Finite Group Databasebeta

← #29723953

Subgroups — #29723953

Order 15616.

940 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839920C211yesmaximal, characteristic
2C2×C4×Dic244C211yesmaximal, characteristic, complemented
3#21449344C211yesmaximal, characteristic
4C22⋊C4×Dic122C211yesmaximal, characteristic
5#21447356C211yesmaximal, characteristic, complemented
6#21447247C211yesmaximal, characteristic
7#21447142C211yesmaximal, characteristic, complemented
8#21447023C211yesmaximal, characteristic
9#21446495C211yesmaximal, characteristic, complemented
10#21446159C211yesmaximal, characteristic, complemented
11#21446159C211yesmaximal, characteristic, complemented
12#21446076C211yesmaximal, characteristic, complemented
13#21445825C211yesmaximal, characteristic, complemented
14#21445755C211yesmaximal, characteristic
15C61×#1923C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C22×Dic244C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23#16839920—24no—
24C4×Dic244—24no—
25C2×#14519711C2211yescharacteristic
26C2×#14519711C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35#16839698—21no—
36#16839696—21no—
37#16839668C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839668C2211yescharacteristic
40#16839666—21no—
41#16839666—21no—
42#16839660C2211yescharacteristic
43#16839655—21no—
44#16839645—21no—
45#16839632C2211yescharacteristic
46#16839622C2211yescharacteristic
47#16839622C2211yescharacteristic
48#16839620—21no—
49C22⋊C4×Dic61—21no—
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4⋊C8×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C2×C4×C8×C61C2211yescharacteristic
58C4×C22⋊C4×C61C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122—22no—
63C22×Dic122—22no—
64C22×Dic122—22no—
65C22×Dic122—22no—
66C22×Dic122—41no—
67C22×Dic122—41no—
68C22×Dic122C2311yescharacteristic
69C22×Dic122C2311yescharacteristic
70C22×Dic122C2311yescharacteristic
71C23×Dic61—21no—
72C2×Dic244—22no—
73C2×Dic244—22no—
74C2×Dic244—22no—
75C2×Dic244—22no—
76C2×Dic244—41no—
77C2×Dic244D414yes—
78#14519756—22no—
79#14519756—41no—
80C4×Dic122—22no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122—41no—
84C4×Dic122—41no—
85C4×Dic122—41no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C22⋊Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104#14519711—22no—
105#14519711—41no—
106#14519702—22no—
107#14519702—22no—
108#14519702—22no—
109#14519702—22no—
110#14519702—22no—
111#14519702—22no—
112#14519702—41no—
113#14519702—41no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C23×C4×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C2×C22⋊C4×C61D411yescharacteristic
129C2×C22⋊C4×C61D411yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132C4⋊C8×C61—41no—
133C22⋊C8×C61—22no—
134C22⋊C8×C61—41no—
135C4×C8×C61—22no—
136#95×C61D411yescharacteristic
137#95×C61D411yescharacteristic
138C24×C61C2×D411yescharacteristic
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—24no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—24no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—44no—
177C2×Dic122—44no—
178C2×Dic122—44no—
179C2×Dic122—42no—
180C2×Dic122—42no—
181C2×Dic122—44no—
182C2×Dic122C2×D412yes—
183C2×Dic122C2×D412yes—
184C2×Dic122#4112yes—
185C2×Dic122#4112yes—
186C2×Dic122#4112yes—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61—21no—
197C2×C8×C61—21no—
198C2×C8×C61C2×D412yes—
199C22⋊C4×C61—41no—
200C22⋊C4×C61Q1614yescomplemented
201C42×C61—21no—
202C42×C61—21no—
203C42×C61—21no—
204C42×C61—21no—
205C22⋊Dic61—42no—
206C22⋊Dic61—42no—
207C22⋊Dic61—81no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—42no—
219C4⋊Dic61—81no—
220#11758196—42no—
221#11758196—42no—
222#11758196—42no—
223#11758196—42no—
224#11758196—81no—
225C4×Dic61—42no—
226C4×Dic61—42no—
227C4×Dic61—42no—
228Dic244—28no—
229Dic244—44no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61C2×Q1612yes—
