344#1879—611no—
345#1879—611no—
346#1862—611no—
347#1862—611no—
348#1862—611no—
349C2×C61#244911yescharacteristic
350C2×C61#310011yescharacteristic
351C2×C61#312411yescharacteristic
352C2×C61#326911yescharacteristic
353C2×C61D4×D811yescharacteristic
354C2×C61Q8×D811yescharacteristic
355C2×C61#338911yescharacteristic
356D61—44no—
357D61—82no—
358C22×D8—611no—
359C2×C4⋊Q8—611no—
360C2×C22⋊Q8—1221no—
361C2×C4×D4—611no—
362C2×C4×D4—611no—
363C2×C4×D4—611no—
364C2×C4×D4—611no—
365C22×C4⋊C4—1221no—
366#474—1224no—
367C4×D8—1224no—
368C2×#107—611no—
369C2×#107—611no—
370C2×#107—611no—
371C2×#107—611no—
372C2×#107—611no—
373C2×C4⋊C8C2×D6111yescharacteristic
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#102—611no—
380C2×C4×C8C2×D6111yescharacteristic
381#396—1221no—
382#393—1221no—
383#390—611no—
384#390—611no—
385#389—611no—
386#389—611no—
387#389—611no—
388#389—611no—
389#385—1221no—
390#385—1221no—
391#385—1221no—
392#384—611no—
393#380—1221no—
394C4×C4⋊C4C2×D6111yescharacteristic
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#336C2×D6111yescharacteristic
400#336C2×D6111yescharacteristic
401#336C2×D6111yescharacteristic
402#336C2×D6111yescharacteristic
403C61#2636811yescharacteristic, complemented, Sylow
404C22×Q8—1221no—
405C22×D4—611no—
406C22×D4—611no—
407C23×C4—1221no—
408C23×C4—1221no—
409C23×C4—1221no—
410C23×C4—1221no—
411C2×D8—614no—
412C2×D8—1222no—
413C2×D8—1222no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C22×C8C22×D6111yescharacteristic
417C22×C8C22×D6111yescharacteristic
418C4⋊Q8—1222no—
419C4⋊Q8—2441no—
420C22⋊Q8—2444no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C4×D4—2441no—
427C4×D4—2441no—
428C4×D4—2441no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—2444no—
443C2×C4⋊C4D24411yescharacteristic
444C2×C4⋊C4D24411yescharacteristic
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C42C22×D6111yescharacteristic
454C2×C42C22×D6111yescharacteristic
455C2×C42C22×D6111yescharacteristic
456#107—1222no—
457#107—1222no—
458#107—1222no—
459#107—1222no—
460#107—2441no—
461C4⋊C8—22no—
462#102—614no—
463#102—1222no—
464#102—1222no—
465#102—1222no—
466#102—1222no—
467#102—2441no—
468C4×C8—22no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95D24411yescharacteristic
474#95D24411yescharacteristic
475C24—1221no—
476C24—1221no—
477C2×Q8—4881no—
478C2×Q8—2442no—
479C2×Q8—2442no—
480C2×D4—612no—
481C2×D4—612no—
482C2×D4—612no—
483C2×D4—612no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C22×C4—1224no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—4881no—
500C22×C4—4881no—
501C22×C4—4881no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4—2442no—
509C22×C4—2442no—
510C22×C4C2×D24411yescharacteristic
511C22×C4C2×D24411yescharacteristic
512C22×C4C2×D24411yescharacteristic
513C22×C4C2×D24411yescharacteristic
514C22×C4C2×D24411yescharacteristic
515C22×C4C2×D24411yescharacteristic
516C22×C4C23×D6111yescharacteristic
517D8—1228no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C2×C8D4×D6112yes—
529C2×C8Q8×D6112yes—
530C4⋊C4—41no—
531C4⋊C4—612no—
532C4⋊C4—612no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—4881no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C4⋊C4—2442no—
549C4⋊C4—2442no—
550C4⋊C4D48814yescomplemented
551C22⋊C4—1224no—
552C22⋊C4—4881no—
553C22⋊C4—2442no—
554C22⋊C4—2442no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—21no—
565C23—1222no—
566C23—1222no—
567C23—1222no—
568C23—1222no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23—2441no—
573C23—2441no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23C22×D24411yescharacteristic
580Q8—4884no—
581D4—614no—
582D4—1222no—
583D4—1222no—
584D4—1222no—
585D4—1222no—
586D4—1222no—
587D4—2441no—
588D4—2441no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—4884no—
607C2×C4—4884no—
608C2×C4—4884no—
609C2×C4—2448no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4C2×D48812yes—
627C2×C4C2×D48812yes—
628C2×C4C2×D48812yes—
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4C2×Q8×D6111yescharacteristic
631C2×C4#41×D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4#1451982311yescharacteristic
634C2×C4#1451982411yescharacteristic
635C8—22no—
636C8—22no—
637C8—41no—
638C8—41no—
639C22—1224no—
640C22—4881no—
641C22—4881no—
642C22—4881no—
643C22—4881no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22—2442no—
648C22—2442no—
649C22—2442no—
650C22—2442no—
651C22—2442no—
652C22—2442no—
653C22#1684027411yescharacteristic
654C22D4×D24411yescharacteristic
655C22Q8×D24411yescharacteristic
656C22C22×D48811yescharacteristic
657C22#1684056211yescharacteristic
658C22#1684056711yescharacteristic
659C22#1684057911yescharacteristic
660C4—22no—
661C4—22no—
662C4—22no—
663C4—41no—
664C4—41no—
665C4—41no—
666C4—4881no—
667C4—4881no—
668C4—4881no—
669C4—4881no—
670C4—2442no—
671C4—2442no—
672C4C22⋊Q8×D6111yescharacteristic
673C4#1684035311yescharacteristic
674C4#1684041011yescharacteristic
675C4#1684046811yescharacteristic
676C2—4882no—
677C2—2444no—
678C2#2144963311yescharacteristic
679C2#2145582011yescharacteristic
680C2#2145608311yescharacteristic
681C2#2145737911yescharacteristic
682C2D4×D48811yescharacteristic
683C2Q8×D48811yescharacteristic
684C2#2145877911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1916C2
socleC23×C61C22×D4
prime radicalC22×D488C22
hypercenter#1916D61
hypocenterC61#26368
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1916

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16840043C2semidirect
3. #21449669C2semidirect
4. C4⋊C4×D244C2semidirect
5. #21447346C2semidirect
6. #21447292C2semidirect
7. #21447206C2semidirect
8. #21447148C2semidirect
9. #21446552C2semidirect
12. #21446069C2semidirect
13. #21446069C2semidirect
14. #21446069C2semidirect
15. C61×#1916C2semidirect
70. C2×D488Q8semidirect
187. C4⋊C4×C61D8semidirect
286. D244C22⋊Q8semidirect
343. #1916D61semidirect
403. C61#26368semidirect
550. C4⋊C4D488semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168400432
C2×C4×D4882
#214496692
C4⋊C4×D2442
#214473462
#214472922
#214472062
#214471482
#214465522
#214461662
#214460692
C61×#19162
#2636861