Finite Group Databasebeta

← #29708856

Subgroups — #29708856

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839798C211yesmaximal, characteristic, complemented
2C22×C22⋊#92418C211yesmaximal, characteristic
3#21448202C211yesmaximal, characteristic
4#21447256C212yesmaximal, complemented
5#21447255C212yesmaximal, complemented
6#21447250C212yesmaximal, complemented
7#21447245C212yesmaximal, complemented
8#21445535C211yesmaximal, characteristic, complemented
9#21445535C211yesmaximal, characteristic, complemented
10#21445535C211yesmaximal, characteristic, complemented
11C61×#2280C211yesmaximal, characteristic
12C23×#92418C2211yescharacteristic
13C23×#92418C2211yescharacteristic
14C2×#14519736C2211yescharacteristic
15C2×#14519736C2211yescharacteristic
16#16839798—24no—
17C2×#14519710C2211yescharacteristic, complemented
18C2×#14519710C2211yescharacteristic, complemented
19C2×#14519708C2211yescharacteristic, complemented
20C2×#14519708C2211yescharacteristic, complemented
21C2×C22⋊#92418—24no—
22C2×C22⋊#92418C2212yes—
23C2×C22⋊#92418C2212yes—
24#16839625—21no—
25#16839622C2212yes—
26#16839622C2212yes—
27#16839618—21no—
28C4×C22⋊Dic61—22no—
29#16839524C2212yescomplemented
30#16839524C2212yescomplemented
31#16839524C2212yescomplemented
32#16839524C2212yescomplemented
33#16839524C2212yescomplemented
34#16839524C2212yescomplemented
35C24×C4×C61C2211yescharacteristic
36C2×#117×C61C2211yescharacteristic
37C2×#117×C61C2211yescharacteristic
38C61×#381—21no—
39C61×#380C2212yes—
40C61×#380C2212yes—
41C61×#379—21no—
42C4×C22⋊C4×C61—22no—
43C22×#92418—22no—
44C22×#92418—22no—
45C22×#92418—22no—
46C22×#92418—22no—
47C22×#92418—22no—
48C22×#92418D414yes—
49C22×#92418C2312yes—
50C22×#92418C2311yescharacteristic
51C22×#92418C2311yescharacteristic
52#14519736—22no—
53#14519736—22no—
54#14519736—22no—
55#14519736—22no—
56C2×C4⋊Dic61C2312yes—
57C2×C4⋊Dic61C2311yescharacteristic
58C2×C4⋊Dic61C2311yescharacteristic
59C2×C22⋊Dic61—21no—
60C2×C22⋊Dic61—21no—
61C2×C22⋊Dic61—22no—
62C2×C22⋊Dic61—22no—
63C2×C4×Dic61—21no—
64C2×C4×Dic61—21no—
65#14519710—22no—
66#14519710—22no—
67#14519708—22no—
68#14519708—22no—
69C22⋊#92418—24no—
70C22⋊#92418—24no—
71#14519684—22no—
72#14519684—22no—
73C25×C61D411yescharacteristic
74C23×C4×C61—22no—
75C23×C4×C61—22no—
76C23×C4×C61—22no—
77C23×C4×C61—22no—
78C23×C4×C61—22no—
79C23×C4×C61—22no—
80C23×C4×C61D412yes—
81C23×C4×C61D411yescharacteristic
82C23×C4×C61C2312yes—
83C23×C4×C61C2311yescharacteristic
84#117×C61—22no—
85#117×C61—22no—
86#117×C61—22no—
87#117×C61—22no—
88C2×C4⋊C4×C61C2312yes—
89C2×C4⋊C4×C61C2311yescharacteristic
90C2×C4⋊C4×C61C2311yescharacteristic
91C2×C22⋊C4×C61—22no—
92C2×C22⋊C4×C61—22no—
93C2×C22⋊C4×C61—21no—
94C2×C22⋊C4×C61—21no—
95C2×C42×C61—21no—
96C2×C42×C61—21no—
97#95×C61—22no—
98#95×C61—22no—
99C24×C61—22no—
100C24×C61—24no—
101C24×C61—22no—
102C24×C61—22no—
103C24×C61—42no—
104C24×C61C2×D412yes—
105C24×C61C2×D411yescharacteristic
106C22×C4×C61—22no—
107C22×C4×C61—22no—
108C22×C4×C61—22no—
109C22×C4×C61—24no—
