Finite Group Databasebeta

← #29595853

Subgroups — #29595853

Order 15616.

760 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21447504C211yesmaximal, characteristic, complemented
2#21447460C212yesmaximal, complemented
3#21446238C212yesmaximal, complemented
4#21446203C211yesmaximal, characteristic, complemented
5C61×#3529C211yesmaximal, characteristic
6#16840529C2211yescharacteristic
7#16839779—21no—
8#16839746C2212yes—
9#16839736C2211yescharacteristic
10#16839726—22no—
11#16839561—21no—
12#16839559—21no—
13#16839552—22no—
14#16839549—22no—
15#16839532—22no—
16#16839522—21no—
17C61×#563—21no—
18C61×#563C411yescharacteristic, complemented
19C61×#562—21no—
20C61×#562—22no—
21C61×H(3,𝔽4)C411yescharacteristic, complemented
22C61×#560—22no—
23C61×#560C412yescomplemented
24C61×#532—21no—
25C61×#532C2211yescharacteristic
26C61×#529—22no—
27C61×#529—21no—
28C61×#525—22no—
29C61×#525C2212yes—
30#14519784—42no—
31C2×#11758194—22no—
32C2×#11758194—21no—
33C2×#11758194—22no—
34#14519736—21no—
35#14519704—42no—
36#14519696—22no—
37#14519694—22no—
38#14519692—22no—
39C22⋊#92418—44no—
40C2×#41×C61—21no—
41C2×#41×C61D411yescharacteristic
42C2×#41×C61C2311yescharacteristic
43C23×C4×C61D412yes—
44C4⋊Q8×C61—21no—
45C4⋊Q8×C61D411yescharacteristic
46#127×C61—21no—
47#127×C61C2×C411yescharacteristic
48#126×C61—22no—
49#126×C61—22no—
50#126×C61—22no—
51#125×C61—22no—
52#125×C61—21no—
53#124×C61—21no—
54#124×C61—21no—
55#124×C61—22no—
56#124×C61—21no—
57#124×C61C2×C411yescharacteristic
58#124×C61D411yescharacteristic
59#123×C61—22no—
60#123×C61—22no—
61#123×C61—22no—
62#123×C61—22no—
63#123×C61—22no—
64#123×C61C2×C412yes—
65#123×C61D412yes—
66C22⋊Q8×C61—22no—
67C22⋊Q8×C61—21no—
68C22⋊Q8×C61—22no—
69C22⋊Q8×C61—22no—
70C22⋊Q8×C61—22no—
71#121×C61—22no—
72#121×C61—21no—
73#121×C61—22no—
74#121×C61—22no—
75#121×C61—22no—
76#121×C61—22no—
77#121×C61—22no—
78#121×C61—22no—
79#120×C61—22no—
80#120×C61—22no—
81#120×C61C2×C412yes—
82#120×C61D412yes—
83C4×Q8×C61—21no—
84C4×D4×C61—22no—
85C4×D4×C61—21no—
86C4×D4×C61—22no—
87C4×D4×C61—22no—
88C4×D4×C61—22no—
89#117×C61—22no—
90#117×C61D412yes—
91C2×C42×C61—21no—
92C2×C42×C61D411yescharacteristic
93C24×C61C22⋊C412yes—
94#41×C61—41no—
95#41×C61—41no—
96#41×C61—41no—
97#41×C61—41no—
98#41×C61—81no—
99#41×C61—81no—
100C2×Q8×C61—21no—
101C2×Q8×C61—21no—
102C2×Q8×C61C22⋊C411yescharacteristic
103C2×Q8×C61C2×D411yescharacteristic
104C2×D4×C61—21no—
105C2×D4×C61—22no—
106C2×D4×C61—21no—
107C2×D4×C61—22no—
108C2×D4×C61—22no—
109C2×D4×C61—22no—
110C2×D4×C61—22no—
111C2×D4×C61—21no—
112C2×D4×C61C22⋊C411yescharacteristic
113C2×D4×C61C22×C411yescharacteristic
114C2×D4×C61C2×D412yes—
115C22×C4×C61—22no—
116C22×C4×C61—21no—
117C22×C4×C61—21no—
118C22×C4×C61—22no—
119C22×C4×C61—22no—
120C22×C4×C61—42no—
121C22×C4×C61—42no—
122C22×C4×C61C22⋊C412yes—
123C22×C4×C61C2×D411yescharacteristic
124C22×C4×C61C2×D412yes—
125C4⋊C4×C61—22no—
126C4⋊C4×C61—22no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—22no—
132C4⋊C4×C61—22no—
133C4⋊C4×C61—22no—
134C4⋊C4×C61—22no—
135C22⋊C4×C61—22no—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—22no—
139C22⋊C4×C61—22no—
140C22⋊C4×C61—22no—
141C22⋊C4×C61—22no—
142C22⋊C4×C61—22no—
143C22⋊C4×C61—22no—
144C22⋊C4×C61—22no—
