Finite Group Databasebeta

← #29737616

Subgroups — #29737616

Order 15616.

802 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449515C211yesmaximal, characteristic
2#21448688C211yesmaximal, characteristic
3#21447380C211yesmaximal, characteristic, complemented
4#21447311C211yesmaximal, characteristic, complemented
5#21447309C211yesmaximal, characteristic, complemented
6#21447146C211yesmaximal, characteristic, complemented
7#21446769C211yesmaximal, characteristic, complemented
8#21446576C211yesmaximal, characteristic, complemented
9#21446393C211yesmaximal, characteristic, complemented
10#21446389C211yesmaximal, characteristic, complemented
11#21446007C211yesmaximal, characteristic
12#21445997C211yesmaximal, characteristic
13#21445905C211yesmaximal, characteristic
14#21445848C211yesmaximal, characteristic
15C2×C61×#340C211yesmaximal, characteristic
16C2×#14519776C2211yescharacteristic
17C2×#14519756—21no—
18C2×#14519736—21no—
19C2×#14519711C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×C22⋊#92418C2211yescharacteristic
26#16839695—21no—
27#16839682—21no—
28#16839661C2211yescharacteristic
29#16839661C2211yescharacteristic
30#16839661C2211yescharacteristic
31#16839658C2211yescharacteristic
32#16839648—21no—
33#16839648—21no—
34#16839635—21no—
35#16839631C2211yescharacteristic
36#16839625—21no—
37#16839622C2211yescharacteristic
38#16839622C2211yescharacteristic
39C4⋊C4×Dic61C2211yescharacteristic
40C4⋊C4×Dic61C2211yescharacteristic
41C4×C22⋊Dic61C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C23×C8×C61C2211yescharacteristic
53C22×C4⋊C4×C61C2211yescharacteristic
54C22×C4⋊C4×C61C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×#92418C2311yescharacteristic
61C22×#92418C2311yescharacteristic
62#14519776—22no—
63#14519776—41no—
64#14519756—44no—
65#14519736—44no—
66C2×C4⋊Dic61—21no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61C2311yescharacteristic
70C2×C4⋊Dic61C2311yescharacteristic
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×#11758196—21no—
74C2×#11758196—21no—
75C2×#11758196—21no—
76C2×#11758196—21no—
77C2×C22⋊Dic61—21no—
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61D411yescharacteristic
80C2×C22⋊Dic61D411yescharacteristic
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61C2311yescharacteristic
84C2×C4×Dic61C2311yescharacteristic
85#14519711—22no—
86#14519711—41no—
87#14519710—22no—
88#14519710—41no—
89#14519706—22no—
90#14519706—41no—
91C22⋊#92418—22no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684D411yescharacteristic
101#14519684D411yescharacteristic
102C23×C4×C61—21no—
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C22×C8×C61—22no—
106C22×C8×C61—22no—
107C22×C8×C61—41no—
108C22×C8×C61C2311yescharacteristic
109C22×C8×C61C2311yescharacteristic
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—41no—
115C2×C4⋊C4×C61Q814yescomplemented
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C24×C61C2×D411yescharacteristic
121C22×C4×C61—21no—
122C22×C4×C61—21no—
123C22×C4×C61—21no—
124C22×C4×C61—21no—
125C22×C4×C61—24no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—42no—
131C22×C4×C61—42no—
132C22×C4×C61—42no—
133C22×C4×C61—42no—
134C22×C4×C61—81no—
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×Q812yes—
137C22×C4×C61#4112yes—
138C22×C4×C61C2411yescharacteristic
139C22×Dic61—21no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61C2×D411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×Dic122—42no—
148C2×Dic122—42no—
149C2×Dic122—81no—
150C2×C8×C61—21no—
151C2×C8×C61—21no—
152C2×C8×C61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—44no—
156C2×C8×C61#4112yes—
157C4⋊C4×C61—24no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61C2×Q812yes—
165C4⋊C4×C61C2×Q812yes—
166C4⋊C4×C61#4112yes—
167C22⋊Dic61—22no—
168C22⋊Dic61—42no—
169C22⋊Dic61—42no—
170C22⋊Dic61D814yes—
171C4⋊Dic61—21no—
172C4⋊Dic61—21no—
173C4⋊Dic61—21no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—42no—
180C4⋊Dic61—42no—
181C4⋊Dic61—42no—
182#11758196—42no—
183#11758196—42no—
