Finite Group Databasebeta

← #29595168

Subgroups — #29595168

Order 15616.

470 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454514C211yesmaximal, characteristic
2#21447460C211yesmaximal, characteristic
3#21445622C211yesmaximal, characteristic, complemented
4#21445621C211yesmaximal, characteristic, complemented
5#21443941C211yesmaximal, characteristic, complemented
6#21443939C211yesmaximal, characteristic, complemented
7C61×#2120C211yesmaximal, characteristic
8C2×#14519784C2211yescharacteristic
9C23×#92418C411yescharacteristic, complemented
10#16839834—21no—
11#16839834—21no—
12D4×#92418—21no—
13D4×#92418—21no—
14#16839789C411yescharacteristic, complemented
15#16839746C2211yescharacteristic
16#16839736C2211yescharacteristic
17#16839729C2211yescharacteristic
18#16839726C411yescharacteristic, complemented
19#16839720—21no—
20#16839718—21no—
21#16839718C411yescharacteristic, complemented
22#16839552C412yescomplemented
23#16839549C412yescomplemented
24#16839534—21no—
25#16839532—21no—
26C61×#525C2211yescharacteristic
27C61×#413C2211yescharacteristic
28C61×#410C2211yescharacteristic
29C61×#352C412yescomplemented
30C61×AΣL(1,𝔽16)C412yescomplemented
31#14519784—41no—
32#14519784—41no—
33C2×#11758194—21no—
34C2×#11758194C2×C411yescharacteristic
35C2×#11758194C2×C411yescharacteristic
36C2×#11758194D411yescharacteristic
37C2×#11758194D411yescharacteristic
38C22×#92418—21no—
39C22×#92418—21no—
40C22×#92418—41no—
41C22×#92418—41no—
42C22×#92418C2×C411yescharacteristic
43C22×#92418C2×C411yescharacteristic
44C22×#92418D411yescharacteristic
45C22×#92418D411yescharacteristic
46#14519736—21no—
47C4⋊#92418—21no—
48C4⋊#92418—21no—
49#14519696—21no—
50#14519694C2×C412yes—
51#14519692—21no—
52#14519692C2×C412yes—
53C22⋊#92418—21no—
54C22⋊#92418—21no—
55C22⋊#92418—21no—
56C22⋊#92418—21no—
57#14519688—21no—
58#14519688—41no—
59#14519687—41no—
60C4×#92418—21no—
61C4×#92418—41no—
62C8×Dic61—41no—
63C2×#41×C61C2311yescharacteristic
64C23×C4×C61C2×C411yescharacteristic
65C2×M16×C61—21no—
66#123×C61C2×C411yescharacteristic
67#123×C61C2×C411yescharacteristic
68C22⋊Q8×C61—21no—
69#121×C61—21no—
70#120×C61C2×C411yescharacteristic
71#120×C61C2×C411yescharacteristic
72C4×D4×C61—21no—
73C4×D4×C61—21no—
74#117×C61—21no—
75#117×C61C2×C411yescharacteristic
76#101×C61—21no—
77#100×C61C2×C412yes—
78#99×C61—21no—
79#99×C61C2×C412yes—
80C24×C61C4211yescharacteristic
81#41×C61—41no—
82#41×C61—41no—
83C2×Q8×C61#4111yescharacteristic
84C2×D4×C61—21no—
85C2×D4×C61—21no—
86C2×D4×C61C22×C411yescharacteristic
87C2×D4×C61C22×C411yescharacteristic
88C2×D4×C61#4111yescharacteristic
89C22×C4×C61—21no—
90C22×C4×C61—21no—
91C22×C4×C61—41no—
92C22×C4×C61—41no—
93C22×C4×C61C4211yescharacteristic
94C22×C4×C61C22×C411yescharacteristic
95C22×C4×C61C2×D411yescharacteristic
96C22×C4×C61C2×D411yescharacteristic
97M16×C61—22no—
98M16×C61—41no—
99C2×C8×C61—41no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—41no—
104C22⋊C4×C61—21no—
105C22⋊C4×C61—21no—
106C22⋊C4×C61—21no—
107C22⋊C4×C61—21no—
108C22⋊C4×C61—22no—
109C22⋊C4×C61C4211yescharacteristic
110C22⋊C4×C61C4211yescharacteristic
111C42×C61—21no—
112C42×C61—41no—
113C22⋊Dic61—22no—
114C4⋊Dic61—41no—
115C4×Dic61—41no—
