Finite Group Databasebeta

← #30599746

Subgroups — #30599746

Order 15616.

376 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 94 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
1Q8×#11758213C211yesmaximal, characteristic
2#21461934C211yesmaximal, characteristic
3#21461374C211yesmaximal, characteristic
4#21461148C211yesmaximal, characteristic
5#21460711C211yesmaximal, characteristic
6#21460651C211yesmaximal, characteristic
7Q8⋊C8×D61C211yesmaximal, characteristic
8#16840812—21no—
9#16840812—22no—
10Q8×#92420—21no—
11#16840781C2211yescharacteristic
12C2×#14519847—21no—
13#16840766—21no—
14#16840766C2211yescharacteristic
15#16840766C2211yescharacteristic
16C4×#11758213—21no—
17C4×#11758213C2211yescharacteristic
18#16840638—21no—
19#16840623—22no—
20#16840612—21no—
21Q8⋊#92420—21no—
22C4×Q8×D61C2211yescharacteristic
23C4⋊C8×D61C2211yescharacteristic
24C4×C8×D61C2211yescharacteristic
25#16839494C411yescharacteristic
26Q8⋊#92418C411yescharacteristic
27#16839492C411yescharacteristic
28C61×Q8⋊C8C411yescharacteristic
29C2×#11758213—21no—
30C2×#11758213—21no—
31C2×#11758213D411yescharacteristic
32C2×#11758213D411yescharacteristic
33#14519847—42no—
34C4⋊#92420—22no—
35C4⋊#92420—22no—
36C4⋊#92420—21no—
37C4⋊#92420—21no—
38C4⋊#92420—21no—
39C4⋊#92420—21no—
40C4⋊#92420D412yes—
41#14519834—21no—
42#14519832—22no—
43#14519832—21no—
44C4×#92420—22no—
45C4×#92420—21no—
46C4×#92420—21no—
47C2×Q8×D61Q811yescharacteristic
48C2×C8×D61—21no—
49C2×C8×D61—21no—
50Q8×Dic61D411yescharacteristic
51C4×Dic122—21no—
52C4×Dic122C2×C411yescharacteristic
53#14519742—21no—
54#14519742D411yescharacteristic
55C4⋊C4×D61—21no—
56C4⋊C4×D61Q811yescharacteristic
57C42×D61—21no—
58C42×D61C2311yescharacteristic
59C4⋊#92418C2×C411yescharacteristic
60#14519706—21no—
61C4×#92418C2×C411yescharacteristic
62C8×Dic61—21no—
63C4×Q8×C61C2×C411yescharacteristic
64C4⋊C8×C61C2×C411yescharacteristic
65C4×C8×C61C2×C411yescharacteristic
66C2×Q8×C61C4⋊C411yescharacteristic
67Q8×D61—22no—
68Q8×D61—41no—
69C2×C4×D61—21no—
70C2×C4×D61—21no—
71C2×C4×D61C2×D411yescharacteristic
72C2×C4×D61C2×Q811yescharacteristic
73C2×C4×D61#4111yescharacteristic
74C2×Dic122—21no—
75C2×Dic122C4⋊C411yescharacteristic
76C2×#92420—21no—
77C2×#92420—21no—
78C2×#92420—22no—
79C2×#92420—41no—
80C2×#92420—41no—
81#11758213—21no—
82#11758213—21no—
83#11758213—21no—
84#11758213—41no—
85#11758213—41no—
86#11758213—41no—
87#11758213—41no—
88#11758213SD1612yes—
89C2×C8×C61—21no—
90C2×C8×C61—21no—
91C4⋊C4×C61—21no—
92C4⋊C4×C61C4⋊C411yescharacteristic
93C42×C61—21no—
94C42×C61C22×C411yescharacteristic
95C4⋊Dic61—21no—
96C4⋊Dic61C4⋊C411yescharacteristic
97#11758196—21no—
98#11758196—21no—
99#11758196—21no—
100C4×Dic61—21no—
101C4×Dic61—21no—
102C4×Dic61—21no—
103C4×Dic61C2×D411yescharacteristic
104C4×Dic61C2×D411yescharacteristic
105C4×Dic61#4111yescharacteristic
106C2×#92418—21no—
107C2×#92418—21no—
108C8×D61—42no—
109C8×D61—41no—
110C8×D61—41no—
111C22×D61C22⋊Q811yescharacteristic
112Q8×C61—21no—
113Q8×C61#10812yescomplemented
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61C2×C4⋊C411yescharacteristic
117C2×C4×C61C4×D411yescharacteristic
118C2×C4×C61C4×D411yescharacteristic
119C2×Dic61—21no—
120C2×Dic61—21no—
121C2×Dic61#12011yescharacteristic
122C2×Dic61C22⋊Q811yescharacteristic
123C2×Dic61#12311yescharacteristic
124C4×D61—21no—
125C4×D61—21no—
126C4×D61—41no—
127C4×D61—42no—
128C4×D61—81no—
129C4×D61C2×SD1611yescharacteristic
130C4×D61#13511yescharacteristic
131C4×D61#13711yescharacteristic
132C4×D61#13711yescharacteristic
133Dic122—21no—
134Dic122—21no—
135Dic122—42no—
136#92420—22no—
137#92420—41no—
138#92420—42no—
139#92420—81no—
140#92420—81no—
141C8×C61—22no—
142C8×C61—41no—
143#92418—41no—
144#92418—41no—
145#8909—611nomaximal, Sylow
146C22×C61#38511yescharacteristic
147C2×D61—21no—
148C2×D61—21no—
149C2×D61#47511yescharacteristic
150C2×D61#47911yescharacteristic
151C4×C61—22no—
