Finite Group Databasebeta

← #30600778

Subgroups — #30600778

Order 15616.

454 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 86 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#21462314C211yesmaximal, characteristic
2#21462007C211yesmaximal, characteristic
3#21461333C211yesmaximal, characteristic
4#21461251C211yesmaximal, characteristic
5#21460673C211yesmaximal, characteristic
6#21460642C211yesmaximal, characteristic
7#21456763C211yesmaximal, characteristic
8#16840780C2211yescharacteristic
9#16840767C2211yescharacteristic
10(C2×#6923)⋊Q8—22no—
11#16840713—21no—
12#16840711—22no—
13#16840706C2211yescharacteristic
14C4⋊C4×#6923—21no—
15#16840698C2211yescharacteristic
16#16840632—21no—
17#16840632—21no—
18#16840622—22no—
19#16840615—22no—
20#16840317C2211yescharacteristic
21C4⋊Q8×D61C2211yescharacteristic
22#16840141C411yescharacteristic, complemented
23#16840113C411yescharacteristic, complemented
24#16840112C411yescharacteristic, complemented
25#16839970—22no—
26#16839916—22no—
27#16839813C2211yescharacteristic
28C61×#494C411yescharacteristic, complemented
29C2×#11758213—21no—
30C2×C4⋊#6923—21no—
31C2×C4⋊#6923—22no—
32C2×C4×#6923—22no—
33C2×C4×#6923D411yescharacteristic
34C2×C4×#6923D411yescharacteristic
35#14519844—21no—
36#14519844—21no—
37#14519844—21no—
38#14519834—21no—
39C4×#92420—21no—
40(C2×#6923)⋊C4—22no—
41(C2×#6923)⋊C4—22no—
42(C2×#6923)⋊C4—21no—
43(C2×#6923)⋊C4—21no—
44(C2×#6923)⋊C4—21no—
45(C2×#6923)⋊C4D412yes—
46C2×Q8×D61D412yes—
47C2×#11758202—22no—
48#14519777C2×C411yescharacteristic
49#14519777C2×C411yescharacteristic
50#14519776—21no—
51#14519776—21no—
52#14519775D412yes—
53#14519768—21no—
54#14519757—21no—
55#14519753—21no—
56Q8×Dic61—21no—
57#14519742—21no—
58#14519735—21no—
59C4⋊C4×D61—21no—
60C4⋊C4×D61—21no—
61C42×D61C2311yescharacteristic
62#14519706—21no—
63#14519702—22no—
64#14519701—22no—
65#14519689—21no—
66C4×#92418C2×C411yescharacteristic
67C2×Q16×C61—22no—
68C4⋊Q8×C61C2×C411yescharacteristic
69C4⋊Q8×C61C2×C411yescharacteristic
70C4×C8×C61C2×C411yescharacteristic
71C22×#6923—21no—
72C22×#6923—21no—
73C22×#6923—22no—
74C2×Q8×C61—21no—
75C2×Q8×C61C22⋊C412yes—
76Q8×D61—82no—
77C2×C4×D61—22no—
78C2×C4×D61C2×D411yescharacteristic
79C2×C4×D61C2×D411yescharacteristic
80C2×C4×D61C2×D411yescharacteristic
81C2×Dic122—22no—
82C2×Dic122—21no—
83C2×Dic122C22⋊C412yes—
84C2×#92420—41no—
85C2×#92420—41no—
86C4⋊#6923—41no—
87C4⋊#6923—41no—
88C4⋊#6923—82no—
89C4×#6923—21no—
90C4×#6923—21no—
91C4×#6923—21no—
92C4×#6923—82no—
93C4×#6923Q1612yescomplemented
94#11758213—41no—
95#11758213—41no—
96Q16×C61—82no—
97C2×C8×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C42×C61C22×C411yescharacteristic
103#11758202—82no—
104#11758198—41no—