110C22×C4×C61—24no—
111C22×C4×C61—24no—
112C22×C4×C61—22no—
113C22×C4×C61—22no—
114C22×C4×C61—22no—
115C22×C4×C61—22no—
116C22×C4×C61—22no—
117C22×C4×C61—22no—
118C22×C4×C61—21no—
119C22×C4×C61—21no—
120C22×C4×C61—22no—
121C22×C4×C61—44no—
122C22×C4×C61—44no—
123C22×C4×C61—44no—
124C22×C4×C61—44no—
125C22×C4×C61C2×D412yes—
126C22×C4×C61C2×D411yescharacteristic
127C22×C4×C61C2×D412yes—
128C22×C4×C61#4112yes—
129C22×C4×C61#4112yes—
130C22×C4×C61#4112yes—
131C22×C4×C61#4112yes—
132C22×C4×C61#4112yes—
133C22×C4×C61C2411yescharacteristic
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C4⋊C4×C61—21no—
138C4⋊C4×C61—21no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—21no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C22⋊C4×C61—44no—
144C22⋊C4×C61—42no—
145C22⋊C4×C61—42no—
146C22⋊C4×C61—44no—
147C42×C61—42no—
148C42×C61—42no—
149C22⋊Dic61—44no—
150C22⋊Dic61—42no—
151C22⋊Dic61—42no—
152C22⋊Dic61—44no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—22no—
158C4⋊Dic61—22no—
159C4×Dic61—42no—
160C4×Dic61—42no—
161C2×#92418—28no—
162C2×#92418—21no—
163C2×#92418—21no—
164C2×#92418—21no—
165C2×#92418—21no—
166C2×#92418—22no—
167C2×#92418—22no—
168C2×#92418—22no—
169C2×#92418—44no—
170C2×#92418C2×D412yes—
171C2×#92418C2×D412yes—
172C23×C61—22no—
173C23×C61—22no—
174C23×C61—28no—
175C23×C61—22no—
176C23×C61—22no—
177C23×C61—22no—
178C23×C61—22no—
179C23×C61—22no—
180C23×C61—22no—
181C23×C61—48no—
182C23×C61—48no—
183C23×C61—44no—
184C23×C61—44no—
185C23×C61#12112yes—
186C23×C61#12312yes—
187C23×C61#12312yes—
188C23×C61#13514yes—
189C23×C61C22×D411yescharacteristic
190C2×C4×C61—22no—
191C2×C4×C61—22no—
192C2×C4×C61—22no—
193C2×C4×C61—44no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61—42no—
197C2×C4×C61—42no—
198C2×C4×C61—48no—
199C2×C4×C61—41no—
200C2×C4×C61—41no—
201C2×C4×C61—42no—
202C2×C4×C61—42no—
203C2×C4×C61—42no—
204C2×C4×C61—42no—
205C2×C4×C61—42no—
206C2×C4×C61#12112yes—
207C2×C4×C61#12314yes—
208C2×C4×C61#12312yes—
209C2×C4×C61#12312yes—
210C2×C4×C61#12414yes—
211C2×C4×C61#12614yes—
212C2×C4×C61#12614yes—
213C2×C4×C61C22×D411yescharacteristic
214C2×C4×C61C2×#4111yescharacteristic
215C2×C4×C61C2×#4111yescharacteristic
216C2×C4×C61C2×#4111yescharacteristic
217C2×C4×C61C2×#4111yescharacteristic
218C2×C4×C61C2×#4111yescharacteristic
219C2×Dic61—41no—
220C2×Dic61—41no—
221C2×Dic61—42no—
222C2×Dic61—42no—
223C2×Dic61—42no—
224C2×Dic61—42no—
225C2×Dic61—42no—
226#92418—22no—
227#92418—22no—
228#92418—42no—
229#23638—611nomaximal, Sylow
230C22×C61—24no—
231C22×C61—24no—
232C22×C61—22no—
233C22×C61—22no—
234C22×C61—24no—
235C22×C61—24no—
236C22×C61—24no—
237C22×C61—416no—
238C22×C61—42no—
239C22×C61—42no—
240C22×C61—42no—
241C22×C61—42no—
242C22×C61#39714yes—
243C22×C61C2×#12111yescharacteristic
244C22×C61C2×#12311yescharacteristic
245C22×C61C2×#12311yescharacteristic
246C22×C61C2×#13512yes—
247C22×C61#57512yes—
248C4×C61—42no—
249C4×C61—82no—
250C4×C61—81no—
251C4×C61—81no—
252C4×C61#39714yes—
253C4×C61C2×#12311yescharacteristic