184#11758196—42no—
185#11758196—81no—
186C4×Dic61—21no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—42no—
191C4×Dic61—42no—
192C2×#92418—21no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C23×C61—21no—
199C23×C61—21no—
200C23×C61—21no—
201C23×C61—21no—
202C23×C61C22⋊Q812yes—
203C23×C61C2×D812yes—
204C23×C61#13712yes—
205C23×C61C22×D411yescharacteristic
206C2×C4×C61—21no—
207C2×C4×C61—21no—
208C2×C4×C61—22no—
209C2×C4×C61—22no—
210C2×C4×C61—22no—
211C2×C4×C61—22no—
212C2×C4×C61—22no—
213C2×C4×C61—22no—
214C2×C4×C61—22no—
215C2×C4×C61—41no—
216C2×C4×C61—41no—
217C2×C4×C61—41no—
218C2×C4×C61—41no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—48no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—84no—
226C2×C4×C61C22⋊Q812yes—
227C2×C4×C61C22×Q811yescharacteristic
228C2×C4×C61C2×#4111yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61C2×#4111yescharacteristic
231C2×C4×C61H(5,𝔽2)11yescharacteristic
232C2×C4×C612−1+411yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245C2×Dic61—41no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61C2×D812yes—
256C2×Dic61C2×D812yes—
257Dic122—84no—
258C8×C61—22no—
259C8×C61—41no—
260#92418—41no—
261#92418—41no—
262#25988—611nomaximal, Sylow
263C22×C61—22no—
264C22×C61—22no—
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—22no—
268C22×C61—22no—
269C22×C61#47414yes—
270C22×C61C2×C22⋊Q811yescharacteristic
271C22×C61#54611yescharacteristic
272C22×C61#54711yescharacteristic
273C22×C61C22×D811yescharacteristic
274C22×C61C2×#13711yescharacteristic
275C22×C61#57711yescharacteristic
276C22×C61#57811yescharacteristic
277C4×C61—22no—
278C4×C61—42no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61—81no—
282C4×C61#53311yescharacteristic
283C4×C61#54411yescharacteristic
284C4×C61#55111yescharacteristic
285C4×C61#55211yescharacteristic
286Dic61—22no—
287Dic61—22no—
288Dic61—41no—
289Dic61—41no—
290Dic61—81no—
291Dic61—81no—
292Dic61—81no—
293Dic61—81no—
294#2440—611no—
295#2344—611no—
296#2099—611no—
297#2086—611no—
298#2086—611no—
299#2057—611no—
300#1988—611no—
301#1947—611no—
302#1918—611no—
303#1918—611no—
304#1852—611no—
305#1850—611no—
306#1823—611no—
307#1808—611no—
308C2×#340D6111yescharacteristic
309C2×C61—24no—
310C2×C61#243811yescharacteristic
311C2×C61C2×#47411yescharacteristic
312C2×C61#308211yescharacteristic
313C2×C61#329411yescharacteristic
314C2×C61#331211yescharacteristic
315C2×C61#333411yescharacteristic
316C2×C61#334411yescharacteristic
317C23×C8C2×D6111yescharacteristic
318C2×C4⋊Q8—611no—
319C2×C22⋊Q8—1221no—
320C2×#117—1221no—
321C22×C4⋊C4C2×D6111yescharacteristic
322C22×C4⋊C4C2×D6111yescharacteristic
323C2×#107—611no—
324C2×#107—611no—
325C2×#107—611no—
326C2×#107—611no—
327C2×C4⋊C8—611no—
328C2×C4⋊C8—611no—
329C2×C22⋊C8—611no—
330#398—1221no—
331#395—1221no—
332#389—611no—
333#389—611no—
334#389—611no—
335#389—611no—
336#387—1221no—
337#387—1221no—
338#385—1221no—
339#384—611no—
340#381—1221no—
341#380—611no—
342#380—611no—
343C4×C4⋊C4—611no—
344C4×C4⋊C4—611no—
345C4×C22⋊C4—611no—
346#340—611no—
347#340—611no—
348#340—611no—
349#340—611no—
350#340—611no—
351#340—611no—
352#340—611no—
353#340—611no—
354#340C2×D6111yescharacteristic
355#340C2×D6111yescharacteristic
356#340C2×D6111yescharacteristic
357#340C2×D6111yescharacteristic
358#336—611no—
359#336—611no—
360C61#2598811yescharacteristic, complemented, Sylow
361C22×Q8—1221no—
362C23×C4—21no—
363C23×C4—21no—
364C23×C4C22×D6111yescharacteristic
365C22×C8—22no—
366C22×C8—22no—
367C22×C8—41no—
368C22×C8—611no—
369C22×C8—611no—
370C22×C8C22×D6111yescharacteristic
371C22×C8C22×D6111yescharacteristic
372C4⋊Q8—1222no—
373C4⋊Q8—2441no—
374C22⋊Q8—2444no—
375#117—2444no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—22no—
378C2×C4⋊C4—22no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—41no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—1221no—
386C2×C4⋊C4—1221no—
387C2×C4⋊C4—1221no—
388C2×C4⋊C4—1221no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4Dic12214yescomplemented
393C2×C4⋊C4C22×D6111yescharacteristic