105C4⋊Dic61—21no—
106C4⋊Dic61—21no—
107C4⋊Dic61—21no—
108C4⋊Dic61—21no—
109C4⋊Dic61—41no—
110#11758196—22no—
111#11758196—22no—
112#11758196—41no—
113C4×Dic61—41no—
114C4×Dic61C2×D411yescharacteristic
115C4×Dic61#4111yescharacteristic
116C4×Dic61#4111yescharacteristic
117C2×#92418—21no—
118C22×D61#12011yescharacteristic
119C2×#6923—22no—
120C2×#6923—41no—
121C2×#6923—41no—
122C2×#6923—42no—
123C2×#6923—41no—
124C2×#6923—41no—
125C2×#6923—42no—
126C2×#6923—41no—
127C2×#6923—42no—
128Q8×C61—42no—
129Q8×C61—42no—
130C2×C4×C61—22no—
131C2×C4×C61—41no—
132C2×C4×C61C2×C22⋊C411yescharacteristic
133C2×C4×C61C4×D411yescharacteristic
134C2×C4×C61C4×D411yescharacteristic
135C2×Dic61—22no—
136C2×Dic61—41no—
137C2×Dic61#12111yescharacteristic
138C2×Dic61#12111yescharacteristic
139C2×Dic61#12311yescharacteristic
140C4×D61—41no—
141C4×D61—41no—
142C4×D61—42no—
143C4×D61—42no—
144C4×D61C2×Q1611yescharacteristic
145C4×D61#13511yescharacteristic
146C4×D61#13711yescharacteristic
147C4×D61#13711yescharacteristic
148Dic122—42no—
149Dic122—82no—
150Dic122—81no—
151#92420—81no—
152#92420—81no—
153C8×C61—41no—
154#92418—41no—
155#9297—611nomaximal, Sylow
156C22×C61#38611yescharacteristic
157C2×D61—21no—
158C2×D61—21no—
159C2×D61#45111yescharacteristic
160C2×D61#45211yescharacteristic
161#6923—42no—
162#6923—82no—
163#6923—81no—
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—42no—
167C4×C61—81no—
168C4×C61C4×Q1611yescharacteristic
169C4×C61#44111yescharacteristic
170Dic61—21no—
171Dic61—21no—
172Dic61—42no—
173Dic61—81no—
174Dic61#46211yescharacteristic
175Dic61#46411yescharacteristic
176#3209—611no—
177#2331—611no—
178#1995—611no—
179#1548—611no—
180#1518—611no—
181#1286—611no—
182#1274—611no—
183C2×C61#187811yescharacteristic
184C2×C61#190511yescharacteristic
185C2×C61#191111yescharacteristic
186D61—21no—
187D61—21no—
188#533—611no—
189C2×C4⋊Q8—611no—
190#494—611no—
191#494#692311yescharacteristic, complemented
192#487—611no—
193#487—611no—
194#462—1222no—
195#451—1222no—
196#431—611no—
197#431—611no—
198#423—611no—
199#391—1222no—
200#389—1221no—
201#387—1222no—
202#382—611no—
203C4×C4⋊C4—1221no—
204#376—611no—
205#339—1221no—
206#339—1221no—
207#333—1222no—
208#328—1222no—
209C61#929711yescharacteristic, complemented, Sylow
210C22×Q8—612no—
211C2×Q16—22no—
212C2×Q16—1222no—
213C2×M16—1221no—
214C4⋊Q8—612no—
215C4⋊Q8—611no—
216C4⋊Q8—611no—
217C4⋊Q8—1221no—
218C4⋊Q8—1221no—
219C4⋊Q8C2×#692311yescharacteristic
220C4⋊Q8C2×#692311yescharacteristic
221#125—1221no—
222C22⋊Q8—1221no—
223C22⋊Q8—1221no—
224C4×Q8—1221no—
225C4×Q8—1221no—
226#117—1221no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4—1221no—
229C2×C4⋊C4—1222no—
230C2×C4⋊C4—1221no—
231C2×C42—611no—
232C2×C42—611no—
233C2×C42—611no—
