Finite Group Databasebeta

← #30616145

Subgroups — #30616145

Order 15616.

982 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840772C211yesmaximal, characteristic
2C2×#16840725C211yesmaximal, characteristic
3#21462299C212yesmaximal
4#21461924C214yesmaximal, complemented
5#21461923C214yesmaximal, complemented
6#21461506C212yesmaximal
7C2×#107×D61C211yesmaximal, characteristic
8C2×D61⋊M16C2211yescharacteristic
9C2×#14519856C2212yes—
10C2×#14519856C2211yescharacteristic
11#16840772—24no—
12C2×#14519846C2212yes—
13C2×#14519845C2212yes—
14#16840725—24no—
15C2×#14519831C2212yes—
16C2×C8×#6923C2212yes—
17#16840713—24no—
18#16840713C2218yes—
19#16840704—24no—
20#16840635C2218yes—
21C22×C8×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2212yes—
23C2×#14519711C411yescharacteristic, complemented
24C2×#14519710C412yescomplemented
25#107×D61C412yescomplemented
26#107×D61C412yescomplemented
27#107×D61C2212yes—
28#107×D61C2212yes—
29C2×#107×C61C411yescharacteristic, complemented
30C2×#11758213—21no—
31D61⋊M16—22no—
32D61⋊M16Q814yes—
33C2×#11758217—21no—
34C2×D61⋊C8C2312yes—
35#14519856—22no—
36#14519856—22no—
37#14519856—24no—
38#14519856—24no—
39C2×C4⋊#6923—28no—
40C2×C4⋊#6923D412yes—
41C2×C4⋊#6923C2314yes—
42C2×C22⋊#6923—22no—
43C2×C22⋊#6923D412yes—
44C2×C4×#6923—22no—
45C2×C4×#6923C2312yes—
46#14519846—24no—
47#14519845—24no—
48#14519831—24no—
49C8×#6923—24no—
50(C2×#6923)⋊C4—28no—
51C22×C4×D61—22no—
52C22×C4×D61C2311yescharacteristic
53C22×#92418C2×C411yescharacteristic
54C2×C8×D61C2×C411yescharacteristic
55C2×C8×D61C2×C411yescharacteristic
56C2×C8×D61D414yes—
57C2×C8×D61Q814yes—
58C2×C8×D61C2311yescharacteristic
59C2×C8×D61C2311yescharacteristic
60C2×C4⋊Dic61C2×C412yes—
61C2×#11758196—22no—
62C4⋊C4×D61—22no—
63C4⋊C4×D61—22no—
64C4⋊C4×D61C2×C414yes—
65C4⋊C4×D61C2314yes—
66#14519711C2×C412yes—
67#14519711C2×C412yes—
68#14519710C2×C414yes—
69#14519710C2×C414yes—
70C22×C8×C61C2×C411yescharacteristic
71C2×C4⋊C4×C61C2×C412yes—
72#107×C61C2×C412yes—
73#107×C61C2×C412yes—
74C23×D61C2×D411yescharacteristic
75C22×#6923—24no—
76C22×#6923C2×D414yes—
77C22×C4×C61—22no—
78C22×C4×C61C22×C411yescharacteristic
79C22×Dic61—22no—
80C22×Dic61#4111yescharacteristic
81C2×C4×D61—24no—
82C2×C4×D61—24no—
83C2×C4×D61—84no—
84C2×C4×D61C22×C411yescharacteristic
85C2×C4×D61C2×D412yes—
86C2×C4×D61C2×D411yescharacteristic
87C2×C4×D61C2×Q812yes—
88C2×C4×D61C2×Q812yes—
89C2×C4×D61#4112yes—
90C2×C4×D61#4111yescharacteristic
91C2×C4×D61C2411yescharacteristic
92C22⋊#6923—24no—
93C22⋊#6923—44no—
94#11758217—24no—
95C2×#92420—22no—
96C4⋊#6923—24no—
97C4⋊#6923—24no—
98C4⋊#6923—24no—
99C4⋊#6923—88no—
100C4×#6923—22no—
101C4×#6923—22no—
102C4×#6923—44no—
103#11758213—24no—
104D61⋊C8—22no—
105D61⋊C8C2×Q814yes—
106C2×C8×C61C4⋊C414yescomplemented
107C2×C8×C61C22×C411yescharacteristic
108C2×C8×C61C22×C411yescharacteristic
109C4⋊C4×C61—22no—
110C4⋊C4×C61C22×C414yes—
111C4⋊Dic61—22no—
112C4⋊Dic61C22×C414yes—
113#11758196—24no—
114#11758196—24no—
115C2×#92418C4⋊C414yescomplemented
