Finite Group Databasebeta

← #30609280

Subgroups — #30609280

Order 15616.

1926 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
1C2×#16840708C211yesmaximal, characteristic, complemented
2C2×#16840706C211yesmaximal, characteristic, complemented
3C2×#16840704C212yesmaximal
4#21461004C214yesmaximal, complemented
5#21461002C214yesmaximal
6#21460997C212yesmaximal, complemented
7C2×#117×D61C211yesmaximal, characteristic
8C22×C4⋊#6923C2212yes—
9C22×C4⋊#6923C2212yes—
10C22×C4⋊#6923C2214yes—
11C22×C22⋊#6923C2214yes—
12#16840708—24no—
13#16840706—24no—
14#16840704—28no—
15C2×(C2×#6923)⋊C4C2214yes—
16C2×(C2×#6923)⋊C4C2214yes—
17C2×(C2×#6923)⋊C4C2218yes—
18C23×C4×D61C2211yescharacteristic
19C2×#14519736C411yescharacteristic, complemented
20C2×#14519735C412yescomplemented
21C2×#14519734C411yescharacteristic, complemented
22C2×#14519734C411yescharacteristic, complemented
23C2×#14519733C412yescomplemented
24#117×D61—24no—
25#117×D61—24no—
26C2×C4⋊C4×D61C2212yes—
27C2×C22⋊C4×D61C2212yes—
28C2×C42×D61C2211yescharacteristic
29C2×C42×D61C2211yescharacteristic
30C2×#117×C61C411yescharacteristic, complemented
31C23×#6923C2314yes—
32C23×#6923C2314yes—
33C2×C4⋊#6923—24no—
34C2×C4⋊#6923—24no—
35C2×C4⋊#6923—24no—
36C2×C4⋊#6923—28no—
37C2×C4⋊#6923—24no—
38C2×C4⋊#6923—24no—
39C2×C4⋊#6923—24no—
40C2×C4⋊#6923—48no—
41C2×C22⋊#6923—28no—
42C2×C22⋊#6923—48no—
43(C2×#6923)⋊C4—28no—
44(C2×#6923)⋊C4—28no—
45(C2×#6923)⋊C4—48no—
46C24×D61C2311yescharacteristic
47C23×Dic61C2×C411yescharacteristic
48C22×C4×D61—22no—
49C22×C4×D61—22no—
50C22×C4×D61—22no—
51C22×C4×D61—22no—
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2×C411yescharacteristic
54C22×C4×D61D414yes—
55C22×C4×D61Q814yes—
56C22×C4×D61C2312yes—
57C22×C4×D61C2312yes—
58C22×C4×D61C2311yescharacteristic
59C22×C4×D61C2311yescharacteristic
60#14519736—24no—
61#14519735—28no—
62#14519734—24no—
63#14519734—24no—
64#14519733—28no—
65C2×C4⋊Dic61C2×C412yes—
66C2×#11758196C2×C412yes—
67C2×#11758196C2×C412yes—
68C4⋊C4×D61—24no—
69C4⋊C4×D61—24no—
70C2×C22⋊Dic61C2×C412yes—
71C2×#11758198C2×C412yes—
72C2×#11758198C2×C412yes—
73C22⋊C4×D61—24no—
74C22⋊C4×D61—24no—
75C2×C4×Dic61C2×C412yes—
76C2×C4×Dic61C2×C411yescharacteristic
77C2×C4×Dic61C2×C412yes—
78C2×C4×Dic61C2×C411yescharacteristic
79C42×D61—22no—
80C42×D61—22no—
81C42×D61D414yes—
82C42×D61D414yes—
83C23×C4×C61C2×C411yescharacteristic
84#117×C61—24no—
85C2×C4⋊C4×C61C2×C412yes—
86C2×C22⋊C4×C61C2×C412yes—
87C2×C42×C61C2×C411yescharacteristic
88C2×C42×C61C2×C411yescharacteristic
89C24×C61C22×C411yescharacteristic
90C23×D61—21no—
91C23×D61—21no—
92C23×D61—21no—
93C23×D61—21no—
94C23×D61—24no—
95C23×D61—24no—
96C23×D61C22×C411yescharacteristic
97C23×D61C2×D412yes—
