Finite Group Databasebeta

← #29737502

Subgroups — #29737502

Order 15616.

928 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840054C211yesmaximal, characteristic, complemented
2C2×#16839770C211yesmaximal, characteristic
3C22⋊C4×Dic122C211yesmaximal, characteristic
4#21447370C211yesmaximal, characteristic, complemented
5#21447211C211yesmaximal, characteristic, complemented
6#21447197C211yesmaximal, characteristic, complemented
7#21446957C211yesmaximal, characteristic, complemented
8#21446086C211yesmaximal, characteristic, complemented
9#21446086C211yesmaximal, characteristic, complemented
10#21446018C211yesmaximal, characteristic
11#21446004C211yesmaximal, characteristic
12#21445850C211yesmaximal, characteristic
13#21445831C211yesmaximal, characteristic, complemented
14#21445753C211yesmaximal, characteristic
15C61×#2439C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C22×#11758194C2211yescharacteristic
18C2×C4×Dic122C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16840054—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27#16839770—24no—
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519700C2211yescharacteristic
36C2×C22⋊#92418C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839666—21no—
40#16839666—21no—
41#16839622C2211yescharacteristic
42#16839620—21no—
43C22⋊C4×Dic61—21no—
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×#117×C61C2211yescharacteristic
52C61×#396—21no—
53C61×#393—21no—
54C61×#391—21no—
55C61×#389C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#387—21no—
58C61×#382C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C4×C22⋊C4×C61C2211yescharacteristic
61C22×Dic122—22no—
62C22×Dic122—22no—
63C22×Dic122—21no—
64C22×Dic122—22no—
65C22×Dic122—41no—
66C22×Dic122—41no—
67C22×Dic122Q814yescomplemented
68C22×Dic122C2311yescharacteristic
69C22×Dic122C2311yescharacteristic
70C23×Dic61—21no—
71C2×#11758194—22no—
72C2×#11758194—22no—
73C2×#11758194—41no—
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76C22×#92418C2311yescharacteristic
77C22×#92418C2311yescharacteristic
78C4×Dic122—22no—
79C4×Dic122—22no—
80C4×Dic122—41no—
81C4×Dic122—41no—
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×C22⋊Dic61—21no—
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89#14519710—22no—
90#14519710—22no—
91#14519708—41no—
92#14519708—41no—
93C4⋊#92418—22no—
94C4⋊#92418—41no—
95#14519700—22no—
96#14519700—22no—
97#14519700—22no—
98#14519700—22no—
99#14519700—22no—
100#14519700—41no—
101#14519700—41no—
102#14519700D414yes—
103C22⋊#92418—22no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109C22×Q8×C61—21no—
110C23×C4×C61C2311yescharacteristic
111C4⋊Q8×C61—22no—
112C4⋊Q8×C61—41no—
113#117×C61—22no—
114#117×C61—41no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61D411yescharacteristic
125C2×C22⋊C4×C61D411yescharacteristic
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132#95×C61—21no—
133#95×C61—21no—
134#95×C61D411yescharacteristic
135#95×C61D411yescharacteristic
136C24×C61C2×D411yescharacteristic
137C2×Q8×C61—42no—
138C2×Q8×C61—42no—
139C2×Q8×C61—81no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×Q812yes—
154C22×C4×C61#4112yes—
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—24no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—28no—
171C2×Dic122—24no—
172C2×Dic122—21no—
173C2×Dic122—44no—
174C2×Dic122—44no—
175C2×Dic122—44no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×Dic122C2×D412yes—
179C2×Dic122C2×Q812yes—
180C2×Dic122C2×Q812yes—
181C2×Dic122#4112yes—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—42no—
190C4⋊C4×C61—42no—
191C4⋊C4×C61—81no—
192C4⋊C4×C61C2×D412yes—
193C22⋊C4×C61—22no—
194C22⋊C4×C61—41no—
195C22⋊C4×C61—42no—
196C22⋊C4×C61—81no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—21no—
