Finite Group Databasebeta

← #29658509

Subgroups — #29658509

Order 15616.

966 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456767C211yesmaximal, characteristic, complemented
2#21455858C211yesmaximal, characteristic, complemented
3#21455616C211yesmaximal, characteristic
4#21445033C214yesmaximal, complemented
5#21444953C214yesmaximal, complemented
6#21444500C211yesmaximal, characteristic, complemented
7#21444248C211yesmaximal, characteristic, complemented
8#21444210C211yesmaximal, characteristic
9C2×C61×#554C211yesmaximal, characteristic
10#16840319C2211yescharacteristic
11#16840141C2212yescomplemented
12#16840112C2212yescomplemented
13#16840097—21no—
14#16840062—22no—
15#16840057C2212yescomplemented
16#16840052C2212yes—
17#16840042C2212yes—
18#16839972C2212yescomplemented
19#16839969—22no—
20#16839918—22no—
21#16839918—21no—
22C4×#11758202—22no—
23#16839815C2211yescharacteristic
24#16839815C2211yescharacteristic
25#16839802—21no—
26#16839786C2211yescharacteristic
27#16839767—21no—
28#16839511C2214yescomplemented
29Q8⋊#92418C2214yescomplemented
30D4⋊#92418C2214yes—
31C61×#554—22no—
32C61×#554—22no—
33C61×#554—22no—
34C61×#554C2214yes—
35C2×C4⋊Q8×C61C2211yescharacteristic
36C2×#125×C61—21no—
37C2×C22⋊Q8×C61—21no—
38C2×C22⋊Q8×C61—22no—
39C2×C4×Q8×C61C2211yescharacteristic
40C2×C4×D4×C61—21no—
41C2×C4×D4×C61C2211yescharacteristic
42C22×C4⋊C4×C61—21no—
43C2×#11758202—22no—
44C2×#11758202—24no—
45C2×#11758194—21no—
46#14519777C2311yescharacteristic
47#14519777C2311yescharacteristic
48#14519769—21no—
49#14519756—21no—
50#14519756—21no—
51C4×Dic122—21no—
52C4×Dic122—21no—
53#14519736—21no—
54#14519710—22no—
55#14519710—22no—
56#14519708—22no—
57C4⋊#92418—21no—
58C4⋊#92418—21no—
59C4⋊#92418C2312yes—
60C4⋊#92418C2312yes—
61#14519701—24no—
62#14519701—22no—
63#14519700—22no—
64#14519698—22no—
65#14519698—22no—
66C22⋊#92418—21no—
67C22⋊#92418—21no—
68C4×#92418—21no—
69C4×#92418C2312yes—
70C22×Q8×C61D411yescharacteristic
71C22×Q8×C61D412yes—
72C22×D4×C61D411yescharacteristic
73C23×C4×C61—22no—
74C23×C4×C61—21no—
75C4⋊Q8×C61—21no—
76C4⋊Q8×C61—21no—
77C4⋊Q8×C61D412yes—
78C4⋊Q8×C61C2312yes—
79#125×C61—24no—
80#125×C61—24no—
81C22⋊Q8×C61—24no—
82C22⋊Q8×C61—28no—
83C22⋊Q8×C61—24no—
84C22⋊Q8×C61—28no—
85C4×Q8×C61—21no—
86C4×Q8×C61—21no—
87C4×Q8×C61D412yes—
88C4×Q8×C61C2312yes—
89C4×D4×C61—24no—
90C4×D4×C61—24no—
91C4×D4×C61—21no—
92C4×D4×C61—21no—
93C4×D4×C61D412yes—
94C4×D4×C61C2312yes—
95C2×C4⋊C4×C61—24no—
96C2×C4⋊C4×C61—22no—
97C2×C4⋊C4×C61—22no—
98C2×C4⋊C4×C61—21no—
99C2×C4⋊C4×C61—21no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—24no—
102C2×C4⋊C4×C61D411yescharacteristic
