Finite Group Databasebeta

← #30601837

Subgroups — #30601837

Order 15616.

422 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 90 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#21461995C211yesmaximal, characteristic
2#21461994C211yesmaximal, characteristic
3C4×#14519845C211yesmaximal, characteristic
4#21461203C211yesmaximal, characteristic
5#21461202C211yesmaximal, characteristic
6#21461029C211yesmaximal, characteristic
7#21456404C211yesmaximal, characteristic
8#16840781C2211yescharacteristic
9C2×#14519845—21no—
10C4×#11758213C2211yescharacteristic
11C4×C4⋊#6923C2211yescharacteristic
12C4×C4⋊#6923C2211yescharacteristic
13#16840636—21no—
14#16840632—22no—
15#16840632—22no—
16#16840616—22no—
17#16840615—22no—
18#92420⋊C8—21no—
19#16840603—21no—
20#16840317C2211yescharacteristic
21#16840317C2211yescharacteristic
22#16840165C411yescharacteristic, complemented
23#16840141C411yescharacteristic, complemented
24#16840113C411yescharacteristic, complemented
25#16840002—21no—
26#16840002—21no—
27#16839854—21no—
28#16839854—21no—
29C4×C8×D61C2211yescharacteristic
30C61×#487C411yescharacteristic, complemented
31C2×#11758213D412yes—
32C2×C4⋊#6923—21no—
33C2×C4⋊#6923—21no—
34C2×C4×#6923—21no—
35C2×C4×#6923—21no—
36#14519845—24no—
37C4⋊#92420—21no—
38#14519834D412yes—
39#14519832—21no—
40C4×#92420—21no—
41C4×#92420—21no—
42(C2×#6923)⋊C4—21no—
43(C2×#6923)⋊C4—21no—
44C2×#11758202—21no—
45C2×#11758202—21no—
46C2×C8×D61D411yescharacteristic
47C2×C8×D61D411yescharacteristic
48#14519777C2×C411yescharacteristic
49#14519777C2×C411yescharacteristic
50#14519768—21no—
51#14519768—21no—
52#14519757—21no—
53#14519757—21no—
54#14519753—21no—
55#14519753—21no—
56Q8×Dic61—21no—
57Q8×Dic61—21no—
58#14519742—21no—
59#14519742—21no—
60#14519735—21no—
61#14519735—21no—
62C42×D61C2311yescharacteristic
63#14519710—21no—
64#14519710—21no—
65#14519702—21no—
66#14519702—21no—
67C4×#92418C2×C411yescharacteristic
68C8×Dic61D411yescharacteristic
69C8×Dic61D411yescharacteristic
70C4⋊Q8×C61C2×C411yescharacteristic
71C4⋊Q8×C61C2×C411yescharacteristic
72#103×C61—21no—
73#103×C61—21no—
74C4×C8×C61C2×C411yescharacteristic
75C22×#6923—41no—
76C22×#6923—41no—
77C2×Q8×C61—21no—
78C2×Q8×C61—21no—
79C2×C4×D61C2×D411yescharacteristic
80C2×C4×D61C2×D411yescharacteristic
81C2×C4×D61C2×D411yescharacteristic
82C2×Dic122—21no—
83C2×Dic122—21no—
84C2×#92420—22no—
85C2×#92420—22no—
86C4⋊#6923—22no—
87C4⋊#6923—22no—
88C4⋊#6923—41no—
89C4⋊#6923—41no—
90C4×#6923—42no—
91C4×#6923—42no—
92#11758213—22no—
93#11758213—22no—
94C2×C8×C61C22⋊C411yescharacteristic
95C2×C8×C61C22⋊C411yescharacteristic
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C42×C61C22×C411yescharacteristic
101#11758202—42no—
102#11758202—42no—
103#11758198—41no—
104#11758198—41no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—41no—
110C4⋊Dic61—41no—
111#11758196—41no—
112#11758196—41no—
113C4×Dic61—41no—
114C4×Dic61—41no—
115C4×Dic61C2×D411yescharacteristic
116C4×Dic61#4111yescharacteristic
117C4×Dic61#4111yescharacteristic
118C2×#92418C22⋊C411yescharacteristic
119C2×#92418C22⋊C411yescharacteristic
120C8×D61—22no—
121C8×D61—41no—
122C22×D61#12711yescharacteristic
123C2×#6923—42no—
124C2×#6923—42no—
125C2×#6923—81no—
126C2×#6923—81no—
127C2×#6923—81no—
128C2×#6923—81no—
129Q8×C61—41no—
130Q8×C61—41no—
131Q8×C61—81no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61C2×C22⋊C411yescharacteristic
135C2×C4×C61C4×D411yescharacteristic
136C2×C4×C61C4×D411yescharacteristic
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61#12111yescharacteristic
140C2×Dic61#12111yescharacteristic
141C2×Dic61#12411yescharacteristic
142C4×D61—21no—
143C4×D61—21no—
144C4×D61—41no—
145C4×D61#13712yes—
146C4×D61#13712yes—
147Dic122—41no—
148Dic122—41no—
149Dic122—81no—
150#92420—42no—
151#92420—42no—
152C8×C61—21no—
153C8×C61—21no—
154#92418—21no—
155#92418#10412yes—
156#9720—611nomaximal, Sylow
157C22×C61#39011yescharacteristic
158C2×D61—21no—
159C2×D61—21no—
160C2×D61#49711yescharacteristic
161C2×D61#49711yescharacteristic
162#6923—82no—
163#6923—82no—
164C4×C61—21no—
165C4×C61—81no—