200C42×C61—21no—
201C42×C61—42no—
202C22⋊Dic61—42no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207#11758196—42no—
208#11758196—42no—
209C4×Dic61—42no—
210C4×Dic61—42no—
211#11758194—44no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418C2×D412yes—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61C22⋊Q812yes—
231C23×C61C22×D411yescharacteristic
232Q8×C61—84no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
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×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61C22⋊Q812yes—
263C2×C4×C61#13712yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×Q811yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C2×C4×C61H(5,𝔽2)11yescharacteristic
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—48no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283Dic122—22no—
284Dic122—22no—
285Dic122—22no—
286Dic122—22no—
287Dic122—22no—
288Dic122—48no—
289Dic122—41no—
290Dic122—41no—
291Dic122C22⋊Q814yes—
292#92418—22no—
293#92418—41no—
294#92418—41no—
295#92418—41no—
296#25979—611nomaximal, Sylow
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61C2×C22⋊Q811yescharacteristic
305C22×C61D4211yescharacteristic
306C22×C61#54611yescharacteristic
307C22×C61C2×#13711yescharacteristic
308C22×C61#57511yescharacteristic
309C22×C61#57711yescharacteristic
310C22×C61#57811yescharacteristic
311C4×C61—22no—
312C4×C61—22no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61—42no—
316C4×C61—41no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61C2×C22⋊Q811yescharacteristic
321C4×C61#54311yescharacteristic
322C4×C61#54611yescharacteristic
323C4×C61#55111yescharacteristic
324Dic61—24no—
325Dic61—42no—
326Dic61—81no—
327Dic61—81no—
328C2×#477—611no—
329C2×#419—611no—
330#2439D6111yescharacteristic
331Q8×C22⋊C4—611no—
332#2097—611no—
333#2068—611no—
334#2066—611no—
335#2024—611no—
336#1865—611no—
337#1865—611no—
338#1854—611no—
339#1851—611no—
340#1809—611no—
341#1805—611no—
342#1787—611no—
343C2×C61—42no—
344C2×C61#243711yescharacteristic
345C2×C61#307811yescharacteristic
346C2×C61#308211yescharacteristic
347C2×C61#327011yescharacteristic
348C2×C61#327511yescharacteristic
349C2×C61#331311yescharacteristic
350C2×C61#332911yescharacteristic
351C23×Q8—611no—
352C22×M16—611no—
353C2×C4⋊Q8C2×D6111yescharacteristic
354C2×C4×Q8—611no—
355C2×C4×Q8—611no—
356C2×#117C2×D6111yescharacteristic
357#477—1224no—
358C2×#107—611no—
359C2×#107—611no—
360C2×#106—611no—
361C2×#106—611no—
362C2×C4⋊C8—611no—
363C2×C4⋊C8—611no—
364#419—1224no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×C22⋊C8—611no—
374#396—21no—
375#393—21no—
376#391—21no—
377#391—611no—
378#391—611no—
379#391—1221no—
380#391—1221no—
381#389C2×D6111yescharacteristic
382#389C2×D6111yescharacteristic
383#387—21no—
384#382C2×D6111yescharacteristic
385#380—611no—
386#380—1221no—
387#380C2×D6111yescharacteristic
388C4×C22⋊C4—1221no—
389C4×C22⋊C4C2×D6111yescharacteristic
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#336—611no—
395#336—611no—
396C61#2597911yescharacteristic, complemented, Sylow
397C22×Q8—21no—
398C22×Q8—614no—
399C22×Q8—611no—
400C22×Q8—611no—
401C22×Q8—1222no—
402C22×Q8—1222no—
403C22×Q8—1222no—
404C22×Q8—1221no—
405C22×Q8—2441no—
406C22×Q8—2441no—
407C23×C4—1221no—
408C23×C4C22×D6111yescharacteristic
409C2×M16—1222no—
410C2×M16—1222no—
411C2×M16—2441no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8—611no—
415C22×C8—611no—
416C4⋊Q8—22no—
417C4⋊Q8—41no—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×Q8—2441no—
421C4×Q8—2441no—
422#117—22no—
423#117—41no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4C22⋊D6111yescharacteristic
439C2×C22⋊C4C22⋊D6111yescharacteristic
440C2×C42—21no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42C22×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445#107—1222no—