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C22⋊C4×C61—22no—
105C2×C22⋊C4×C61—21no—
106C2×C42×C61—21no—
107C2×C42×C61C2311yescharacteristic
108C24×C61—21no—
109C2×Q8×C61—21no—
110C2×Q8×C61—22no—
111C2×Q8×C61—42no—
112C2×Q8×C61—82no—
113C2×Q8×C61C2×D412yes—
114C2×Q8×C61C2×D414yes—
115C2×D4×C61—21no—
116C2×D4×C61—22no—
117C2×D4×C61Q1614yescomplemented
118C2×D4×C61C2×D412yes—
119C22×C4×C61—21no—
120C22×C4×C61—21no—
121C22×C4×C61—21no—
122C22×C4×C61—22no—
123C22×C4×C61—28no—
124C22×C4×C61—22no—
125C22×C4×C61—22no—
126C22×C4×C61—44no—
127C22×C4×C61—84no—
128C22×C4×C61C2×D411yescharacteristic
129C22×C4×C61C2×D411yescharacteristic
130C22×C4×C61C2×D411yescharacteristic
131C2×Dic122—21no—
132C2×Dic122—21no—
133C4⋊C4×C61—28no—
134C4⋊C4×C61—28no—
135C4⋊C4×C61—21no—
136C4⋊C4×C61—22no—
137C4⋊C4×C61—22no—
138C4⋊C4×C61—22no—
139C4⋊C4×C61—21no—
140C4⋊C4×C61—22no—
141C4⋊C4×C61—22no—
142C4⋊C4×C61—22no—
143C4⋊C4×C61C2×D412yes—
144C4⋊C4×C61C2×D412yes—
145C22⋊C4×C61—28no—
146C22⋊C4×C61—22no—
147C22⋊C4×C61—22no—
148C42×C61—22no—
149C42×C61—22no—
150C42×C61C2×D411yescharacteristic
151C42×C61C2×D411yescharacteristic
152C42×C61C2×D411yescharacteristic
153C42×C61C2411yescharacteristic
154C22⋊Dic61—41no—
155#11758202—44no—
156#11758202—84no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61—21no—
161#11758196—41no—
162#11758196—41no—
163C4×Dic61—41no—
164#11758194—42no—
165C2×#92418—22no—
166C2×#92418—22no—
167C2×#92418—22no—
168C23×C61—22no—
169C23×C61—24no—
170C23×C61—42no—
171C23×C61—42no—
172C23×C61#12011yescharacteristic
173Q8×C61—22no—
174Q8×C61—22no—
175Q8×C61—42no—
176Q8×C61—44no—
177Q8×C61—82no—
178D4×C61—22no—
179D4×C61—24no—
180D4×C61C2×Q1614yes—
181C2×C4×C61—22no—
182C2×C4×C61—22no—
183C2×C4×C61—22no—
184C2×C4×C61—22no—
185C2×C4×C61—24no—
186C2×C4×C61—41no—
187C2×C4×C61—44no—
188C2×C4×C61—42no—
189C2×C4×C61—42no—
190C2×C4×C61—44no—
191C2×C4×C61—42no—
192C2×C4×C61—44no—
193C2×C4×C61—88no—
194C2×C4×C61#12011yescharacteristic
195C2×C4×C61#12011yescharacteristic
196C2×C4×C61#12011yescharacteristic
197C2×C4×C61C2×Q1612yes—
198C2×C4×C61#13712yes—
199C2×C4×C61C22×D411yescharacteristic
200C2×C4×C61C22×D411yescharacteristic
201C2×C4×C61C22×D411yescharacteristic
202C2×Dic61—41no—
203C2×Dic61—41no—
204Dic122—42no—
205Dic122—42no—
206#92418—42no—
207#92418—42no—
208#92418—42no—
209#18749—611nomaximal, Sylow
210C22×C61—21no—
211C22×C61—21no—
212C22×C61—24no—
213C22×C61—48no—
214C22×C61—42no—
215C22×C61—42no—
216C22×C61#45112yes—
217C22×C61C2×#12011yescharacteristic
218C4×C61—21no—
219C4×C61—21no—
220C4×C61—21no—
221C4×C61—42no—
222C4×C61—42no—
223C4×C61—42no—
224C4×C61—44no—
225C4×C61#45112yes—
226C4×C61C22×Q1611yescharacteristic