145C22⋊C4×C61—22no—
146C22⋊C4×C61C22⋊C412yes—
147C22⋊C4×C61C22⋊C412yes—
148C42×C61—21no—
149C42×C61—21no—
150C42×C61—22no—
151C42×C61—21no—
152C42×C61C22⋊C411yescharacteristic
153C42×C61C22⋊C411yescharacteristic
154C22⋊Dic61—41no—
155C4⋊Dic61—41no—
156C4×Dic61—41no—
157#11758194—22no—
158#11758194—42no—
159#11758194—41no—
160#11758194—42no—
161#11758194—42no—
162C2×#92418—24no—
163C2×#92418—41no—
164C2×#92418—42no—
165C23×C61—22no—
166C23×C61—21no—
167C23×C61—42no—
168C23×C61—82no—
169C23×C61C2×C22⋊C412yes—
170C23×C61#12011yescharacteristic
171Q8×C61—41no—
172Q8×C61—41no—
173Q8×C61—81no—
174Q8×C61—81no—
175D4×C61—41no—
176D4×C61—42no—
177D4×C61—41no—
178D4×C61—81no—
179D4×C61—82no—
180D4×C61—82no—
181D4×C61—82no—
182D4×C61—82no—
183D4×C61—81no—
184D4×C61—82no—
185D4×C61—81no—
186C2×C4×C61—22no—
187C2×C4×C61—21no—
188C2×C4×C61—22no—
189C2×C4×C61—21no—
190C2×C4×C61—22no—
191C2×C4×C61—21no—
192C2×C4×C61—22no—
193C2×C4×C61—21no—
194C2×C4×C61—21no—
195C2×C4×C61—22no—
196C2×C4×C61—42no—
197C2×C4×C61—41no—
198C2×C4×C61—42no—
199C2×C4×C61—41no—
200C2×C4×C61—42no—
201C2×C4×C61—82no—
202C2×C4×C61—81no—
203C2×C4×C61—82no—
204C2×C4×C61C2×C22⋊C411yescharacteristic
205C2×C4×C61#12012yes—
206C2×C4×C61#12011yescharacteristic
207C2×Dic61—81no—
208#92418—42no—
209#92418—82no—
210#12766—611nomaximal, Sylow
211C22×C61—21no—
212C22×C61—22no—
213C22×C61—41no—
214C22×C61—42no—
215C22×C61—42no—
216C22×C61—42no—
217C22×C61—42no—
218C22×C61—44no—
219C22×C61—42no—
220C22×C61—41no—
221C22×C61—82no—
222C22×C61#37911yescharacteristic
223C4×C61—22no—
224C4×C61—21no—
225C4×C61—42no—
226C4×C61—41no—
227C4×C61—42no—
228C4×C61—41no—
229C4×C61—42no—
230C4×C61—41no—
231C4×C61B(4,𝔽2)11yescharacteristic
232C4×C61#45811yescharacteristic
233Dic61—161no—
234#3529D6111yescharacteristic
235#2132—611no—
236#2120—612no—
237#1891—612no—
238#1884—611no—
239C2×C61—21no—
240C2×C61—41no—
241C2×C61—42no—
242C2×C61—81no—
243C2×C61—82no—
244C2×C61#179011yescharacteristic
245#568—611no—
246#563—21no—
247#563Dic6111yescharacteristic, complemented
248#562—21no—
249#562—22no—
250H(3,𝔽4)Dic6111yescharacteristic, complemented
251#560—22no—
252#560Dic6112yescomplemented
253#532—21no—
254#532C2×D6111yescharacteristic
255#529—22no—
256#529—21no—
257#525—22no—
258#525C2×D6112yes—
259#421—1221no—
260#413—612no—
261#410—611no—
262#407—1222no—
263#356—1221no—
264#355—1221no—
265#352—1222no—
266AΣL(1,𝔽16)—1222no—
267#344—1222no—
268#339—1221no—
269C61#1276611yescharacteristic, complemented, Sylow
270C2×#41—21no—
271C2×#41C22⋊D6111yescharacteristic
272C2×#41C22×D6111yescharacteristic
273C23×C4C22⋊D6112yes—
274#131—2442no—
275C2×M16—1222no—
276C2×M16—1221no—
277C2×M16—1222no—
278C4⋊Q8—21no—
279C4⋊Q8C22⋊D6111yescharacteristic
280#127—21no—
281#127C2×Dic6111yescharacteristic
282#126—22no—
283#126—22no—
284#126—22no—
285#125—22no—
286#125—21no—
287#124—21no—
288#124—21no—
289#124—22no—
290#124—21no—
291#124C2×Dic6111yescharacteristic
292#124C22⋊D6111yescharacteristic
293#123—22no—
294#123—22no—
295#123—22no—
296#123—22no—
297#123—22no—
298#123C2×Dic6112yes—
299#123C22⋊D6112yes—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302C22⋊Q8—21no—
303C22⋊Q8—22no—
304C22⋊Q8—22no—
305#121—22no—
306#121—21no—
307#121—22no—
