Finite Group Databasebeta

← #29742970

Subgroups — #29742970

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×#16840040C211yesmaximal, characteristic, complemented
2C2×D8×Dic61C211yesmaximal, characteristic
3#21449672C211yesmaximal, characteristic, complemented
4D4×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447347C211yesmaximal, characteristic, complemented
6#21447292C211yesmaximal, characteristic, complemented
7#21447203C211yesmaximal, characteristic, complemented
8#21447146C211yesmaximal, characteristic, complemented
9#21446551C211yesmaximal, characteristic, complemented
10#21446375C211yesmaximal, characteristic, complemented
11#21446169C211yesmaximal, characteristic
12#21446068C211yesmaximal, characteristic, complemented
13#21446066C211yesmaximal, characteristic, complemented
14#21446064C211yesmaximal, characteristic, complemented
15C61×#1879C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519754—21no—
18C2×D4×Dic61C2211yescharacteristic
19C2×D4×Dic61C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C22×C4⋊Dic61—21no—
22#16840040—24no—
23D8×Dic61—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519711C2211yescharacteristic
26C2×#14519710C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519698C2211yescharacteristic
31C2×#14519698C2211yescharacteristic
32C2×#14519698C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839685—21no—
36#16839674—21no—
37#16839665C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839637—21no—
43#16839637—21no—
44#16839634—21no—
45#16839632C2211yescharacteristic
46#16839622—21no—
47C4×C4⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×D8×C61C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#102×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#390C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C23×Dic61—21no—
63C23×Dic61—21no—
64C23×Dic61—21no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519776—22no—
68#14519776—41no—
69#14519754—44no—
70D4×Dic61—22no—
71D4×Dic61—22no—
72D4×Dic61—22no—
73D4×Dic61—41no—
74D4×Dic61—41no—
75D4×Dic61—41no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—44no—
81C2×C4⋊Dic61D411yescharacteristic
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97#14519711—22no—
98#14519711—22no—
99#14519710—22no—
100#14519710—22no—
101#14519710—41no—
102#14519706—22no—
103#14519698—22no—
104#14519698—22no—
105#14519698—41no—
106#14519698D414yes—
107C8×Dic61—22no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C22×D4×C61C2311yescharacteristic
115C22×D4×C61C2311yescharacteristic
116C23×C4×C61—21no—
117C2×D8×C61—22no—
118C2×D8×C61—22no—
119C2×D8×C61Q814yescomplemented
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C4×D4×C61—22no—
123C4×D4×C61—22no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C22⋊C4×C61—21no—
129C2×C42×C61C2311yescharacteristic
130#102×C61—22no—
131#102×C61—22no—
132C24×C61—21no—
133C24×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61C2×D412yes—
143C2×D4×C61C2×Q812yes—
144C2×D4×C61C2×Q812yes—
145C2×D4×C61#4112yes—
146C22×C4×C61—24no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—42no—
150C22×C4×C61—42no—
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2411yescharacteristic
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—42no—
164C22×Dic61—81no—
165C22×Dic61—81no—
166C22×Dic61—81no—
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C2×Dic122—42no—
172C2×Dic122—42no—
173C2×Dic122—81no—
174D8×C61—28no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61C2×Q812yes—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61#4112yes—
185C22⋊C4×C61—24no—
186C22⋊C4×C61—42no—
187C22⋊C4×C61—81no—
188C42×C61—21no—
189C42×C61—21no—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—42no—
194C22⋊Dic61—42no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61—42no—
204C4⋊Dic61—42no—
205C4⋊Dic61—42no—
206C4⋊Dic61—81no—
207C4⋊Dic61D814yescomplemented
208C4⋊Dic61C2×D412yes—
209#11758196—42no—
210#11758196—42no—