227C4×C61C2×#13711yescharacteristic
228C4×C61#57512yes—
229Dic61—81no—
230Dic61—81no—
231C2×#554D6111yescharacteristic
232#3209—611no—
233#3104—611no—
234#3077—611no—
235#1644—614no—
236#1632—614no—
237#1548—611no—
238#1487—611no—
239#1478—611no—
240C2×C61—22no—
241C2×C61—44no—
242C2×C61C2×#45111yescharacteristic
243C2×C61#300511yescharacteristic
244C2×C61#301911yescharacteristic
245#554—22no—
246#554—22no—
247#554—22no—
248#554C2×D6114yes—
249#533—611no—
250C2×C4⋊Q8C2×D6111yescharacteristic
251C2×#125—21no—
252C2×C22⋊Q8—21no—
253C2×C22⋊Q8—22no—
254C2×C4×Q8C2×D6111yescharacteristic
255C2×C4×D4—21no—
256C2×C4×D4C2×D6111yescharacteristic
257C22×C4⋊C4—21no—
258#494—612no—
259#487—612no—
260#484—1221no—
261#478—1222no—
262#477—612no—
263#476—612no—
264#474—612no—
265#462—612no—
266#462—1222no—
267#451—1221no—
268#451—1222no—
269C4×Q16—1222no—
270#431—611no—
271#431—611no—
272#428—1221no—
273#423—611no—
274#418—1221no—
275#333—614no—
276Q8⋊C8—614no—
277D4⋊C8—614no—
278C61#1874911yescharacteristic, complemented, Sylow
279C22×Q8C22⋊D6111yescharacteristic
280C22×Q8C22⋊D6112yes—
281C22×D4C22⋊D6111yescharacteristic
282C23×C4—22no—
283C23×C4—21no—
284C2×Q16—1224no—
285C2×Q16—1222no—
286C2×M16—1221no—
287C4⋊Q8—21no—
288C4⋊Q8—21no—
289C4⋊Q8—611no—
290C4⋊Q8—611no—
291C4⋊Q8C22⋊D6112yes—
292C4⋊Q8C22×D6112yes—
293#125—24no—
294#125—24no—
295#125—1221no—
296C22⋊Q8—24no—
297C22⋊Q8—28no—
298C22⋊Q8—24no—
299C22⋊Q8—28no—
300C22⋊Q8—1221no—
301C22⋊Q8—1221no—
302C4×Q8—21no—
303C4×Q8—21no—
304C4×Q8—1221no—
305C4×Q8—1221no—
306C4×Q8C22⋊D6112yes—
307C4×Q8C22×D6112yes—
308C4×D4—24no—
309C4×D4—24no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4C22⋊D6112yes—
313C4×D4C22×D6112yes—
314#117—1221no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—24no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4C22⋊D6111yescharacteristic
323C2×C4⋊C4C22⋊D6111yescharacteristic
324C2×C22⋊C4—22no—
325C2×C22⋊C4—21no—
326C2×C42—21no—
327C2×C42C22×D6111yescharacteristic
328#107—1222no—
329#107—1222no—
330#106—1222no—
331C4⋊C8—612no—
332C4⋊C8—612no—
333C4⋊C8—1221no—
334C4⋊C8—1221no—
335#103—1222no—
336#103—1224no—
337#103—1222no—
338#102—1222no—
339#102—1222no—
340C22⋊C8—1221no—
341C22⋊C8—1221no—
342C4×C8—612no—
343C4×C8—1221no—
344C24—21no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—42no—
348C2×Q8—82no—
349C2×Q8—1221no—
350C2×Q8—1221no—
351C2×Q8C2×C22⋊D6112yes—
352C2×Q8C2×C22⋊D6114yes—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4#1175820214yescomplemented
356C2×D4C2×C22⋊D6112yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—28no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—44no—
365C22×C4—84no—