237C23×C61C22×D411yescharacteristic
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—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61C2×Q1612yes—
256C2×C4×C61C2×Q1612yes—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C2×C4×C61H(5,𝔽2)11yescharacteristic
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—48no—
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—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288Dic122—22no—
289Dic122—22no—
290Dic122—22no—
291Dic122—22no—
292Dic122—22no—
293Dic122—22no—
294Dic122—22no—
295Dic122—48no—
296Dic122—41no—
297Dic122—41no—
298Dic122—41no—
299Dic122—41no—
300Dic122#12314yes—
301C8×C61—22no—
302C8×C61—41no—
303C8×C61—41no—
304C8×C61—41no—
305#24906—611nomaximal, Sylow
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—22no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—41no—
313C22×C61#52511yescharacteristic
314C22×C61D4211yescharacteristic
315C22×C61#54611yescharacteristic
316C22×C61C22×Q1611yescharacteristic
317C22×C61#57511yescharacteristic
318C22×C61#57711yescharacteristic
319C22×C61#57811yescharacteristic
320C4×C61—22no—
321C4×C61—22no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—41no—
325C4×C61C2×#12311yescharacteristic
326C4×C61#53811yescharacteristic
327C4×C61#54611yescharacteristic
328C4×C61#55111yescharacteristic
329Dic61—24no—
330Dic61—42no—
331Dic61—42no—
332Dic61—81no—
333Dic61—81no—
334Dic61—81no—
335Dic61—81no—
336Dic61—81no—
337C2×#451—611no—
338C2×C4×Q16—611no—
339#2423—611no—
340Q8×C22⋊C4—611no—
341#2095—611no—
342#2076—611no—
343#2056—611no—
344#2036—611no—
345#1933—611no—
346#1923D6111yescharacteristic
347#1878—611no—
348#1878—611no—
349#1863—611no—
350#1803—611no—
351#1787—611no—
352C2×C61—42no—
353C2×C61#241811yescharacteristic
354C2×C61#310411yescharacteristic
355C2×C61#312111yescharacteristic
356C2×C61#327111yescharacteristic
357C2×C61D4×Q1611yescharacteristic
358C2×C61#329911yescharacteristic
359C2×C61#332711yescharacteristic
360C23×Q8—611no—
361C22×Q16—611no—
362C22×Q16—611no—
363C2×C22⋊Q8—611no—
364C2×C4×Q8—611no—
365C2×C4×Q8—611no—
366C2×C4×Q8—611no—
367#451—1224no—
368C4×Q16—1224no—
369C2×#107—611no—
370C2×#107—611no—
371C2×C4⋊C8C2×D6111yescharacteristic
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103—611no—
380C2×C22⋊C8C2×D6111yescharacteristic
381C2×C22⋊C8C2×D6111yescharacteristic
382C2×C4×C8C2×D6111yescharacteristic
383#399—1221no—
384#398—1221no—
385#391—611no—
386#391—611no—
387#391—611no—
388#391—1221no—
389#391—1221no—
390#389—611no—
391#388—1221no—
392#387—1221no—
393#384—611no—
394#380—611no—
395#380—611no—
396#380—1221no—
397C4×C22⋊C4—1221no—
398C4×C22⋊C4C2×D6111yescharacteristic
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#336C2×D6111yescharacteristic
404#336C2×D6111yescharacteristic
405C61#2490611yescharacteristic, complemented, Sylow
406C22×Q8—611no—
407C22×Q8—611no—
408C22×Q8—611no—
409C22×Q8—1222no—
410C22×Q8—1222no—
411C22×Q8—1222no—
412C22×Q8—1222no—
413C22×Q8—1221no—
414C22×Q8—2441no—
415C22×Q8—2441no—
416C23×C4—1221no—
417C23×C4C22×D6111yescharacteristic
418C2×Q16—614no—
419C2×Q16—1222no—
420C2×Q16—1222no—
421C2×Q16—1222no—
422C2×Q16—1222no—
423C2×Q16—2441no—
424C22×C8C22×D6111yescharacteristic
425C22×C8C22×D6111yescharacteristic
426C22×C8C22×D6111yescharacteristic
427C22×C8C22×D6111yescharacteristic
428C22⋊Q8—1222no—
429C22⋊Q8—2441no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—1222no—
433C4×Q8—2441no—
434C4×Q8—2441no—
435C4×Q8—2441no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4D24411yescharacteristic
452C2×C22⋊C4D24411yescharacteristic
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42C22×D6111yescharacteristic
457C2×C42C22×D6111yescharacteristic
458#107—1222no—
459#107—2441no—
460C4⋊C8—41no—
461#103—1222no—
462#103—1222no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103—1222no—
467#103—2441no—
468#103—2441no—