254C4×C61C2×#12411yescharacteristic
255C4×C61C2×#12611yescharacteristic
256C4×C61C2×#12611yescharacteristic
257Dic61—82no—
258Dic61—81no—
259Dic61—81no—
260C2×#427—611no—
261C22×C22⋊C8—611no—
262#2280—611no—
263#2280D6111yescharacteristic
264#2077—612no—
265#2077—612no—
266#2076—612no—
267#2075—612no—
268#1735—611no—
269#1735—611no—
270#1735—611no—
271C2×C61—24no—
272C2×C61—44no—
273C2×C61C2×#39711yescharacteristic
274C2×C61#305011yescharacteristic
275C2×C61#305411yescharacteristic
276C2×C61#309112yes—
277C2×C61#309112yes—
278C24×C4C2×D6111yescharacteristic
279C23×C8—611no—
280C23×C8—611no—
281C2×#117—611no—
282C2×#117—611no—
283C2×#117C2×D6111yescharacteristic
284C2×#117C2×D6111yescharacteristic
285#427—1224no—
286C2×#107—611no—
287C2×#107—611no—
288C2×#106—611no—
289C2×#106—611no—
290C2×C22⋊C8—612no—
291C2×C22⋊C8—612no—
292C2×C22⋊C8—1224no—
293#381—21no—
294#381—1221no—
295#380—612no—
296#380—612no—
297#380C2×D6112yes—
298#380C2×D6112yes—
299#379—21no—
300#379—1221no—
301C4×C22⋊C4—22no—
302C4×C22⋊C4—1222no—
303#340—612no—
304#340—612no—
305#340—612no—
306#340—612no—
307#340—612no—
308#340—612no—
309C61#2363811yescharacteristic, complemented, Sylow
310C25C22⋊D6111yescharacteristic
311C23×C4—22no—
312C23×C4—22no—
313C23×C4—22no—
314C23×C4—22no—
315C23×C4—22no—
316C23×C4—22no—
317C23×C4C22⋊D6111yescharacteristic
318C23×C4C22⋊D6112yes—
319C23×C4C22×D6112yes—
320C23×C4C22×D6111yescharacteristic
321C22×C8—614no—
322C22×C8—611no—
323C22×C8—611no—
324C22×C8—612no—
325C22×C8—1222no—
326C22×C8—1222no—
327C22×C8—1222no—
328C22×C8—1222no—
329C22×C8—1222no—
330#117—22no—
331#117—22no—
332#117—22no—
333#117—22no—
334#117—1222no—
335#117—1222no—
336#117—1222no—
337#117—1222no—
338C2×C4⋊C4—612no—
339C2×C4⋊C4—611no—
340C2×C4⋊C4—611no—
341C2×C4⋊C4C22×D6112yes—
342C2×C4⋊C4C22×D6111yescharacteristic
343C2×C4⋊C4C22×D6111yescharacteristic
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—1221no—
349C2×C22⋊C4—1221no—
350C2×C22⋊C4—1222no—
351C2×C22⋊C4—1222no—
352C2×C42—21no—
353C2×C42—21no—
354C2×C42—1221no—
355C2×C42—1221no—
356#107—1222no—
357#107—1222no—
358#106—1222no—
359#106—1222no—
360C22⋊C8—1224no—
361C22⋊C8—1224no—
362#95—22no—
363#95—22no—
364#95—1222no—
365#95—1222no—
366C24—22no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—42no—
371C24C2×C22⋊D6112yes—
372C24C2×C22⋊D6111yescharacteristic
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—44no—
392C22×C4—1221no—
393C22×C4—1221no—
394C22×C4—1222no—
395C22×C4#1175822212yes—
396C22×C4#1175822212yes—
397C22×C4D4×D6112yes—
398C22×C4#1175822412yes—
399C22×C4#1175822412yes—
400C22×C4#1175822412yes—
401C22×C4C2×C22⋊D6112yes—
402C22×C4C2×C22⋊D6111yescharacteristic
403C22×C4C23×D6111yescharacteristic
404C2×C8—612no—
405C2×C8—612no—
406C2×C8—1221no—
407C2×C8—1221no—
408C2×C8—1221no—
409C2×C8—1221no—
410C2×C8—1228no—
411C2×C8—1222no—
412C2×C8—1222no—
413C2×C8—1222no—
414C2×C8—2444no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—1221no—
422C4⋊C4—1221no—
423C4⋊C4—1221no—