366C22×C4C2×C22⋊D6111yescharacteristic
367C22×C4C2×C22⋊D6111yescharacteristic
368C22×C4C2×C22⋊D6111yescharacteristic
369Q16—4884no—
370Q16—2444no—
371M16—2442no—
372C2×C8—1222no—
373C2×C8—1222no—
374C2×C8—1222no—
375C4⋊C4—28no—
376C4⋊C4—28no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—1221no—
386C4⋊C4—1221no—
387C4⋊C4—1221no—
388C4⋊C4—1221no—
389C4⋊C4—2441no—
390C4⋊C4—2441no—
391C4⋊C4C2×C22⋊D6112yes—
392C4⋊C4C2×C22⋊D6112yes—
393C22⋊C4—28no—
394C22⋊C4—22no—
395C22⋊C4—22no—
396C22⋊C4—2441no—
397C42—22no—
398C42—22no—
399C42—2441no—
400C42C2×C22⋊D6111yescharacteristic
401C42C2×C22⋊D6111yescharacteristic
402C42C2×C22⋊D6111yescharacteristic
403C42C23×D6111yescharacteristic
404C23—22no—
405C23—24no—
406C23—42no—
407C23—42no—
408C23#1451974611yescharacteristic
409Q8—22no—
410Q8—22no—
411Q8—44no—
412Q8—42no—
413Q8—82no—
414Q8—2442no—
415Q8—2442no—
416D4—22no—
417D4—24no—
418D4C2×#1175820214yes—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—24no—
424C2×C4—41no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—88no—
432C2×C4—2441no—
433C2×C4—2441no—
434C2×C4#1451974611yescharacteristic
435C2×C4#1451974611yescharacteristic
436C2×C4#1451974611yescharacteristic
437C2×C4C2×#1175820212yes—
438C2×C4#1451980812yes—
439C2×C4C22×C22⋊D6111yescharacteristic
440C2×C4C22×C22⋊D6111yescharacteristic
441C2×C4C22×C22⋊D6111yescharacteristic
442C8—2442no—
443C8—2442no—
444C8—2442no—
445C22—21no—
446C22—21no—
447C22—24no—
448C22—48no—
449C22—42no—
450C22—42no—
451C22#1683991812yes—
452C22C2×#1451974611yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—42no—
457C4—44no—
458C4—42no—
459C4—42no—
460C4—4881no—
461C4—4881no—
462C4#1683991812yes—
463C4C22×#1175820211yescharacteristic
464C4C2×#1451980811yescharacteristic
465C4#1684056012yes—
466C2—22no—
467C2—44no—
468C2C2×#1683991811yescharacteristic
469C2#2145506111yescharacteristic
470C2#2145517611yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#554C2
socleC22×C61C2×#120
prime radicalC22×D4×C61D4
hypercenterC2×#554D61
hypocenterC61#18749
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #21456767C2semidirect
2. #21455858C2semidirect
4. #21445033C2semidirect
5. #21444953C2semidirect
6. #21444500C2semidirect
7. #21444248C2semidirect
11. #16840141C22semidirect
12. #16840112C22semidirect
15. #16840057C22semidirect
18. #16839972C22semidirect
28. #16839511C22semidirect
29. Q8⋊#92418C22semidirect
117. C2×D4×C61Q16semidirect
278. C61#18749semidirect
355. C2×D4#11758202semidirect

Maximal subgroups

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

subgroupindex
#214567672
#214558582
#214556162
#214450332
#214449532
#214445002
#214442482
#214442102
C2×C61×#5542
#1874961