Finite Group Databasebeta

← #30602027

Subgroups — #30602027

Order 15616.

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

normal subgroups only

This group has 484 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#21461698C211yesmaximal, characteristic
2#21461555C211yesmaximal, characteristic
3#21461219C212yesmaximal
4#21461099C212yesmaximal
5C2×#16840000C211yesmaximal, characteristic
6C2×#14519847—21no—
7#16840733C2212yes—
8D61⋊(C22⋊C8)C2211yescharacteristic
9C2×#14519831—21no—
10C4×C22⋊#6923C2211yescharacteristic
11#16840638—21no—
12(C2×#6923)⋊C8—21no—
13#16840616—24no—
14#16840608—24no—
15C22×C8×D61C2211yescharacteristic
16C2×#14519753C2212yes—
17#16840000—21no—
18#16840000—21no—
19#16840000—21no—
20#16840000—21no—
21C2×#14519711C411yescharacteristic, complemented
22C2×#14519701C412yescomplemented
23C22×Q16×C61C411yescharacteristic, complemented
24C2×#11758213—22no—
25C2×D61⋊C8—21no—
26C2×C22⋊#6923—21no—
27C2×C4×#6923—21no—
28#14519847—42no—
29#14519834—21no—
30#14519834—22no—
31C22⋊#92420—22no—
32C22⋊#92420D412yes—
33#14519831—42no—
34(C2×#6923)⋊C4D412yes—
35C22×C4×D61C2311yescharacteristic
36C22×#92418C2×C411yescharacteristic
37C2×C8×D61—42no—
38C2×C8×D61D411yescharacteristic
39C2×C8×D61D411yescharacteristic
40C2×C8×D61D411yescharacteristic
41C2×C8×D61D411yescharacteristic
42#14519753—22no—
43#14519753—22no—
44#14519753—22no—
45#14519753—22no—
46C2×C4⋊Dic61C2×C412yes—
47C2×#11758196—22no—
48C2×#11758198—22no—
49#14519711—21no—
50#14519711—21no—
51#14519701—22no—
52#14519701—22no—
53C22×Q8×C61C2×C412yes—
54C2×Q16×C61—21no—
55C2×Q16×C61—21no—
56C2×Q16×C61—42no—
57C22×C8×C61C2×C411yescharacteristic
58C23×D61#4111yescharacteristic
59C22×#6923—22no—
60C2×Q8×C61—22no—
61C2×Q8×C61—22no—
62C2×Q8×C61—42no—
63C2×Q8×C61—42no—
64C22×C4×C61—22no—
65C22×C4×C61C22×C411yescharacteristic
66C22×Dic61—22no—
67C22×Dic61C2×D411yescharacteristic
68C2×C4×D61—41no—
69C2×C4×D61—41no—
70C2×C4×D61C2×D411yescharacteristic
71C2×C4×D61C2×D411yescharacteristic
72C2×C4×D61C2×D411yescharacteristic
73C2×C4×D61#4111yescharacteristic
74C22⋊#6923—42no—
75C2×#92420—22no—
76C2×#92420—42no—
77C4×#6923—41no—
78C4×#6923—41no—
79#11758213—42no—
80#11758213—42no—
81D61⋊C8—41no—
82D61⋊C8—41no—
83Q16×C61—42no—
84Q16×C61—42no—
85C2×C8×C61—22no—
86C2×C8×C61C22⋊C411yescharacteristic
87C2×C8×C61C22⋊C411yescharacteristic
88#11758198—42no—
89#11758198—42no—
90C4⋊Dic61—22no—
91C4⋊Dic61—22no—
92#11758196—42no—
93#11758196—42no—
94C2×#92418—41no—
95C2×#92418C22⋊C411yescharacteristic
96C2×#92418C22⋊C411yescharacteristic
97C8×D61—22no—
98C8×D61—22no—
99C8×D61—41no—
100C8×D61—41no—
101C23×C61C4×D411yescharacteristic
102C22×D61—22no—
103C22×D61—22no—
104C22×D61—41no—
105C22×D61#12111yescharacteristic
106C22×D61#12411yescharacteristic
107C2×#6923—42no—
108C2×#6923—42no—
109C2×#6923—42no—
110Q8×C61—42no—
111Q8×C61—42no—
112Q8×C61—82no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—42no—
116C2×C4×C61—82no—
117C2×C4×C61C2×C22⋊C411yescharacteristic
118C2×C4×C61C4×D411yescharacteristic
119C2×Dic61—41no—
120C2×Dic61—42no—
121C2×Dic61—44no—
122C2×Dic61#12111yescharacteristic
123C2×Dic61#12711yescharacteristic
124C4×D61—21no—
125C4×D61—21no—
126C4×D61—21no—
127C4×D61—21no—
128C4×D61—41no—
129C4×D61—41no—
130#92420—42no—
131#92420—82no—
132C8×C61—21no—
133C8×C61#9912yes—
134#92418—21no—
135#92418—21no—
136#9817—611nomaximal, Sylow
137C22×C61—21no—
138C22×C61—21no—
139C22×C61#39011yescharacteristic
140C22×C61#44412yes—
141C2×D61—21no—
142C2×D61—21no—
143C2×D61—21no—
144C2×D61—21no—
145C2×D61—21no—
146C2×D61—44no—
147C2×D61#47312yes—
148#6923—82no—
149C4×C61—21no—
150C4×C61—82no—
151C4×C61C2×#9911yescharacteristic
152C4×C61#41011yescharacteristic