152C4×C61—21no—
153C4×C61—41no—
154C4×C61C2×#10811yescharacteristic
155C4×C61#43011yescharacteristic
156C4×C61C4×SD1611yescharacteristic
157C4×C61#44111yescharacteristic
158Dic61—21no—
159Dic61—21no—
160Dic61—41no—
161Dic61—41no—
162Dic61#44811yescharacteristic
163Dic61#45211yescharacteristic
164Q8×M16—611no—
165#1950—611no—
166#1566—611no—
167#1487—611no—
168C2×Q8⋊C8—611no—
169#1301—611no—
170#1278—611no—
171C2×C61#185511yescharacteristic
172C2×C61#186111yescharacteristic
173C2×C61#186511yescharacteristic
174D61—21no—
175D61—21no—
176C2×C4×Q8—611no—
177#446—1222no—
178#446—1221no—
179C8×Q8—1221no—
180#431—611no—
181C2×#108—1221no—
182#423—611no—
183#423—611no—
184#423—1221no—
185C2×C4⋊C8—611no—
186C4×M16—611no—
187C4×M16—1221no—
188C2×C4×C8—611no—
189#341—1221no—
190#334—1222no—
191Q8⋊C8—611no—
192Q8⋊C8—611no—
193Q8⋊C8—611no—
194Q8⋊C8—1221no—
195Q8⋊C8—1221no—
196Q8⋊C8#692311yescharacteristic
197C61#890911yescharacteristic, complemented, Sylow
198C22×Q8—611no—
199C2×M16—611no—
200C2×M16—611no—
201C2×M16—1221no—
202C2×M16—1221no—
203C22×C8—1221no—
204C22×C8—1221no—
205C4×Q8—611no—
206C4×Q8—611no—
207C4×Q8—611no—
208C4×Q8—1221no—
209C4×Q8—1221no—
210C4×Q8C2×#692311yescharacteristic
211C2×C4⋊C4—611no—
212C2×C4⋊C4—1221no—
213C2×C42—611no—
214C2×C42—1221no—
215#108—2442no—
216C4⋊C8—612no—
217C4⋊C8—611no—
218C4⋊C8—1221no—
219C4⋊C8—1221no—
220C4⋊C8—1221no—
221C4⋊C8—1221no—
222C4⋊C8—1221no—
223C4⋊C8—1222no—
224C4⋊C8—1222no—
225C4⋊C8C2×#692311yescharacteristic
226C22⋊C8—1221no—
227#97—1221no—
228#97—1222no—
229C4×C8—611no—
230C4×C8—1221no—
231C4×C8—1221no—
232C4×C8—1221no—
233C4×C8—1222no—
234C4×C8C2×#692311yescharacteristic
235C2×Q8—611no—
236C2×Q8—1221no—
237C2×Q8—1222no—
238C2×Q8—2441no—
239C2×Q8C4⋊#692311yescharacteristic
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—1221no—
244C22×C4—1221no—
245M16—612no—
246M16—1221no—
247M16—1221no—
248M16—1221no—
249M16—2441no—
250M16—2441no—
251M16—2441no—
252M16—2441no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—1221no—
256C2×C8—1222no—
257C2×C8—1221no—
258C2×C8—1221no—
259C2×C8—1221no—
260C2×C8—2442no—
261C2×C8—2441no—
262C2×C8—2441no—
263C2×C8—2441no—
264C2×C8—2441no—
265C4⋊C4—21no—
266C4⋊C4—611no—
267C4⋊C4—1221no—
268C4⋊C4—1221no—
269C4⋊C4—1221no—
270C4⋊C4—1221no—
271C4⋊C4C4⋊#692311yescharacteristic
272C42—21no—
273C42—611no—
274C42—611no—
275C42—611no—
276C42—1221no—
277C42—1221no—
278C42—1221no—
279C42C22×#692311yescharacteristic
280C23—611no—
281Q8—21no—
282Q8—1221no—
283Q8—1221no—
284Q8—2442no—
285Q8#1451984712yescomplemented
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—611no—
289C2×C4—611no—
290C2×C4—611no—
291C2×C4—611no—
292C2×C4—611no—
293C2×C4—611no—
294C2×C4—611no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—4881no—
300C2×C4—2441no—
301C2×C4—2442no—
302C2×C4C2×C4⋊#692311yescharacteristic
303C2×C4D4×#692311yescharacteristic
304C2×C4D4×#692311yescharacteristic
305C8—22no—
306C8—41no—
307C8—1222no—
308C8—4881no—
309C8—4881no—
310C8—2441no—
311C8—2441no—
312C8—2442no—
313C8—2441no—
314C22—611no—
315C22—611no—
316C22—1221no—
317C22—1221no—
318C22#1684070911yescharacteristic
319C4—22no—
320C4—21no—
321C4—41no—
322C4—611no—
323C4—611no—
324C4—1221no—
325C4—1221no—
326C4—2441no—
327C4—2441no—
328C4C2×#1451984711yescharacteristic
329C4#1684077711yescharacteristic
330C4SD16×#692311yescharacteristic
331C4#1684080211yescharacteristic
332C2—1221no—
333C2—1221no—
334C2#2146177211yescharacteristic
335C2#2146178011yescharacteristic
336C2#2146178511yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×Q8⋊C8C4
socleC22×C61#385
prime radicalC22×D61C22⋊Q8
cosocleC42×D61C23
hypercenterQ8⋊C8#6923
hypocenterC61#8909
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
113. Q8×C61#108semidirect
197. C61#8909semidirect
285. Q8#14519847semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582132
#214619342
#214613742
#214611482
#214607112
#214606512
Q8⋊C8×D612
#890961