234C2×C42—1222no—
235C4⋊C8—1221no—
236C4⋊C8—1221no—
237C4⋊C8—1221no—
238C4⋊C8—1221no—
239#103—1222no—
240#103—1222no—
241C22⋊C8—1221no—
242C22⋊C8—1221no—
243C4×C8—611no—
244C4×C8—1221no—
245C4×C8C2×#692311yescharacteristic
246#95—612no—
247#95—1221no—
248#95—1221no—
249#95—1221no—
250#95—1222no—
251#95—1222no—
252C2×Q8—21no—
253C2×Q8—612no—
254C2×Q8—1222no—
255C2×Q8—1221no—
256C2×Q8—4882no—
257C2×Q8C22⋊#692312yes—
258C22×C4—611no—
259C22×C4—611no—
260C22×C4—611no—
261C22×C4—1222no—
262C22×C4—1221no—
263C22×C4—1222no—
264C22×C4—1221no—
265Q16—82no—
266Q16—4882no—
267M16—2441no—
268M16—2441no—
269C2×C8—21no—
270C2×C8—1221no—
271C2×C8—2441no—
272C2×C8—2441no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—1222no—
278C4⋊C4—1222no—
279C4⋊C4—1221no—
280C4⋊C4—1221no—
281C4⋊C4—1221no—
282C4⋊C4—1221no—
283C4⋊C4—4882no—
284C4⋊C4—2441no—
285C4⋊C4—2441no—
286C4⋊C4—2441no—
287C4⋊C4—2441no—
288C22⋊C4—2441no—
289C42—611no—
290C42—611no—
291C42—611no—
292C42—612no—
293C42—1221no—
294C42—1221no—
295C42—1221no—
296C42—4882no—
297C42—2441no—
298C42C22×#692311yescharacteristic
299C23—611no—
300Q8—42no—
301Q8—42no—
302Q8—4881no—
303Q8—4882no—
304Q8—2442no—
305C2×C4—22no—
306C2×C4—41no—
307C2×C4—611no—
308C2×C4—611no—
309C2×C4—611no—
310C2×C4—611no—
311C2×C4—611no—
312C2×C4—611no—
313C2×C4—611no—
314C2×C4—1222no—
315C2×C4—1222no—
316C2×C4—2441no—
317C2×C4—2441no—
318C2×C4—2442no—
319C2×C4—2442no—
320C2×C4—2441no—
321C2×C4—2442no—
322C2×C4—2442no—
323C2×C4—2441no—
324C2×C4—2442no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4C2×C22⋊#692311yescharacteristic
330C2×C4D4×#692311yescharacteristic
331C2×C4D4×#692311yescharacteristic
332C8—41no—
333C8—4881no—
334C8—4881no—
335C8—2441no—
336C22—611no—
337C22—611no—
338C22—1221no—
339C22—1221no—
340C22#1684071011yescharacteristic
341C4—21no—
342C4—21no—
343C4—42no—
344C4—81no—
345C4—611no—
346C4—611no—
347C4—1221no—
348C4—1221no—
349C4—4881no—
350C4—4882no—
351C4—4881no—
352C4—2442no—
353C4—2442no—
354C4Q16×#692311yescharacteristic
355C4#1684080211yescharacteristic
356C2—1221no—
357C2—1221no—
358C2#2146181311yescharacteristic
359C2#2146185411yescharacteristic
360C2#2146186511yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#494C4
socleC22×C61#386
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#494#6923
hypocenterC61#9297
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
22. #16840141C4semidirect
23. #16840113C4semidirect
24. #16840112C4semidirect
28. C61×#494C4semidirect
93. C4×#6923Q16semidirect
191. #494#6923semidirect
209. C61#9297semidirect

Maximal subgroups

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

subgroupindex
#214623142
#214620072
#214613332
#214612512
#214606732
#214606422
#214567632
#929761