446#107—1222no—
447#106—2441no—
448#106—2441no—
449C4⋊C8—1222no—
450C4⋊C8—2441no—
451#103—614no—
452#103—1222no—
453#103—1222no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#103—2441no—
458#103—2441no—
459C22⋊C8—1222no—
460#95—21no—
461#95—21no—
462#95—21no—
463#95—21no—
464#95—21no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95C22⋊D6111yescharacteristic
471#95C22⋊D6111yescharacteristic
472C24C2×C22⋊D6111yescharacteristic
473C2×Q8—42no—
474C2×Q8—42no—
475C2×Q8—81no—
476C2×Q8—612no—
477C2×Q8—612no—
478C2×Q8—612no—
479C2×Q8—612no—
480C2×Q8—1228no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1224no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—2442no—
491C2×Q8—2442no—
492C2×Q8—2444no—
493C2×Q8—2444no—
494C2×Q8—2444no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1224no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—2442no—
509C22×C4—2442no—
510C22×C4Q8×D6112yes—
511C22×C4#1175822612yes—
512C22×C4C2×C22⋊D6111yescharacteristic
513C22×C4C2×C22⋊D6111yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C23×D6111yescharacteristic
518M16—2444no—
519C2×C8—612no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—81no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4D4×D6112yes—
548C22⋊C4—22no—
549C22⋊C4—41no—
550C22⋊C4—42no—
551C22⋊C4—81no—
552C22⋊C4—2442no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—42no—
558C42—2442no—
559C42—2442no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1451975312yes—
567C23C22×C22⋊D6111yescharacteristic
568Q8—84no—
569Q8—614no—
570Q8—1222no—
571Q8—1222no—
572Q8—1222no—
573Q8—1222no—
574Q8—1222no—
575Q8—2448no—
576Q8—2441no—
577Q8—2441no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—1222no—
608C2×C4—1222no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—2448no—
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#1451975312yes—
621C2×C4#1451980712yes—
622C2×C4C2×D4×D6111yescharacteristic
623C2×C4C2×Q8×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4C2×#1175822611yescharacteristic
626C2×C4#1451982211yescharacteristic
627C2×C4#1451982311yescharacteristic
628C8—1222no—
629C8—2441no—
630C8—2441no—
631C8—2441no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22C2×#1451975311yescharacteristic
640C22D4×C22⋊D6111yescharacteristic
641C22#1684040011yescharacteristic
642C22C2×#1451980711yescharacteristic
643C22#1684056011yescharacteristic
644C22#1684057311yescharacteristic
645C22#1684057711yescharacteristic
646C4—22no—
647C4—22no—
648C4—41no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—81no—
653C4—81no—
654C4—81no—
655C4—1224no—
656C4—4881no—
657C4—4881no—
658C4—2442no—
659C4C22⋊Q8×D6111yescharacteristic
660C4#1684038711yescharacteristic
661C4#1684040311yescharacteristic
662C4#1684043911yescharacteristic
663C2—42no—
664C2#2144948211yescharacteristic
665C2#2145562011yescharacteristic
666C2#2145565511yescharacteristic
667C2#2145738511yescharacteristic
668C2#2145744611yescharacteristic
669C2#2145787911yescharacteristic
670C2#2145810711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2439C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#2439D61
hypocenterC61#25979
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
1. C2×#16840054C2semidirect
4. #21447370C2semidirect
5. #21447211C2semidirect
6. #21447197C2semidirect
7. #21446957C2semidirect
8. #21446086C2semidirect
9. #21446086C2semidirect
13. #21445831C2semidirect
67. C22×Dic122Q8semidirect
396. C61#25979semidirect

Maximal subgroups

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

subgroupindex
C2×#168400542
C2×#168397702
C22⋊C4×Dic1222
#214473702
#214472112
#214471972
#214469572
#214460862
#214460182
#214460042
#214458502
#214458312
#214457532
C61×#24392
#2597961