Finite Group Databasebeta

← #29576863

Subgroups — #29576863

Order 15616.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455628C211yesmaximal, characteristic
2#21454927C211yesmaximal, characteristic
3#21446743C211yesmaximal, characteristic, complemented
4#21445836C211yesmaximal, characteristic
5#21443721C211yesmaximal, characteristic, complemented
6#21443693C211yesmaximal, characteristic, complemented
7C61×#1549C211yesmaximal, characteristic, complemented
8C4×#11758222C2211yescharacteristic
9#16840054C412yescomplemented
10#16840049C412yescomplemented
11#16840044—21no—
12#16840039—21no—
13C8×Dic122—21no—
14C8×D244—21no—
15#16839825—22no—
16#16839814—21no—
17#16839802—21no—
18#16839786C2211yescharacteristic
19C2×C4⋊#92418C2211yescharacteristic
20C4⋊C8×D61—21no—
21C2×#14519703—21no—
22#16839761—21no—
23C2×#14519701—22no—
24C4×C4⋊Dic61C2211yescharacteristic
25#16839523—21no—
26C2×C4⋊Q8×C61C2211yescharacteristic
27C61×#431C2211yescharacteristic
28C61×#423C2211yescharacteristic
29C61×#332—21no—
30C61×#332—21no—
31C2×#11758222D411yescharacteristic
32C2×#11758194—21no—
33C22×#92418D412yes—
34C2×C8×D61—22no—
35C4×Dic122C2×C411yescharacteristic
36C4×Dic122D411yescharacteristic
37C4×D244C2×C411yescharacteristic
38C4×D244D411yescharacteristic
39C4×C22⋊D61—21no—
40C4×C22⋊D61—21no—
41#14519736D411yescharacteristic
42#14519734—21no—
43C2×C4⋊Dic61—21no—
44C2×C4×Dic61—21no—
45C42×D61—21no—
46#14519710—22no—
47#14519708—22no—
48C4⋊#92418—21no—
49C4⋊#92418—21no—
50C4⋊#92418—21no—
51C4⋊#92418C2×C412yes—
52#14519706—22no—
53#14519703—42no—
54#14519701—44no—
55#14519700—22no—
56#14519697—22no—
57#14519689—22no—
58C8×Dic61—22no—
59#14519684—21no—
60C22×Q8×C61—21no—
61C2×M16×C61—21no—
62C4⋊Q8×C61—21no—
63C4⋊Q8×C61—21no—
64C4⋊Q8×C61—21no—
65C4⋊Q8×C61C2×C412yes—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61—21no—
68C2×C42×C61C2311yescharacteristic
69C4⋊C8×C61—21no—
70C4⋊C8×C61—21no—
71C4⋊C8×C61D411yescharacteristic
72C4⋊C8×C61D411yescharacteristic
73C22⋊C8×C61—21no—
74C4×C8×C61—21no—
75C2×Q8×C61—22no—
76C2×Q8×C61—42no—
77C2×Q8×C61—42no—
78C2×Q8×C61—41no—
79C22×C4×C61—41no—
80C22×C4×C61C2×D411yescharacteristic
81C22×C4×C61C2×D411yescharacteristic
82C22×C4×C61#4111yescharacteristic
83C2×C22⋊D61—21no—
84C22×Dic61—41no—
85#11758222—21no—
86#11758222—21no—
87#11758222SD1612yes—
88#11758222Q1612yes—
89C2×D244C22⋊C411yescharacteristic
90C2×C4×D61—21no—
91C2×C4×D61—21no—
92C2×C4×D61—21no—
93C2×Dic122C22⋊C411yescharacteristic
94M16×C61—41no—
95M16×C61—41no—
96C2×C8×C61—21no—
97C2×C8×C61—21no—
98C2×C8×C61—41no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C42×C61C22×C411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61C2×D411yescharacteristic
106C42×C61#4111yescharacteristic
107C22⋊Dic61—21no—
108#11758198—21no—
109#11758198—21no—
110C4⋊Dic61—22no—
111C4⋊Dic61—41no—
112C4⋊Dic61C22⋊C411yescharacteristic
113C4⋊Dic61C22⋊C411yescharacteristic
114#11758196—21no—
115#11758196—21no—
116C4×Dic61—21no—
117C4×Dic61—21no—
118C4×Dic61—21no—
119C4×Dic61—41no—
120C4×Dic61—41no—
121#11758194—41no—
122#11758194—41no—
123C2×#92418—22no—
124C2×#92418—22no—
125C2×#92418—42no—