98C23×D61C2×Q812yes—
99C23×D61C2411yescharacteristic
100C22×#6923—24no—
101C22×#6923—24no—
102C22×#6923—24no—
103C22×#6923—24no—
104C22×#6923—24no—
105C22×#6923—24no—
106C22×#6923—28no—
107C22×#6923—28no—
108C22×#6923—28no—
109C22×#6923—28no—
110C22×C4×C61—22no—
111C22×C4×C61—22no—
112C22×C4×C61C4⋊C414yescomplemented
113C22×C4×C61C22×C412yes—
114C22×C4×C61C22×C412yes—
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C22×Dic61—22no—
118C22×Dic61—22no—
119C22×Dic61C4⋊C414yescomplemented
120C22×Dic61C22×C412yes—
121C22×Dic61C22×C412yes—
122C22×Dic61C22×C411yescharacteristic
123C22×Dic61C22×C411yescharacteristic
124C2×C4×D61—21no—
125C2×C4×D61—21no—
126C2×C4×D61—21no—
127C2×C4×D61—21no—
128C2×C4×D61—24no—
129C2×C4×D61—24no—
130C2×C4×D61—22no—
131C2×C4×D61—22no—
132C2×C4×D61—22no—
133C2×C4×D61—22no—
134C2×C4×D61—28no—
135C2×C4×D61—28no—
136C2×C4×D61—28no—
137C2×C4×D61—28no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—24no—
141C2×C4×D61—24no—
142C2×C4×D61C4⋊C414yescomplemented
143C2×C4×D61C4⋊C414yescomplemented
144C2×C4×D61C2×D414yes—
145C2×C4×D61C2×D414yes—
146C2×C4×D61C2×D412yes—
147C2×C4×D61C2×D412yes—
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×D412yes—
150C2×C4×D61C2×Q812yes—
151C2×C4×D61C2×Q812yes—
152C2×C4×D61#4112yes—
153C2×C4×D61#4112yes—
154C22⋊#6923—416no—
155C4⋊#6923—416no—
156C4⋊#6923—48no—
157C4⋊#6923—48no—
158C4⋊C4×C61—24no—
159C22⋊C4×C61—24no—
160C42×C61—22no—
161C42×C61C22⋊C414yescomplemented
162C22⋊Dic61—24no—
163#11758198—24no—
164#11758198—24no—
165C4⋊Dic61—24no—
166#11758196—24no—
167#11758196—24no—
168C4×Dic61—22no—
169C4×Dic61—24no—
170C4×Dic61C22⋊C418yescomplemented
171C4×Dic61C22⋊C414yescomplemented
172C23×C61—21no—
173C23×C61—21no—
174C23×C61—24no—
175C23×C61C2×C4⋊C412yes—
176C23×C61C23×C411yescharacteristic
177C22×D61—21no—
178C22×D61—21no—
179C22×D61—24no—
180C22×D61—22no—
181C22×D61—22no—
182C22×D61—22no—
183C22×D61—22no—
184C22×D61—24no—
185C22×D61—24no—
186C22×D61—28no—
187C22×D61—42no—
188C22×D61—42no—
189C22×D61—44no—
190C22×D61—42no—
191C22×D61—42no—
192C22×D61—44no—
193C22×D61C2×C4⋊C412yes—
194C22×D61C22×D411yescharacteristic
195C22×D61C22×D411yescharacteristic
196C22×D61C22×D411yescharacteristic
197C22×D61C22×Q811yescharacteristic
198C22×D61C2×#4111yescharacteristic
199C22×D61C2×#4111yescharacteristic
200C22×D61H(5,𝔽2)14yes—
201C22×D612−1+414yes—
202C2×#6923—44no—
203C2×#6923—44no—
204C2×#6923—48no—
205C2×#6923—44no—
206C2×#6923—44no—
207C2×#6923—44no—
208C2×#6923—44no—
209C2×#6923—48no—
210C2×#6923—48no—
211C2×#6923—48no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—24no—
215C2×C4×C61—28no—
216C2×C4×C61—22no—
217C2×C4×C61—24no—
218C2×C4×C61C2×C22⋊C414yes—
219C2×C4×C61C2×C22⋊C412yes—