166C4×C61—81no—
167C4×C61C2×#10411yescharacteristic
168C4×C61#42111yescharacteristic
169C4×C61#44112yes—
170Dic61—21no—
171Dic61—21no—
172Dic61—81no—
173Dic61—81no—
174Dic61#47012yes—
175#3170—611no—
176#1986—611no—
177#1986—611no—
178C4×#106—611no—
179#1503—611no—
180#1503—611no—
181#1451—611no—
182C2×C61#197111yescharacteristic
183C2×C61#197611yescharacteristic
184C2×C61#198011yescharacteristic
185D61—21no—
186D61—21no—
187#533—611no—
188#533—611no—
189#500—611no—
190#494—611no—
191#487—611no—
192#487#692311yescharacteristic, complemented
193#467—1221no—
194#467—1221no—
195C4×Q16—1221no—
196C4×Q16—1221no—
197#431—611no—
198C2×#106—1221no—
199C4×M16—611no—
200C2×C4×C8—611no—
201C4×C4⋊C4—611no—
202C4×C4⋊C4—611no—
203#340—1221no—
204#339—1222no—
205#339—1222no—
206#328—1222no—
207#328—1222no—
208#322—1221no—
209#322—1221no—
210C61#972011yescharacteristic, complemented, Sylow
211C2×Q16—1221no—
212C2×Q16—1221no—
213C2×M16—612no—
214C22×C8—611no—
215C22×C8—611no—
216C4⋊Q8—611no—
217C4⋊Q8—611no—
218C4⋊Q8C2×#692311yescharacteristic
219C4⋊Q8C2×#692311yescharacteristic
220#125—1221no—
221#125—1221no—
222C22⋊Q8—1221no—
223C22⋊Q8—1221no—
224C22⋊Q8—1221no—
225C22⋊Q8—1221no—
226C4×Q8—1221no—
227C4×Q8—1221no—
228C4×Q8—1221no—
229C4×Q8—1221no—
230#117—1221no—
231#117—1221no—
232C2×C4⋊C4—1221no—
233C2×C4⋊C4—1221no—
234C2×C42—611no—
235C2×C42—1221no—
236C2×C42—1221no—
237#107—1221no—
238#107—1221no—
239#106—1224no—
240C4⋊C8—1221no—
241#103—21no—
242#103—21no—
243#103—1221no—
244#103—1221no—
245C22⋊C8—612no—
246#97—1221no—
247C4×C8—611no—
248C4×C8—611no—
249C4×C8—611no—
250C4×C8—1221no—
251C4×C8—1221no—
252C4×C8C2×#692311yescharacteristic
253#95—1221no—
254#95—1221no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—1221no—
258C2×Q8—1221no—
259C22×C4—611no—
260C22×C4—611no—
261C22×C4—611no—
262C22×C4—2441no—
263C22×C4—2441no—
264Q16—2442no—
265Q16—2442no—
266M16—1222no—
267M16—1222no—
268C2×C8—611no—
269C2×C8—611no—
270C2×C8—1222no—
271C2×C8—1222no—
272C2×C8—1222no—
273C2×C8—2441no—
274C2×C8C22⋊#692311yescharacteristic
275C2×C8C22⋊#692311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—1222no—
281C4⋊C4—1222no—
282C4⋊C4—1221no—
283C4⋊C4—1221no—
284C4⋊C4—1221no—
285C4⋊C4—1221no—
286C4⋊C4—2441no—
287C4⋊C4—2441no—
288C4⋊C4—2441no—
289C4⋊C4—2441no—
290C4⋊C4—2441no—
291C4⋊C4—2441no—
292C22⋊C4—2441no—
293C22⋊C4—2441no—
294C42—611no—
295C42—611no—
296C42—611no—
297C42—2441no—
298C42—2441no—
299C42—2442no—
300C42—2442no—
301C42C22×#692311yescharacteristic
302C23—611no—
303Q8—41no—
304Q8—41no—
305Q8—81no—
306Q8—4881no—
307Q8—2441no—
308Q8—2441no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—611no—
312C2×C4—612no—
313C2×C4—612no—
314C2×C4—611no—
315C2×C4—611no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—4881no—
319C2×C4—4881no—
320C2×C4—4881no—
321C2×C4—4881no—
322C2×C4—2441no—
323C2×C4—2442no—
324C2×C4—2442no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4C2×C22⋊#692311yescharacteristic
328C2×C4D4×#692311yescharacteristic
329C2×C4D4×#692311yescharacteristic
330C8—21no—
331C8—21no—
332C8—612no—
333C8—1221no—
334C8—2442no—
335C8—2442no—
336C22—611no—
337C22—611no—
338C22—1221no—
339C22—1221no—
340C22#1684071411yescharacteristic
341C4—21no—
342C4—81no—
343C4—81no—
344C4—612no—
345C4—1221no—
346C4—1221no—
347C4—4882no—
348C4—4882no—
349C4—4881no—
350C4—4881no—
351C4D61⋊#10411yescharacteristic
352C4#1684075911yescharacteristic
353C4#1684080212yes—
354C2—1221no—
355C2—1221no—
356C2#2146196711yescharacteristic
357C2#2146197711yescharacteristic
358C2#2146198411yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#487C4
socleC22×C61#390
prime radicalC22×D61#127
cosocleC42×D61C23
hypercenter#487#6923
hypocenterC61#9720
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
22. #16840165C4semidirect
23. #16840141C4semidirect
24. #16840113C4semidirect
30. C61×#487C4semidirect
192. #487#6923semidirect
210. C61#9720semidirect

Maximal subgroups

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

subgroupindex
#214619952
#214619942
C4×#145198452
#214612032
#214612022
#214610292
#214564042
#972061