126C8×D61—82no—
127C23×C61#12111yescharacteristic
128C22×D61—21no—
129Q8×C61—44no—
130Q8×C61—81no—
131Q8×C61—81no—
132C2×C4×C61—21no—
133C2×C4×C61—21no—
134C2×C4×C61—41no—
135C2×C4×C61—42no—
136C2×C4×C61—82no—
137C2×C4×C61C2×C22⋊C411yescharacteristic
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×D411yescharacteristic
140C2×C4×C61#12011yescharacteristic
141C2×C4×C61#12111yescharacteristic
142C2×C4×C61#12311yescharacteristic
143C2×C4×C61C2×SD1611yescharacteristic
144C2×C4×C61C2×Q1611yescharacteristic
145C2×C4×C61#13511yescharacteristic
146C2×C4×C61#13511yescharacteristic
147C22⋊D61—22no—
148C22⋊D61—22no—
149C2×Dic61—21no—
150C2×Dic61—21no—
151C2×Dic61—21no—
152C2×Dic61—42no—
153C2×Dic61—81no—
154C2×Dic61—81no—
155D244—21no—
156D244#10312yescomplemented
157C4×D61—22no—
158C4×D61—22no—
159C4×D61—41no—
160C4×D61—41no—
161C4×D61—81no—
162Dic122—21no—
163Dic122#10312yescomplemented
164C8×C61—41no—
165C8×C61—41no—
166C8×C61—81no—
167#92418—42no—
168#92418—42no—
169#9317—611nomaximal, Sylow
170C22×C61—21no—
171C22×C61—21no—
172C22×C61#38611yescharacteristic
173C22×C61#46511yescharacteristic
174C22×C61#46711yescharacteristic
175C2×D61—22no—
176C2×D61—41no—
177C2×D61—41no—
178C4×C61—21no—
179C4×C61—21no—
180C4×C61—21no—
181C4×C61—82no—
182C4×C61C2×#10311yescharacteristic
183C4×C61#41611yescharacteristic
184C4×C61#45311yescharacteristic
185C4×C61#45411yescharacteristic
186C4×C61#46511yescharacteristic
187C4×C61#46711yescharacteristic
188Dic61—22no—
189Dic61—41no—
190Dic61—41no—
191Dic61—82no—
192#3079—611no—
193#2988—611no—
194#1981—611no—
195#1806—611no—
196#1549D6111yescharacteristic, complemented
197#1303—611no—
198#1292—611no—
199C2×C61—21no—
200C2×C61—21no—
201C2×C61#188311yescharacteristic
202C2×C61#189211yescharacteristic
203C2×C61#190211yescharacteristic
204D61—42no—
205C2×C4⋊Q8C2×D6111yescharacteristic
206C4×#41—611no—
207#477—612no—
208#476—612no—
209#475—1221no—
210#474—1221no—
211C8×Q8—1221no—
212C8×D4—1221no—
213C8×D4—1222no—
214#431—1221no—
215#431C2×D6111yescharacteristic
216#428—1221no—
217#423—611no—
218#423C2×D6111yescharacteristic
219C2×C4⋊C8—611no—
220C2×C4⋊C8—1221no—
221C2×#104—1221no—
222#417—1221no—
223C2×#103—1222no—
224C4×C4⋊C4—611no—
225#339—1221no—
226#332—21no—
227#332—21no—
228C61#931711yescharacteristic, complemented, Sylow
229C2×#41—611no—
230C22×Q8—21no—
231C2×M16—21no—
232C2×M16—1221no—
233C22×C8—612no—
234C22×C8—1222no—
235C4⋊Q8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8C4×D6112yes—
239C4×Q8—611no—
240C4×Q8—611no—
241C4×D4—611no—
242C4×D4—611no—
243C4×D4—1221no—
244C4×D4—1221no—
245#117—611no—
246#117—1221no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—1221no—
250C2×C42—1221no—
251C2×C42—1221no—
252C2×C42C22×D6111yescharacteristic
253#107—1222no—
254#106—1222no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—612no—
258C4⋊C8—1222no—
259C4⋊C8—1221no—
260C4⋊C8—1221no—
261C4⋊C8—1221no—
262C4⋊C8C22⋊D6111yescharacteristic
263C4⋊C8C22⋊D6111yescharacteristic
264#104—2442no—
265#103—1222no—
266#103—2444no—
267#102—1222no—
268C22⋊C8—21no—
269C22⋊C8—1222no—
270C4×C8—21no—
271C4×C8—1222no—
272#95—1221no—
273#41—612no—