116#11758194—22no—
117#11758194—41no—
118#11758194—41no—
119#11758194—41no—
120#11758194—41no—
121#11758194—41no—
122C2×#92418—21no—
123C2×#92418—21no—
124C2×#92418—21no—
125C2×#92418—21no—
126C2×#92418—21no—
127C2×#92418—21no—
128C2×#92418—21no—
129C2×#92418—21no—
130C2×#92418—42no—
131C2×#92418—41no—
132C2×#92418—41no—
133C2×#92418—41no—
134C2×#92418—41no—
135C2×#92418C22⋊C411yescharacteristic
136C2×#92418C22⋊C411yescharacteristic
137C2×#92418C22⋊C411yescharacteristic
138C2×#92418C22⋊C411yescharacteristic
139C23×C61—21no—
140C23×C61—41no—
141C23×C61—42no—
142C23×C61C2×C4211yescharacteristic
143C23×C61C4×D411yescharacteristic
144C23×C61C4×D411yescharacteristic
145Q8×C61—41no—
146D4×C61—41no—
147D4×C61—41no—
148D4×C61—41no—
149D4×C61—42no—
150D4×C61—42no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—21no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—21no—
160C2×C4×C61—42no—
161C2×C4×C61—41no—
162C2×C4×C61—41no—
163C2×C4×C61—41no—
164C2×C4×C61—42no—
165C2×C4×C61C2×C22⋊C411yescharacteristic
166C2×C4×C61#11711yescharacteristic
167C2×C4×C61C4×D411yescharacteristic
168C2×C4×C61C4×D411yescharacteristic
169C2×Dic61—42no—
170C8×C61—42no—
171#92418—21no—
172#92418—21no—
173#92418—21no—
174#92418—42no—
175#92418—41no—
176#92418—41no—
177#92418#9911yescharacteristic
178#92418#9911yescharacteristic
179#12623—611nomaximal, Sylow
180C22×C61—22no—
181C22×C61—21no—
182C22×C61—21no—
183C22×C61—21no—
184C22×C61—41no—
185C22×C61—41no—
186C22×C61—41no—
187C22×C61—41no—
188C22×C61—42no—
189C22×C61—42no—
190C22×C61—41no—
191C22×C61C4×C22⋊C411yescharacteristic
192C4×C61—21no—
193C4×C61—21no—
194C4×C61—21no—
195C4×C61—41no—
196C4×C61—41no—
197C4×C61—41no—
198C4×C61—82no—
199C4×C61C2×#9911yescharacteristic
200C4×C61#41011yescharacteristic
201Dic61—82no—
202#2929—611no—
203#2120—611no—
204#2120D6111yescharacteristic
205#1757—611no—
206#1757—611no—
207#1392—611no—
208#1391—611no—
209C2×C61—21no—
210C2×C61—41no—
211C2×C61—41no—
212C2×C61—42no—
213C2×C61—41no—
214C2×C61C4×#9911yescharacteristic
215C2×#131—611no—
216C23×C8—611no—
217#525C2×D6111yescharacteristic
218#435—1221no—
219#435—1221no—
220C8×D4—1221no—
221C8×D4—1221no—
222#424—611no—
223#413—611no—
224#413C2×D6111yescharacteristic
225#410—611no—
226#410C2×D6111yescharacteristic
227#408—611no—
228#407—611no—
229#405—611no—
230#405—1221no—
231#405—1221no—
232#352—612no—
233#352Dic6112yescomplemented
234AΣL(1,𝔽16)—612no—
235AΣL(1,𝔽16)Dic6112yescomplemented
236#344—1221no—
237#344—1221no—
238C61#1262311yescharacteristic, complemented, Sylow
239C2×#41C22×D6111yescharacteristic
240C23×C4C4×D6111yescharacteristic
241#131—2441no—
242#131—2441no—
243C2×M16—21no—
244C2×M16—611no—
245C2×M16—611no—
246C2×M16—611no—
247C2×M16—611no—
248C2×M16—1221no—
249C22×C8—611no—
250C22×C8—611no—
251C22×C8—611no—
252C22×C8—611no—
253C22×C8—1221no—
254C22×C8—1221no—
255C22×C8—2441no—
256C22×C8—2441no—
257#123C4×D6111yescharacteristic
258#123C2×Dic6111yescharacteristic
259C22⋊Q8—21no—
260#121—21no—
261#120C4×D6111yescharacteristic
262#120C2×Dic6111yescharacteristic
263C4×D4—21no—
264C4×D4—21no—
265#117—21no—
266#117—1221no—
267#117C4×D6111yescharacteristic
268C4⋊C8—1221no—
269C4⋊C8—1221no—
270#101—21no—
271#101—1221no—