220C2×C4×C61C2×C4⋊C412yes—
221C2×C4×C61C2×C4⋊C412yes—
222C2×C4×C61#11712yes—
223C2×Dic61—21no—
224C2×Dic61—21no—
225C2×Dic61—24no—
226C2×Dic61—28no—
227C2×Dic61—22no—
228C2×Dic61—24no—
229C2×Dic61C2×C22⋊C414yes—
230C2×Dic61C2×C22⋊C412yes—
231C2×Dic61C2×C4⋊C412yes—
232C2×Dic61C2×C4⋊C412yes—
233C2×Dic61#11712yes—
234C4×D61—22no—
235C4×D61—22no—
236C4×D61—22no—
237C4×D61—22no—
238C4×D61—24no—
239C4×D61—24no—
240C4×D61—28no—
241C4×D61—28no—
242C4×D61—28no—
243C4×D61—42no—
244C4×D61—42no—
245C4×D61—44no—
246C4×D61#12114yes—
247C4×D61#12114yes—
248C4×D61C22⋊Q814yes—
249C4×D61C22⋊Q814yes—
250#14340—611nomaximal, Sylow
251C22×C61—24no—
252C22×C61—22no—
253C22×C61—42no—
254C22×C61—42no—
255C22×C61C22×C22⋊C411yescharacteristic
256C22×C61C22×C4⋊C411yescharacteristic
257C22×C61C2×#11711yescharacteristic
258C22×C61#52014yes—
259C2×D61—24no—
260C2×D61—22no—
261C2×D61—42no—
262C2×D61—42no—
263C2×D61—42no—
264C2×D61—44no—
265C2×D61—42no—
266C2×D61—44no—
267C2×D61—44no—
268C2×D61C2×#12111yescharacteristic
269C2×D61C2×#12111yescharacteristic
270C2×D61C2×C22⋊Q811yescharacteristic
271C2×D61C2×C22⋊Q811yescharacteristic
272C2×D61#53512yes—
273C2×D61#53512yes—
274C2×D61#53612yes—
275C2×D61#53612yes—
276C2×D61#53714yes—
277C2×D61#54012yes—
278C2×D61#54112yes—
279C2×D61#54214yes—
280C2×D61#54414yes—
281#6923—84no—
282#6923—84no—
283C4×C61—24no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61#38014yes—
287Dic61—24no—
288Dic61—22no—
289Dic61—42no—
290Dic61#38014yes—
291C22×#117—611no—
292C2×#384—611no—
293C2×#382—611no—
294C2×#380—612no—
295#1441—614no—
296#1439—614no—
297C4⋊#95—612no—
298C2×C61—42no—
299C2×C61C2×#38011yescharacteristic
300C2×C61#231112yes—
301C2×C61#231412yes—
302C2×C61#231712yes—
303D61—42no—
304D61#261911yescharacteristic
305D61#262012yes—
306D61#262111yescharacteristic
307D61#262812yes—
308D61#262912yes—
309C24×C4—611no—
310C2×#117—611no—
311C2×#117—612no—
312C2×#117—612no—
313C2×#117—611no—
314C2×#117—611no—
315C2×#117—1224no—
316C2×#117—1224no—
317C2×#117#692311yescharacteristic, complemented
318C22×C4⋊C4—612no—
319C22×C4⋊C4—612no—
320C22×C4⋊C4—612no—
321C22×C4⋊C4—614no—
322C22×C22⋊C4—612no—
323C22×C22⋊C4—614no—
324C22×C42—611no—
325C22×C42—611no—
326#384—1224no—
327#382—1224no—
328#380—1228no—
329C2×#95—614no—
330C2×#95—614no—
331C2×#95—618no—
332C61#1434011yescharacteristic, complemented, Sylow
333C25—611no—
334C23×C4—611no—
335C23×C4—611no—
336C23×C4—614no—
337C23×C4—614no—
338C23×C4—612no—
339C23×C4—611no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—612no—
343C23×C4—614no—
344C23×C4—614no—
345C23×C4—1222no—
346C23×C4—1222no—
347C23×C4—1222no—
348C23×C4—1222no—
349C23×C4C2×#692311yescharacteristic
350#117—24no—