394C2×C4⋊C4C22×D6111yescharacteristic
395C2×C4⋊C4C22×D6111yescharacteristic
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C22⋊C4—611no—
398C2×C22⋊C4—611no—
399C2×C22⋊C4—1221no—
400C2×C22⋊C4—1221no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—1221no—
404C2×C42—1221no—
405#107—1222no—
406#107—1222no—
407#107—2441no—
408#107—2441no—
409C4⋊C8—1222no—
410C4⋊C8—2441no—
411C22⋊C8—1222no—
412#95—611no—
413#95—611no—
414#95—1221no—
415#95—1221no—
416#95—1221no—
417#95—1221no—
418#95—1221no—
419#95—1221no—
420#95—1221no—
421#95—1221no—
422C24D4×D6111yescharacteristic
423C2×Q8—4881no—
424C2×Q8—2442no—
425C2×Q8—2442no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—81no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4C2×Dic12212yes—
449C22×C4D4×D6111yescharacteristic
450C22×C4#1175822412yes—
451C22×C4C23×D6111yescharacteristic
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—44no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8#1175822212yes—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—42no—
472C4⋊C4—1221no—
473C4⋊C4—1221no—
474C4⋊C4—1221no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—4881no—
481C4⋊C4—2442no—
482C4⋊C4—2442no—
483C4⋊C4—2442no—
484C4⋊C4—2442no—
485C4⋊C4—2442no—
486C4⋊C4—2442no—
487C4⋊C4C2×Dic12212yes—
488C4⋊C4C2×Dic12212yes—
489C4⋊C4#1175822212yes—
490C22⋊C4—614no—
491C22⋊C4—1222no—
492C22⋊C4—2442no—
493C22⋊C4—2442no—
494C42—1221no—
495C42—1221no—
496C42—1221no—
497C42—1221no—
498C42—2442no—
499C42—2442no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23#1451975412yes—
505C23D8×D6112yes—
506C23#1451980612yes—
507C23C2×D4×D6111yescharacteristic
508Q8—4884no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—48no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—84no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—1221no—
532C2×C4—1221no—
533C2×C4—1221no—
534C2×C4—1221no—
535C2×C4—1221no—
536C2×C4—1221no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—2441no—
542C2×C4—2441no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2441no—
546C2×C4—2441no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4#1451975412yes—
554C2×C4C22×Dic12211yescharacteristic
555C2×C4C2×#1175822411yescharacteristic
556C2×C4C2×#1175822211yescharacteristic
557C2×C4C2×#1175822211yescharacteristic
558C2×C4#1451982211yescharacteristic
559C2×C4#1451982511yescharacteristic
560C8—22no—
561C8—41no—
562C8—2441no—
563C8—2441no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22#1684004014yes—
571C22C2×#1451975411yescharacteristic
572C22#1684040211yescharacteristic
573C22#1684041311yescharacteristic
574C22C2×D8×D6111yescharacteristic
575C22C2×#1451980611yescharacteristic
576C22#1684056811yescharacteristic
577C22#1684057511yescharacteristic
578C4—22no—
579C4—42no—
580C4—42no—
581C4—42no—
582C4—81no—
583C4—1222no—
584C4—1222no—
585C4—4881no—
586C4—4881no—
587C4—4881no—
588C4—4881no—
589C4—2441no—
590C4—2441no—
591C4#1684031911yescharacteristic
592C4#1684039311yescharacteristic
593C4#1684043811yescharacteristic
594C4#1684044711yescharacteristic
595C2—24no—
596C2#2144949511yescharacteristic
597C2C2×#1684004011yescharacteristic
598C2#2145565711yescharacteristic
599C2#2145762811yescharacteristic
600C2#2145786611yescharacteristic
601C2#2145816411yescharacteristic
602C2#2145827011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#340C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#340D61
hypocenterC61#25988
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. #21447380C2semidirect
4. #21447311C2semidirect
5. #21447309C2semidirect
6. #21447146C2semidirect
7. #21446769C2semidirect
8. #21446576C2semidirect
9. #21446393C2semidirect
10. #21446389C2semidirect
115. C2×C4⋊C4×C61Q8semidirect
360. C61#25988semidirect
392. C2×C4⋊C4Dic122semidirect

Maximal subgroups

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

subgroupindex
#214495152
#214486882
#214473802
#214473112
#214473092
#214471462
#214467692
#214465762
#214463932
#214463892
#214460072
#214459972
#214459052
#214458482
C2×C61×#3402
#2598861