274#41—612no—
275#41—1221no—
276#41—1221no—
277C2×Q8—22no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—41no—
281C2×Q8—611no—
282C2×D4—611no—
283C2×D4—1221no—
284C22×C4—41no—
285C22×C4—1221no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—2441no—
289C22×C4#1175822211yescharacteristic
290C22×C4D4×D6111yescharacteristic
291C22×C4D4×D6111yescharacteristic
292M16—41no—
293M16—41no—
294M16—2441no—
295M16—2441no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—41no—
299C2×C8—1222no—
300C2×C8—1222no—
301C2×C8—4882no—
302C2×C8—2442no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—611no—
308C4⋊C4—611no—
309C4⋊C4—1222no—
310C4⋊C4—1221no—
311C4⋊C4—1221no—
312C4⋊C4—2441no—
313C22⋊C4—1221no—
314C22⋊C4—1221no—
315C22⋊C4—1221no—
316C42—1221no—
317C42—1221no—
318C42—1221no—
319C42—2441no—
320C42—2441no—
321C42C2×C4×D6111yescharacteristic
322C42D4×D6111yescharacteristic
323C42#1175822411yescharacteristic
324C42C2×C22⋊D6111yescharacteristic
325C23—1221no—
326C23#1451975011yescharacteristic
327Q8—44no—
328Q8—81no—
329Q8—81no—
330Q8—612no—
331Q8—1221no—
332D4—612no—
333D4—1222no—
334D4—1222no—
335D4—1221no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—82no—
341C2×C4—1221no—
342C2×C4—1222no—
343C2×C4—1222no—
344C2×C4—1221no—
345C2×C4—1221no—
346C2×C4—4881no—
347C2×C4—4881no—
348C2×C4—4881no—
349C2×C4—2442no—
350C2×C4—2441no—
351C2×C4—2441no—
352C2×C4C22⋊C4×D6111yescharacteristic
353C2×C4#1451973711yescharacteristic
354C2×C4C4×C22⋊D6111yescharacteristic
355C2×C4#1451974511yescharacteristic
356C2×C4#1451974811yescharacteristic
357C2×C4#1451975911yescharacteristic
358C2×C4SD16×D6111yescharacteristic
359C2×C4Q16×D6111yescharacteristic
360C2×C4#1451979811yescharacteristic
361C2×C4#1451980011yescharacteristic
362C8—41no—
363C8—41no—
364C8—81no—
365C8—2442no—
366C8—2442no—
367C22—21no—
368C22—21no—
369C22—1222no—
370C22—2441no—
371C22—2441no—
372C22#1683963811yescharacteristic
373C22#1683999411yescharacteristic
374C22#1684000211yescharacteristic
375C4—21no—
376C4—21no—
377C4—21no—
378C4—82no—
379C4—1222no—
380C4—4882no—
381C4—2441no—
382C4—2441no—
383C4#103×D6111yescharacteristic
384C4#1683975711yescharacteristic
385C4#1683992811yescharacteristic
386C4#1683993311yescharacteristic
387C4#1683998811yescharacteristic
388C4#1684000011yescharacteristic
389C2—21no—
390C2—21no—
391C2—2442no—
392C2#2144619211yescharacteristic
393C2#2144623911yescharacteristic
394C2#2144629511yescharacteristic
subgroupG/H
centerC22#16839638
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1549C2
socleC22×C61#386
prime radicalC2×#11758222D4
cosocleC2×C42×C61C23
hypercenter#1549D61
hypocenterC61#9317
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
3. #21446743C2semidirect
5. #21443721C2semidirect
6. #21443693C2semidirect
7. C61×#1549C2semidirect
9. #16840054C4semidirect
10. #16840049C4semidirect
156. D244#103semidirect
163. Dic122#103semidirect
196. #1549D61semidirect
228. C61#9317semidirect

Maximal subgroups

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

subgroupindex
#214556282
#214549272
#214467432
#214458362
#214437212
#214436932
C61×#15492
#931761