351#117—1224no—
352#117—1228no—
353#117—1228no—
354#117—1224no—
355#117—1224no—
356C2×C4⋊C4—612no—
357C2×C4⋊C4—612no—
358C2×C4⋊C4—612no—
359C2×C4⋊C4—1224no—
360C2×C4⋊C4—1224no—
361C2×C4⋊C4—1224no—
362C2×C4⋊C4—1224no—
363C2×C4⋊C4—1224no—
364C2×C4⋊C4—1224no—
365C2×C4⋊C4—1224no—
366C2×C4⋊C4—1224no—
367C2×C4⋊C4—1228no—
368C2×C4⋊C4—2448no—
369C2×C4⋊C4C2×#692312yes—
370C2×C22⋊C4—612no—
371C2×C22⋊C4—612no—
372C2×C22⋊C4—612no—
373C2×C22⋊C4—1224no—
374C2×C22⋊C4—1224no—
375C2×C22⋊C4—1228no—
376C2×C22⋊C4—2448no—
377C2×C22⋊C4C2×#692312yes—
378C2×C42—612no—
379C2×C42—611no—
380C2×C42—614no—
381C2×C42—614no—
382C2×C42—612no—
383C2×C42—611no—
384C2×C42—1222no—
385C2×C42—1222no—
386C2×C42C2×#692311yescharacteristic
387C2×C42C2×#692311yescharacteristic
388#95—1228no—
389#95—1228no—
390#95—2448no—
391C24—611no—
392C24—612no—
393C24—612no—
394C24—611no—
395C24—1221no—
396C24—1221no—
397C24—1224no—
398C24—1221no—
399C24—1221no—
400C24—1224no—
401C24C22×#692311yescharacteristic
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—611no—
405C22×C4—612no—
406C22×C4—612no—
407C22×C4—612no—
408C22×C4—612no—
409C22×C4—612no—
410C22×C4—612no—
411C22×C4—614no—
412C22×C4—612no—
413C22×C4—611no—
414C22×C4—614no—
415C22×C4—614no—
416C22×C4—612no—
417C22×C4—614no—
418C22×C4—614no—
419C22×C4—612no—
420C22×C4—612no—
421C22×C4—1222no—
422C22×C4—1222no—
423C22×C4—1228no—
424C22×C4—1228no—
425C22×C4—1228no—
426C22×C4—1228no—
427C22×C4—1221no—
428C22×C4—1221no—
429C22×C4—1224no—
430C22×C4—1221no—
431C22×C4—1221no—
432C22×C4—1224no—
433C22×C4—1224no—
434C22×C4—1222no—
435C22×C4—1222no—
436C22×C4—1224no—
437C22×C4—1222no—
438C22×C4—1222no—
439C22×C4—1222no—
440C22×C4—1222no—
441C22×C4—1224no—
442C22×C4—1224no—
443C22×C4—1224no—
444C22×C4—1228no—
445C22×C4—1228no—
446C22×C4—1228no—
447C22×C4—1224no—
448C22×C4—1224no—
449C22×C4—1224no—
450C22×C4—1228no—
451C22×C4C4⋊#692314yescomplemented
452C22×C4C22×#692312yes—
453C22×C4C22×#692312yes—
454C22×C4C22×#692311yescharacteristic
455C22×C4C22×#692311yescharacteristic
456C4⋊C4—24no—
457C4⋊C4—1224no—
458C4⋊C4—1224no—
459C4⋊C4—1224no—
460C4⋊C4—2448no—
461C4⋊C4—2448no—
462C4⋊C4—24416no—
463C22⋊C4—24no—
464C22⋊C4—1224no—
465C22⋊C4—1224no—
466C22⋊C4—1224no—
467C22⋊C4—24416no—
468C42—22no—
469C42—618no—
470C42—614no—
471C42—1224no—
472C42—1222no—
473C42C22⋊#692314yescomplemented
474C23—21no—
475C23—21no—
476C23—24no—
477C23—611no—
478C23—611no—
479C23—611no—
480C23—611no—
481C23—611no—
482C23—611no—
483C23—614no—
484C23—614no—
485C23—612no—
486C23—1224no—
487C23—1222no—
488C23—1222no—
489C23—1222no—
490C23—1222no—
491C23—1224no—
492C23—1228no—
493C23—1221no—
494C23—1221no—
495C23—1224no—
496C23—2442no—
497C23—2444no—
498C23—2442no—