153Dic61—21no—
154Dic61—21no—
155Dic61—82no—
156C2×#467—611no—
157#1810—611no—
158#1752—611no—
159#1508—612no—
160#1472—612no—
161C2×C61—22no—
162C2×C61#196411yescharacteristic
163C2×C61#197511yescharacteristic
164C2×C61#197911yescharacteristic
165D61—22no—
166D61—41no—
167C22×Q16#692311yescharacteristic, complemented
168C23×C8—611no—
169C2×C22⋊Q8—612no—
170#467—1221no—
171#467—1221no—
172#467—1221no—
173#467—1221no—
174C2×#108—1221no—
175C2×#107—611no—
176C2×#103—612no—
177#407—612no—
178C2×C22⋊C8—611no—
179C2×#97—1221no—
180C4×C22⋊C4—611no—
181#341—1221no—
182#336—1221no—
183#328—1224no—
184#324—1224no—
185C61#981711yescharacteristic, complemented, Sylow
186C22×Q8C2×#692312yes—
187C23×C4—611no—
188C2×Q16—21no—
189C2×Q16—21no—
190C2×Q16—42no—
191C2×M16—1222no—
192C22×C8—611no—
193C22×C8—611no—
194C22×C8—611no—
195C22×C8—611no—
196C22×C8—611no—
197C22×C8—1221no—
198C22×C8—2442no—
199C22×C8C2×#692311yescharacteristic
200C22⋊Q8—1222no—
201C22⋊Q8—1222no—
202C22⋊Q8—1222no—
203C22⋊Q8—1222no—
204C2×C4⋊C4—612no—
205C2×C4⋊C4—1222no—
206C2×C22⋊C4—1222no—
207C2×C22⋊C4—1221no—
208C2×C42—1221no—
209#108—2442no—
210#107—1221no—
211#107—1221no—
212#103—1222no—
213#103—1222no—
214C22⋊C8—612no—
215C22⋊C8—1221no—
216C22⋊C8—1222no—
217C22⋊C8—1222no—
218#97—2442no—
219#95—612no—
220C24—611no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—42no—
225C22×C4—22no—
226C22×C4—611no—
227C22×C4—611no—
228C22×C4—611no—
229C22×C4—611no—
230C22×C4—611no—
231C22×C4—1222no—
232C22×C4—1222no—
233C22×C4—2441no—
234C22×C4—2441no—
235C22×C4C22×#692311yescharacteristic
236Q16—42no—
237Q16—42no—
238M16—2442no—
239M16—2442no—
240C2×C8—22no—
241C2×C8—611no—
242C2×C8—611no—
243C2×C8—1222no—
244C2×C8—1222no—
245C2×C8—1222no—
246C2×C8—2441no—
247C2×C8—2441no—
248C2×C8—2441no—
249C2×C8—2442no—
250C2×C8—2441no—
251C2×C8—2441no—
252C2×C8C22⋊#692311yescharacteristic
253C2×C8C22⋊#692311yescharacteristic
254C4⋊C4—1222no—
255C4⋊C4—1222no—
256C4⋊C4—2442no—
257C4⋊C4—2442no—
258C22⋊C4—2442no—
259C22⋊C4—2442no—
260C22⋊C4—2442no—
261C42—2441no—
262C42—2441no—
263C23—611no—
264C23—611no—
265C23—1222no—
266C23—1222no—
267C23—2441no—
268C23D4×#692311yescharacteristic
269Q8—42no—
270Q8—42no—
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—42no—
275C2×C4—82no—
276C2×C4—611no—
277C2×C4—611no—
278C2×C4—1221no—
279C2×C4—1221no—
280C2×C4—1221no—
281C2×C4—1221no—
282C2×C4—2441no—
283C2×C4—2441no—
284C2×C4—2441no—
285C2×C4—2442no—
286C2×C4—2442no—
287C2×C4—2442no—
288C2×C4—2442no—
289C2×C4—2444no—
290C2×C4C2×C22⋊#692311yescharacteristic
291C2×C4D4×#692311yescharacteristic
292C8—21no—
293C8—1221no—
294C8—1221no—
295C8—4882no—
296C8—2442no—
297C8#1451983512yes—
298C22—21no—
299C22—21no—
300C22—612no—
301C22—1221no—
302C22—1221no—
303C22—1221no—
304C22—1221no—
305C22—1221no—
306C22—2444no—
307C22#1684071411yescharacteristic
308C22#1684080812yes—
309C4—21no—
310C4—82no—
311C4—1221no—
312C4—1221no—
313C4—4882no—
314C4—4882no—
315C4C2×#1451983511yescharacteristic
316C4#1684074211yescharacteristic
317C2—22no—
318C2—1222no—
319C2—2441no—
320C2#2146195811yescharacteristic
321C2#2146197311yescharacteristic
322C2#2146198111yescharacteristic
subgroupG/H
centerC22#16840714
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC22×Q16×C61C4
socleC22×C61#390
prime radicalC23×D61#41
cosocleC22×C4×D61C23
hypercenterC22×Q16#6923
hypocenterC61#9817
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
21. C2×#14519711C4semidirect
22. C2×#14519701C4semidirect
23. C22×Q16×C61C4semidirect
167. C22×Q16#6923semidirect
185. C61#9817semidirect

Maximal subgroups

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

subgroupindex
#214616982
#214615552
#214612192
#214610992
C2×#168400002
#981761