469C22⋊C8—22no—
470C22⋊C8—41no—
471C4×C8—22no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481#95D24411yescharacteristic
482#95D24411yescharacteristic
483C24C2×D24411yescharacteristic
484C2×Q8—612no—
485C2×Q8—612no—
486C2×Q8—612no—
487C2×Q8—612no—
488C2×Q8—612no—
489C2×Q8—1224no—
490C2×Q8—1221no—
491C2×Q8—1221no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8—1221no—
502C2×Q8—1221no—
503C2×Q8—2442no—
504C2×Q8—2442no—
505C2×Q8—2444no—
506C2×Q8—2444no—
507C2×Q8—2444no—
508C2×Q8—2444no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—1221no—
516C22×C4—1224no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—2442no—
525C22×C4—2442no—
526C22×C4C2×D24411yescharacteristic
527C22×C4C2×D24411yescharacteristic
528C22×C4C2×D24411yescharacteristic
529C22×C4C2×D24411yescharacteristic
530C22×C4C2×D24411yescharacteristic
531C22×C4C23×D6111yescharacteristic
532Q16—1228no—
533Q16—2444no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—21no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8D4×D6112yes—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—4881no—
557C4⋊C4—4881no—
558C4⋊C4—2442no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4—2442no—
562C4⋊C4—2442no—
563C22⋊C4—41no—
564C22⋊C4—4881no—
565C22⋊C4—2442no—
566C22⋊C4—2442no—
567C22⋊C4Dic24414yescomplemented
568C42—21no—
569C42—21no—
570C42—21no—
571C42—21no—
572C42—2442no—
573C42—2442no—
574C42—2442no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23C2×Dic24412yes—
582C23C22×D24411yescharacteristic
583Q8—614no—
584Q8—1222no—
585Q8—1222no—
586Q8—1222no—
587Q8—1222no—
588Q8—1222no—
589Q8—1222no—
590Q8—1222no—
591Q8—2448no—
592Q8—2441no—
593Q8—2441no—
594Q8—2441no—
595Q8—2441no—
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—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—2448no—
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×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4C2×Dic24412yes—
639C2×C4C2×Dic24412yes—
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4#41×D6111yescharacteristic
642C2×C4#41×D6111yescharacteristic
643C2×C4#41×D6111yescharacteristic
644C2×C4#1451982211yescharacteristic
645C2×C4#1451982311yescharacteristic
646C8—22no—
647C8—41no—
648C8—41no—
649C8—41no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22#1684027411yescharacteristic
658C22D4×D24411yescharacteristic
659C22#1684040711yescharacteristic
660C22C22×Dic24411yescharacteristic
661C22#1684056211yescharacteristic
662C22#1684057411yescharacteristic
663C22#1684057911yescharacteristic
664C4—22no—
665C4—22no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—1224no—
670C4—4881no—
671C4—4881no—
672C4—4881no—
673C4—4881no—
674C4—4881no—
675C4—2442no—
676C4—2442no—
677C4#123×D6111yescharacteristic
678C4#1684034611yescharacteristic
679C4#1684040911yescharacteristic
680C4#1684043711yescharacteristic
681C2—42no—
682C2#2144928611yescharacteristic
683C2#2145585911yescharacteristic
684C2#2145605011yescharacteristic
685C2#2145740911yescharacteristic
686C2D4×Dic24411yescharacteristic
687C2#2145769611yescharacteristic
688C2#2145808911yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1923C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1923D61
hypocenterC61#24906
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1923

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic244C2semidirect
5. #21447356C2semidirect
7. #21447142C2semidirect
9. #21446495C2semidirect
10. #21446159C2semidirect
11. #21446159C2semidirect
12. #21446076C2semidirect
13. #21445825C2semidirect
200. C22⋊C4×C61Q16semidirect
405. C61#24906semidirect
567. C22⋊C4Dic244semidirect

Maximal subgroups

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

subgroupindex
C2×#168399202
C2×C4×Dic2442
#214493442
C22⋊C4×Dic1222
#214473562
#214472472
#214471422
#214470232
#214464952
#214461592
#214460762
#214458252
#214457552
C61×#19232
#2490661