272#100—612no—
273#100C2×Dic6112yes—
274#99—21no—
275#99—612no—
276#99—1221no—
277#99C2×Dic6112yes—
278C22⋊C8—1221no—
279C22⋊C8—1221no—
280C22⋊C8—1221no—
281C22⋊C8—1221no—
282#97—1221no—
283#97—2441no—
284#97—2441no—
285C4×C8—1221no—
286C4×C8—2441no—
287C4×C8—2441no—
288C24C4×Dic6111yescharacteristic
289#41—41no—
290#41—41no—
291C2×Q8#1175822411yescharacteristic
292C2×D4—21no—
293C2×D4—21no—
294C2×D4C2×C4×D6111yescharacteristic
295C2×D4#1175822411yescharacteristic
296C2×D4C22×Dic6111yescharacteristic
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4C4×Dic6111yescharacteristic
302C22×C4C2×C4×D6111yescharacteristic
303C22×C4D4×D6111yescharacteristic
304C22×C4D4×D6111yescharacteristic
305M16—22no—
306M16—41no—
307M16—1222no—
308M16—2441no—
309M16—2441no—
310M16—2441no—
311M16—2441no—
312M16—2441no—
313C2×C8—41no—
314C2×C8—611no—
315C2×C8—611no—
316C2×C8—611no—
317C2×C8—611no—
318C2×C8—1221no—
319C2×C8—1221no—
320C2×C8—1221no—
321C2×C8—1221no—
322C2×C8—1221no—
323C2×C8—1221no—
324C2×C8—1221no—
325C2×C8—1221no—
326C2×C8—2441no—
327C2×C8—2441no—
328C2×C8—2442no—
329C2×C8—2441no—
330C2×C8—2441no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—41no—
335C4⋊C4—2441no—
336C22⋊C4—21no—
337C22⋊C4—21no—
338C22⋊C4—21no—
339C22⋊C4—21no—
340C22⋊C4—22no—
341C22⋊C4—1222no—
342C22⋊C4C4×Dic6111yescharacteristic
343C22⋊C4C4×Dic6111yescharacteristic
344C42—21no—
345C42—41no—
346C42—2441no—
347C23—21no—
348C23—41no—
349C23—42no—
350C23C2×C4×Dic6111yescharacteristic
351C23#1451973711yescharacteristic
352C23D4×Dic6111yescharacteristic
353Q8—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—42no—
358D4—42no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—2442no—
374C2×C4C22⋊C4×D6111yescharacteristic
375C2×C4#1451973311yescharacteristic
376C2×C4#1451973711yescharacteristic
377C2×C4D4×Dic6111yescharacteristic
378C8—42no—
379C8—611no—
380C8—611no—
381C8—1221no—
382C8—1221no—
383C8—1221no—
384C8—2442no—
385C8—2441no—
386C8—2441no—
387C22—22no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22C22⋊C4×Dic6111yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—82no—
406C4—4882no—
407C4#99×D6111yescharacteristic
408C4#1683973411yescharacteristic
409C2—21no—
410C2—41no—
411C2—41no—
412C2—42no—
413C2—41no—
414C2#99×Dic6111yescharacteristic
subgroupG/H
centerC4#99×D61
commutatorC23×C61C2×C42
FrattiniC2×#41C22×D61
FittingC61×#2120C2
socleC2×C61C4×#99
prime radicalC61×#525C22
cosocleC2×#41×C61C23
hypercenter#2120D61
hypocenterC61#12623
LCSG ▷ C23×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2120

Complements

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

NH ≅ G/Ntype
3. #21445622C2semidirect
4. #21445621C2semidirect
5. #21443941C2semidirect
6. #21443939C2semidirect
9. C23×#92418C4semidirect
14. #16839789C4semidirect
18. #16839726C4semidirect
21. #16839718C4semidirect
22. #16839552C4semidirect
23. #16839549C4semidirect
29. C61×#352C4semidirect
30. C61×AΣL(1,𝔽16)C4semidirect
233. #352Dic61semidirect
235. AΣL(1,𝔽16)Dic61semidirect
238. C61#12623semidirect

Maximal subgroups

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

subgroupindex
#214545142
#214474602
#214456222
#214456212
#214439412
#214439392
C61×#21202
#1262361