211#11758196—42no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418C2×D412yes—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—22no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61—41no—
231C23×C61—41no—
232C23×C61—41no—
233C23×C61—41no—
234C23×C61—41no—
235C23×C61—41no—
236C23×C61C22×D411yescharacteristic
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—22no—
240D4×C61—22no—
241D4×C61—22no—
242D4×C61—41no—
243D4×C61—41no—
244D4×C61C22⋊Q814yescomplemented
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—48no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61C2×D812yes—
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C22×Q811yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×C4×C612−1+411yescharacteristic
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—84no—
287C2×Dic61—84no—
288C2×Dic61—84no—
289C2×Dic61C2×D812yes—
290C2×Dic61C2×D812yes—
291Dic122—84no—
292C8×C61—22no—
293C8×C61—41no—
294#92418—22no—
295#92418—41no—
296#26368—611nomaximal, Sylow
297C22×C61—24no—
298C22×C61—42no—
299C22×C61—42no—
300C22×C61—42no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61—42no—
307C22×C61—81no—
308C22×C61—81no—
309C22×C61—81no—
310C22×C61—81no—
311C22×C61#52511yescharacteristic
312C22×C61D4211yescharacteristic
313C22×C61D4×Q811yescharacteristic
314C22×C61C22×D811yescharacteristic
315C22×C61#57511yescharacteristic
316C22×C61#57611yescharacteristic
317C22×C61#57811yescharacteristic
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—41no—
321C4×C61—81no—
322C4×C61C2×C22⋊Q811yescharacteristic
323C4×C61#53911yescharacteristic
324C4×C61#54711yescharacteristic
325C4×C61#55411yescharacteristic
326Dic61—22no—
327Dic61—22no—
328Dic61—41no—
329Dic61—41no—
330Dic61—42no—
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#1916—611no—
344#1879—611no—
345#1879D6111yescharacteristic
346#1862—611no—
347#1862—611no—
348#1862—611no—
349C2×C61—44no—
350C2×C61—82no—
351C2×C61#244911yescharacteristic
352C2×C61#310011yescharacteristic
353C2×C61#312411yescharacteristic
354C2×C61#326911yescharacteristic
355C2×C61D4×D811yescharacteristic
356C2×C61Q8×D811yescharacteristic
357C2×C61#338911yescharacteristic
358C22×D8C2×D6111yescharacteristic
359C2×C4⋊Q8—611no—
360C2×C22⋊Q8—1221no—
361C2×C4×D4—611no—
362C2×C4×D4—611no—
363C2×C4×D4—611no—
364C2×C4×D4C2×D6111yescharacteristic
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⋊C8—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102C2×D6111yescharacteristic
379C2×#102C2×D6111yescharacteristic
380C2×C4×C8—611no—
381#396—1221no—
382#393—1221no—
383#390—611no—
384#390C2×D6111yescharacteristic
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⋊C4—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340C2×D6111yescharacteristic
399#336—611no—
400#336—611no—
401#336—611no—
402#336C2×D6111yescharacteristic
403C61#2636811yescharacteristic, complemented, Sylow
404C22×Q8—1221no—
405C22×D4C22×D6111yescharacteristic
406C22×D4C22×D6111yescharacteristic
407C23×C4—21no—
408C23×C4—1221no—
409C23×C4—1221no—
410C23×C4—1221no—
411C2×D8—22no—
412C2×D8—22no—
413C2×D8Dic12214yescomplemented
414C22×C8—611no—
415C22×C8—611no—
416C22×C8C22×D6111yescharacteristic
417C22×C8C22×D6111yescharacteristic
418C4⋊Q8—1222no—
419C4⋊Q8—2441no—
420C22⋊Q8—2444no—
421C4×D4—22no—
422C4×D4—22no—
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⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C22⋊C4—21no—
446C2×C22⋊C4—21no—
447C2×C22⋊C4—21no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C42—611no—
454C2×C42—611no—
455C2×C42C22×D6111yescharacteristic
456#107—1222no—
457#107—1222no—
458#107—1222no—
459#107—1222no—
460#107—2441no—
461C4⋊C8—1222no—
462#102—22no—
463#102—22no—
464#102—614no—
465#102—1222no—
466#102—1222no—
467#102—2441no—
468C4×C8—1222no—
469#95—611no—
470#95—611no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475C24—21no—
476C24—21no—
477C2×Q8—4881no—
478C2×Q8—2442no—
479C2×Q8—2442no—
480C2×D4—21no—
481C2×D4—21no—