308#121—22no—
309#121—22no—
310#121—22no—
311#121—22no—
312#121—22no—
313#120—22no—
314#120—22no—
315#120C2×Dic6112yes—
316#120C22⋊D6112yes—
317C4×Q8—21no—
318C4×D4—21no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323#117—22no—
324#117—1221no—
325#117C22⋊D6112yes—
326C2×C42—21no—
327C2×C42C22⋊D6111yescharacteristic
328#104—2442no—
329#101—1222no—
330#100—1222no—
331#99—1222no—
332C22⋊C8—2444no—
333C24C22⋊Dic6112yes—
334#41—41no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—81no—
339#41—81no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8C22⋊Dic6111yescharacteristic
343C2×Q8C2×C22⋊D6111yescharacteristic
344C2×D4—21no—
345C2×D4—22no—
346C2×D4—21no—
347C2×D4—22no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—21no—
352C2×D4C22⋊Dic6111yescharacteristic
353C2×D4C22×Dic6111yescharacteristic
354C2×D4C2×C22⋊D6112yes—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4C22⋊Dic6112yes—
363C22×C4C2×C22⋊D6111yescharacteristic
364C22×C4C2×C22⋊D6112yes—
365M16—1222no—
366M16—2442no—
367M16—2442no—
368M16—2442no—
369M16—2441no—
370C2×C8—1224no—
371C2×C8—2442no—
372C2×C8—2441no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—21no—
377C4⋊C4—22no—
378C4⋊C4—21no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—2441no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—22no—
395C22⋊C4—2441no—
396C22⋊C4C22⋊Dic6112yes—
397C22⋊C4C22⋊Dic6112yes—
398C42—21no—
399C42—21no—
400C42—22no—
401C42—21no—
402C42—2441no—
403C42C22⋊Dic6111yescharacteristic
404C42C22⋊Dic6111yescharacteristic
405C23—22no—
406C23—21no—
407C23—42no—
408C23—82no—
409C23C2×C22⋊Dic6112yes—
410C23#1451974611yescharacteristic
411Q8—41no—
412Q8—41no—
413Q8—81no—
414Q8—81no—
415D4—41no—
416D4—42no—
417D4—41no—
418D4—81no—
419D4—82no—
420D4—82no—
421D4—82no—
422D4—82no—
423D4—81no—
424D4—82no—
425D4—81no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—4881no—
445C2×C4C2×C22⋊Dic6111yescharacteristic
446C2×C4#1451974612yes—
447C2×C4#1451974611yescharacteristic
448C8—4882no—
449C8—2442no—
450C22—22no—
451C22—21no—
452C22—41no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—44no—
458C22—42no—
459C22—41no—
460C22—82no—
461C22#1683961811yescharacteristic
462C4—22no—
463C4—21no—
464C4—42no—
465C4—41no—
466C4—42no—
467C4—41no—
468C4—42no—
469C4—41no—
470C4—9761no—
471C4#1683994911yescharacteristic
472C4#1683995211yescharacteristic
473C2—21no—
474C2—41no—
475C2—42no—
476C2—82no—
477C2—81no—
478C2#2144576711yescharacteristic
subgroupG/H
centerC4#16839949
commutatorC2×D4×C61C22×C4
FrattiniC2×#41C22×D61
Fitting, prime radicalC61×#3529C2
socleC2×C61#1790
cosocleC2×#41×C61C23
hypercenter#3529D61
hypocenterC61#12766
LCSG ▷ C2×D4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #3529

Complements

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

NH ≅ G/Ntype
1. #21447504C2semidirect
2. #21447460C2semidirect
3. #21446238C2semidirect
4. #21446203C2semidirect
18. C61×#563C4semidirect
21. C61×H(3,𝔽4)C4semidirect
23. C61×#560C4semidirect
247. #563Dic61semidirect
250. H(3,𝔽4)Dic61semidirect
252. #560Dic61semidirect
269. C61#12766semidirect

Maximal subgroups

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

subgroupindex
#214475042
#214474602
#214462382
#214462032
C61×#35292
#1276661