424C4⋊C4—1221no—
425C4⋊C4—1222no—
426C4⋊C4—1222no—
427C22⋊C4—44no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—2444no—
432C22⋊C4—2442no—
433C22⋊C4—2442no—
434C22⋊C4—2444no—
435C42—42no—
436C42—42no—
437C42—2442no—
438C42—2442no—
439C23—22no—
440C23—22no—
441C23—28no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—48no—
449C23—48no—
450C23—44no—
451C23—44no—
452C23#1451974912yes—
453C23#1451975812yes—
454C23#1451976012yes—
455C23#1451979714yes—
456C23C22×C22⋊D6111yescharacteristic
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—44no—
461C2×C4—48no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—2441no—
474C2×C4—2441no—
475C2×C4—2442no—
476C2×C4—2442no—
477C2×C4—2442no—
478C2×C4—2442no—
479C2×C4—2442no—
480C2×C4#1451974912yes—
481C2×C4#1451975812yes—
482C2×C4#1451975914yes—
483C2×C4#1451976012yes—
484C2×C4#1451976314yes—
485C2×C4#1451977014yes—
486C2×C4#1451977014yes—
487C2×C4C2×D4×D6111yescharacteristic
488C2×C4C2×#1175822411yescharacteristic
489C2×C4C2×#1175822411yescharacteristic
490C2×C4C2×#1175822411yescharacteristic
491C2×C4C2×#1175822211yescharacteristic
492C2×C4C2×#1175822211yescharacteristic
493C8—1222no—
494C8—1222no—
495C8—2442no—
496C22—22no—
497C22—22no—
498C22—24no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—24no—
503C22—416no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22#1683969014yes—
509C22C2×#1451974911yescharacteristic
510C22C2×#1451975811yescharacteristic
511C22C2×#1451976011yescharacteristic
512C22C2×#1451979712yes—
513C22#1684056012yes—
514C4—42no—
515C4—81no—
516C4—81no—
517C4—82no—
518C4—4881no—
519C4—4881no—
520C4—4882no—
521C4#1683969014yes—
522C4C2×#1451975911yescharacteristic
523C4C2×#1451976311yescharacteristic
524C4C2×#1451977011yescharacteristic
525C4C2×#1451977011yescharacteristic
526C2—24no—
527C2—44no—
528C2C2×#1683969011yescharacteristic
529C2#2145539711yescharacteristic
530C2#2145542511yescharacteristic
531C2#2145572812yes—
532C2#2145573212yes—
subgroupG/H
centerC22×C4C2×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2280C2
socleC23×C61C22×D4
prime radicalC25×C61D4
hypercenter#2280D61
hypocenterC61#23638
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ C24×C4 ◁ #2280

Complements

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

NH ≅ G/Ntype
1. C2×#16839798C2semidirect
4. #21447256C2semidirect
5. #21447255C2semidirect
6. #21447250C2semidirect
7. #21447245C2semidirect
8. #21445535C2semidirect
9. #21445535C2semidirect
10. #21445535C2semidirect
17. C2×#14519710C22semidirect
18. C2×#14519710C22semidirect
19. C2×#14519708C22semidirect
20. C2×#14519708C22semidirect
29. #16839524C22semidirect
30. #16839524C22semidirect
31. #16839524C22semidirect
32. #16839524C22semidirect
33. #16839524C22semidirect
34. #16839524C22semidirect
309. C61#23638semidirect

Maximal subgroups

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

subgroupindex
C2×#168397982
C22×C22⋊#924182
#214482022
#214472562
#214472552
#214472502
#214472452
#214455352
C61×#22802
#2363861