Finite Group Databasebeta

← #29737505

Subgroups — #29737505

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×#16840057C211yesmaximal, characteristic, complemented
2C2×#16839771C211yesmaximal, characteristic
3#21449510C211yesmaximal, characteristic
4#21447367C211yesmaximal, characteristic, complemented
5#21447214C211yesmaximal, characteristic, complemented
6#21447200C211yesmaximal, characteristic, complemented
7#21446960C211yesmaximal, characteristic, complemented
8#21446087C211yesmaximal, characteristic, complemented
9#21446087C211yesmaximal, characteristic, complemented
10#21446016C211yesmaximal, characteristic
11#21446004C211yesmaximal, characteristic
12#21445850C211yesmaximal, characteristic
13#21445834C211yesmaximal, characteristic, complemented
14#21445754C211yesmaximal, characteristic
15Q8×C22⋊C4×C61C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#16840057—24no—
20C2×#14519710C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26#16839771—24no—
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519701C2211yescharacteristic
33C2×#14519701C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×C22⋊#92418C2211yescharacteristic
36#16839685—21no—
37#16839674—21no—
38#16839666—21no—
39#16839660C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839649—21no—
42#16839628C2211yescharacteristic
43#16839622C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C23×Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C61×#391—21no—
54C61×#391—21no—
55C61×#391C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#380—21no—
58C61×#380C2211yescharacteristic
59C4×C22⋊C4×C61—21no—
60C4×C22⋊C4×C61C2211yescharacteristic
61C22×Dic122—21no—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194—41no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519777—22no—
70#14519777—41no—
71#14519736—22no—
72#14519736—41no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83#14519710—22no—
84#14519710—22no—
85#14519708—41no—
86#14519708—41no—
87C4⋊#92418—22no—
88C4⋊#92418—41no—
89#14519701—22no—
90#14519701—22no—
91#14519701—22no—
92#14519701—22no—
93#14519701—22no—
94#14519701—41no—
95#14519701—41no—
96#14519701D414yes—
97C22⋊#92418—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103C22×Q8×C61—21no—
104C22×Q8×C61—22no—
105C22×Q8×C61—22no—
106C22×Q8×C61—22no—
107C22×Q8×C61—41no—
108C22×Q8×C61—41no—
109C22×Q8×C61Q814yescomplemented
110C22×Q8×C61C2311yescharacteristic
111C22×Q8×C61C2311yescharacteristic
112C23×C4×C61—21no—
113C23×C4×C61C2311yescharacteristic
114C4×Q8×C61—22no—
115C4×Q8×C61—22no—
116C4×Q8×C61—41no—
117C4×Q8×C61—41no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61D411yescharacteristic
124C2×C22⋊C4×C61D411yescharacteristic
125C2×C42×C61—21no—
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—24no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—28no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—42no—
148C2×Q8×C61—42no—
149C2×Q8×C61—44no—
150C2×Q8×C61—44no—
151C2×Q8×C61—44no—
152C2×Q8×C61C2×D412yes—
153C2×Q8×C61C2×Q812yes—
154C2×Q8×C61C2×Q812yes—
155C2×Q8×C61#4112yes—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—24no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—42no—
166C22×C4×C61—42no—
167C22×C4×C61C2×D411yescharacteristic
168C22×C4×C61C2×D411yescharacteristic
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×Q812yes—
173C22×C4×C61#4112yes—
174C22×C4×C61C2411yescharacteristic
175C22×Dic61—21no—
176C22×Dic61—21no—
177C22×Dic61—21no—
178C22×Dic61—21no—
179C2×Dic122—42no—
180C2×Dic122—42no—
181C2×Dic122—81no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—42no—
187C4⋊C4×C61—42no—
188C22⋊C4×C61—22no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—42no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195C42×C61—42no—
196C42×C61—42no—
197C22⋊Dic61—42no—
198C22⋊Dic61—81no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—42no—
207C4⋊Dic61C2×D412yes—
208#11758196—42no—
209#11758196—81no—
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—22no—
233Q8×C61—22no—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—48no—
238Q8×C61—41no—
239Q8×C61—41no—
240Q8×C61C22⋊Q814yes—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—48no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61C22⋊Q812yes—
272C2×C4×C61#13712yes—
273C2×C4×C61C22×D411yescharacteristic
274C2×C4×C61C22×Q811yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61C2×#4111yescharacteristic
277C2×C4×C61H(5,𝔽2)11yescharacteristic
278C2×C4×C61H(5,𝔽2)11yescharacteristic
279C2×Dic61—22no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291Dic122—84no—
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—24no—
313C4×C61—22no—
314C4×C61—41no—
315C4×C61—42no—
316C4×C61—41no—
317C4×C61—41no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61C2×C22⋊Q811yescharacteristic