499C23—2442no—
500C23—2444no—
501C23—2442no—
502C23C2×C4⋊#692312yes—
503C23C23×#692311yescharacteristic
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—22no—
509C2×C4—24no—
510C2×C4—612no—
511C2×C4—612no—
512C2×C4—612no—
513C2×C4—614no—
514C2×C4—614no—
515C2×C4—614no—
516C2×C4—614no—
517C2×C4—614no—
518C2×C4—612no—
519C2×C4—1228no—
520C2×C4—1224no—
521C2×C4—1228no—
522C2×C4—1224no—
523C2×C4—1222no—
524C2×C4—1222no—
525C2×C4—1222no—
526C2×C4—1222no—
527C2×C4—1228no—
528C2×C4—1221no—
529C2×C4—1221no—
530C2×C4—1224no—
531C2×C4—1228no—
532C2×C4—1222no—
533C2×C4—1224no—
534C2×C4—2442no—
535C2×C4—2444no—
536C2×C4—2442no—
537C2×C4—2444no—
538C2×C4—2444no—
539C2×C4—2448no—
540C2×C4—2444no—
541C2×C4—2444no—
542C2×C4—2448no—
543C2×C4—2448no—
544C2×C4—2448no—
545C2×C4—2444no—
546C2×C4—2444no—
547C2×C4C2×C22⋊#692314yes—
548C2×C4C2×C22⋊#692312yes—
549C2×C4C2×C4⋊#692312yes—
550C2×C4C2×C4⋊#692312yes—
551C2×C4#1451985612yes—
552C22—24no—
553C22—22no—
554C22—42no—
555C22—42no—
556C22—612no—
557C22—614no—
558C22—612no—
559C22—612no—
560C22—614no—
561C22—612no—
562C22—612no—
563C22—612no—
564C22—614no—
565C22—611no—
566C22—611no—
567C22—611no—
568C22—611no—
569C22—1224no—
570C22—1222no—
571C22—2442no—
572C22—2444no—
573C22—2442no—
574C22—2444no—
575C22—2444no—
576C22—2442no—
577C22—2442no—
578C22C22×C22⋊#692311yescharacteristic
579C22C22×C4⋊#692311yescharacteristic
580C22C2×#1451985611yescharacteristic
581C22#1684083114yes—
582C4—24no—
583C4—22no—
584C4—42no—
585C4—614no—
586C4—1224no—
587C4—1222no—
588C4—4884no—
589C4—4884no—
590C4—2442no—
591C4#1684070414yes—
592C2—42no—
593C2—611no—
594C2—612no—
595C2—611no—
596C2—612no—
597C2—612no—
598C2—2442no—
599C2C2×#1684070411yescharacteristic
600C2#2146229412yes—
601C2#2146229712yes—
602C2#2146230012yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×#117×C61C4
prime radicalC24×D61C23
cosocleC23×D61C24
hypercenterC2×#117#6923
hypocenterC61#14340
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×#16840708C2semidirect
2. C2×#16840706C2semidirect
4. #21461004C2semidirect
6. #21460997C2semidirect
19. C2×#14519736C4semidirect
20. C2×#14519735C4semidirect
21. C2×#14519734C4semidirect
22. C2×#14519734C4semidirect
23. C2×#14519733C4semidirect
30. C2×#117×C61C4semidirect
112. C22×C4×C61C4⋊C4semidirect
119. C22×Dic61C4⋊C4semidirect
142. C2×C4×D61C4⋊C4semidirect
143. C2×C4×D61C4⋊C4semidirect
161. C42×C61C22⋊C4semidirect
170. C4×Dic61C22⋊C4semidirect
171. C4×Dic61C22⋊C4semidirect
317. C2×#117#6923semidirect
332. C61#14340semidirect
451. C22×C4C4⋊#6923semidirect
473. C42C22⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407082
C2×#168407062
C2×#168407042
#214610042
#214610022
#214609972
C2×#117×D612
#1434061