482C2×D4—21no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×Dic12212yes—
489C2×D4C2×Dic12212yes—
490C2×D4D4×D6112yes—
491C2×D4#1175822412yes—
492C22×C4—24no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—42no—
496C22×C4—42no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—611no—
500C22×C4—611no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—4881no—
506C22×C4—4881no—
507C22×C4—4881no—
508C22×C4—2442no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4D4×D6111yescharacteristic
515C22×C4D4×D6111yescharacteristic
516C22×C4C23×D6111yescharacteristic
517D8—28no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—612no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8C2×Dic12212yes—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—614no—
534C4⋊C4—612no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—4881no—
543C4⋊C4—2441no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C4⋊C4—2442no—
549C4⋊C4—2442no—
550C4⋊C4#1175822412yes—
551C22⋊C4—24no—
552C22⋊C4—42no—
553C22⋊C4—81no—
554C22⋊C4—2442no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C42—21no—
560C42—21no—
561C42—1221no—
562C42—1221no—
563C42—1221no—
564C42—1221no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—41no—
577C23—41no—
578C23—41no—
579C23C2×D4×D6111yescharacteristic
580Q8—4884no—
581D4—22no—
582D4—22no—
583D4—22no—
584D4—22no—
585D4—22no—
586D4—41no—
587D4—41no—
588D4#1451975414yescomplemented
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—48no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—612no—
602C2×C4—612no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1222no—
614C2×C4—4884no—
615C2×C4—4884no—
616C2×C4—4884no—
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×C4—2441no—
627C2×C4—2441no—
628C2×C4D8×D6112yes—
629C2×C4C2×D4×D6111yescharacteristic
630C2×C4C22×Dic12211yescharacteristic
631C2×C4C2×#1175822411yescharacteristic
632C2×C4C2×#1175822411yescharacteristic
633C2×C4#1451982311yescharacteristic
634C2×C4#1451982411yescharacteristic
635C8—22no—
636C8—41no—
637C8—1222no—
638C8—2441no—
639C22—24no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—42no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22#1684026811yescharacteristic
654C22#1684039611yescharacteristic
655C22D4×Dic12211yescharacteristic
656C22C2×D8×D6111yescharacteristic
657C22#1684055811yescharacteristic
658C22#1684056611yescharacteristic
659C22#1684057611yescharacteristic
660C4—22no—
661C4—42no—
662C4—41no—
663C4—81no—
664C4—1222no—
665C4—1222no—
666C4—4881no—
667C4—4881no—
668C4—4881no—
669C4—2442no—
670C4—2441no—
671C4—2441no—
672C4C2×#1451975411yescharacteristic
673C4#1684035611yescharacteristic
674C4#1684041211yescharacteristic
675C4#1684046911yescharacteristic
676C2—44no—
677C2—82no—
678C2#2144963411yescharacteristic
679C2#2145581411yescharacteristic
680C2#2145607411yescharacteristic
681C2#2145738011yescharacteristic
682C2#2145749111yescharacteristic
683C2D8×Dic12211yescharacteristic
684C2#2145877811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1879C2
socleC23×C61C22×D4
prime radicalC22×D8×C61C22
hypercenter#1879D61
hypocenterC61#26368
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1879

Complements

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

NH ≅ G/Ntype
1. C2×#16840040C2semidirect
3. #21449672C2semidirect
4. D4×C4⋊Dic61C2semidirect
5. #21447347C2semidirect
6. #21447292C2semidirect
7. #21447203C2semidirect
8. #21447146C2semidirect
9. #21446551C2semidirect
10. #21446375C2semidirect
12. #21446068C2semidirect
13. #21446066C2semidirect
14. #21446064C2semidirect
119. C2×D8×C61Q8semidirect
207. C4⋊Dic61D8semidirect
244. D4×C61C22⋊Q8semidirect
403. C61#26368semidirect
413. C2×D8Dic122semidirect
588. D4#14519754semidirect

Maximal subgroups

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

subgroupindex
C2×#168400402
C2×D8×Dic612
#214496722
D4×C4⋊Dic612
#214473472
#214472922
#214472032
#214471462
#214465512
#214463752
#214461692
#214460682
#214460662
#214460642
C61×#18792
#2636861