321C4×C61#54311yescharacteristic
322C4×C61#54611yescharacteristic
323C4×C61#55111yescharacteristic
324Dic61—42no—
325Dic61—81no—
326Dic61—81no—
327Dic61—81no—
328C2×#477—611no—
329C2×#419—611no—
330#2439—611no—
331Q8×C22⋊C4D6111yescharacteristic
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×Q8C2×D6111yescharacteristic
352C22×M16—611no—
353C2×C4⋊Q8—611no—
354C2×C4×Q8C2×D6111yescharacteristic
355C2×C4×Q8C2×D6111yescharacteristic
356C2×#117—611no—
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—1221no—
375#393—1221no—
376#391—21no—
377#391—21no—
378#391—1221no—
379#391C2×D6111yescharacteristic
380#391C2×D6111yescharacteristic
381#389—611no—
382#389—611no—
383#387—1221no—
384#382—611no—
385#380—21no—
386#380—611no—
387#380C2×D6111yescharacteristic
388C4×C22⋊C4—21no—
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—22no—
398C22×Q8—21no—
399C22×Q8—22no—
400C22×Q8—22no—
401C22×Q8—41no—
402C22×Q8—41no—
403C22×Q8—1221no—
404C22×Q8Dic12214yescomplemented
405C22×Q8C22×D6111yescharacteristic
406C22×Q8C22×D6111yescharacteristic
407C23×C4—21no—
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—1222no—
417C4⋊Q8—2441no—
418C4×Q8—22no—
419C4×Q8—22no—
420C4×Q8—41no—
421C4×Q8—41no—
422#117—1222no—
423#117—2441no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4C22⋊D6111yescharacteristic
439C2×C22⋊C4C22⋊D6111yescharacteristic
440C2×C42—21no—
441C2×C42—21no—
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—21no—
474C2×Q8—21no—
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—24no—
478C2×Q8—28no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—42no—
484C2×Q8—42no—
485C2×Q8—44no—
486C2×Q8—44no—
487C2×Q8—44no—
488C2×Q8—4881no—
489C2×Q8—2442no—
490C2×Q8—2442no—
491C2×Q8C2×Dic12212yes—
492C2×Q8C2×Dic12212yes—
493C2×Q8D4×D6112yes—
494C2×Q8#1175822412yes—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—24no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—42no—
505C22×C4—42no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4C2×Dic12212yes—
511C22×C4#1175822212yes—
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—42no—
536C4⋊C4—42no—
537C4⋊C4—612no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—4881no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C22⋊C4—22no—
549C22⋊C4—41no—
550C22⋊C4—42no—
551C22⋊C4—4881no—
552C22⋊C4—2442no—
553C42—21no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—42no—
558C42—42no—
559C42—2442no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1451975612yes—
567C23C22×C22⋊D6111yescharacteristic
568Q8—22no—
569Q8—22no—
570Q8—22no—
571Q8—22no—
572Q8—22no—
573Q8—48no—
574Q8—41no—
575Q8—41no—
576Q8—4884no—
577Q8#1451975414yes—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
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—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—1222no—
609C2×C4—2441no—
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#1451975612yes—
621C2×C4#1451980812yes—
622C2×C4C2×D4×D6111yescharacteristic
623C2×C4C22×Dic12211yescharacteristic
624C2×C4C2×#1175822411yescharacteristic
625C2×C4C2×#1175822211yescharacteristic
626C2×C4#1451982211yescharacteristic
627C2×C4#1451982211yescharacteristic
628C8—1222no—
629C8—2441no—
630C8—2441no—
631C8—2441no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22C2×#1451975611yescharacteristic
640C22D4×C22⋊D6111yescharacteristic
641C22#1684040611yescharacteristic
642C22C2×#1451980811yescharacteristic
643C22#1684056011yescharacteristic
644C22#1684057011yescharacteristic
645C22#1684057711yescharacteristic
646C4—22no—
647C4—24no—
648C4—22no—
649C4—42no—
650C4—41no—
651C4—41no—
652C4—41no—
653C4—81no—
654C4—81no—
655C4—4881no—
656C4—4881no—
657C4—4881no—
658C4—2442no—
659C4C2×#1451975411yescharacteristic
660C4#1684038811yescharacteristic
661C4#1684040211yescharacteristic
662C4#1684043811yescharacteristic
663C2—42no—
664C2#2144948011yescharacteristic
665C2#2145562411yescharacteristic
666C2#2145566111yescharacteristic
667C2#2145739011yescharacteristic
668C2#2145745111yescharacteristic
669C2#2145788511yescharacteristic
670C2#2145811311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingQ8×C22⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterQ8×C22⋊C4D61
hypocenterC61#25979
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16840057C2semidirect
4. #21447367C2semidirect
5. #21447214C2semidirect
6. #21447200C2semidirect
7. #21446960C2semidirect
8. #21446087C2semidirect
9. #21446087C2semidirect
13. #21445834C2semidirect
109. C22×Q8×C61Q8semidirect
396. C61#25979semidirect
404. C22×Q8Dic122semidirect

Maximal subgroups

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

subgroupindex
C2×#168400572
C2×#168397712
#214495102
#214473672
#214472142
#214472002
#214469602
#214460872
#214460162
#214460042
#214458502
#214458342
#214457542
Q8×C22⋊C4×C612
#2597961