116C2×#92418C22×C411yescharacteristic
117C2×#92418C22×C411yescharacteristic
118C8×D61C4⋊C414yescomplemented
119C8×D61C4⋊C414yescomplemented
120C8×D61C2×D412yes—
121C8×D61C2×D412yes—
122C8×D61C2×Q812yes—
123C8×D61C2×Q812yes—
124C23×C61C4×D411yescharacteristic
125C22×D61—22no—
126C22×D61—22no—
127C22×D61C4×D411yescharacteristic
128C22×D61#12112yes—
129C22×D61C4⋊Q812yes—
130C22×D61C22×D411yescharacteristic
131C2×#6923—24no—
132C2×#6923—24no—
133C2×#6923—24no—
134C2×#6923—44no—
135C2×#6923—44no—
136C2×#6923—44no—
137C2×C4×C61—24no—
138C2×C4×C61—82no—
139C2×C4×C61C2×C4⋊C412yes—
140C2×C4×C61C4×D411yescharacteristic
141C2×C4×C61C4×Q812yes—
142C2×C4×C61C23×C411yescharacteristic
143C2×Dic61—24no—
144C2×Dic61—82no—
145C2×Dic61C4×D411yescharacteristic
146C2×Dic61C22⋊Q812yes—
147C2×Dic61#12512yes—
148C2×Dic61C2×#4111yescharacteristic
149C4×D61—48no—
150C4×D61—84no—
151C4×D61C2×C4⋊C412yes—
152C4×D61C4×Q812yes—
153C4×D61#12112yes—
154C4×D61C22⋊Q812yes—
155C4×D61#12512yes—
156C4×D61C4⋊Q812yes—
157C4×D61C22×D411yescharacteristic
158C4×D61C22×Q811yescharacteristic
159C4×D61C22×Q811yescharacteristic
160C4×D61C2×#4111yescharacteristic
161#92420—24no—
162C8×C61C2×C4⋊C412yes—
163C8×C61C2×C4⋊C412yes—
164#92418C2×C4⋊C412yes—
165#92418C2×C4⋊C412yes—
166#21158—611nomaximal, Sylow
167C22×C61—21no—
168C22×C61—21no—
169C22×C61#38912yes—
170C22×C61C2×C4×D411yescharacteristic
171C2×D61—21no—
172C2×D61—21no—
173C2×D61—24no—
174C2×D61—24no—
175C2×D61#38912yes—
176C2×D61C2×#12111yescharacteristic
177C2×D61C2×C4⋊Q811yescharacteristic
178C2×D61#57612yes—
179C2×D61#57812yes—
180#6923—44no—
181#6923—84no—
182C4×C61—44no—
183C4×C61#38912yes—
184C4×C61C22×C4⋊C411yescharacteristic
185C4×C61C2×C4×Q811yescharacteristic
186Dic61—44no—
187Dic61#38912yes—
188Dic61C2×C22⋊Q811yescharacteristic
189Dic61C2×#12511yescharacteristic
190C2×#427—611no—
191C22×#107—611no—
192C2×#405—611no—
193#2316—612no—
194#1943—614no—
195#1942—614no—
196#1736—612no—
197C2×C61—22no—
198C2×C61C2×#38911yescharacteristic
199C2×C61#294811yescharacteristic
200C2×C61#295011yescharacteristic
201D61—22no—
202D61#306112yes—
203D61#316412yes—
204C22×M16—611no—
205C23×C8—611no—
206C2×#117—611no—
207C2×#117—612no—
208C22×C4⋊C4—612no—
209#427—1224no—
210C2×#107—611no—
211C2×#107—612no—
212C2×#107—612no—
213C2×#107—612no—
214C2×#107—612no—
215C2×#107—612no—
216C2×#107—612no—
217C2×#107#692311yescharacteristic, complemented
218C2×#106—612no—
219#405—1224no—
220C2×#97—612no—
221C2×C4×C8—612no—
222#389—618no—
223#389—1224no—
224#380—1224no—
225#340—618no—
226C61#2115811yescharacteristic, complemented, Sylow
227C23×C4—611no—
228C23×C4—1222no—
229C2×M16—614no—
230C2×M16—1221no—
231C2×M16—1222no—
232C2×M16—1221no—
233C22×C8—611no—
234C22×C8—614no—
235C22×C8—614no—
236C22×C8—611no—
237C22×C8—611no—
238C22×C8—611no—
239C22×C8—611no—
240C22×C8—612no—
241C22×C8C2×#692311yescharacteristic
242#117—1222no—
243#117—1222no—
244#117—1224no—
245#117—1224no—
246C2×C4⋊C4—612no—
247C2×C4⋊C4—614no—
248C2×C4⋊C4—614no—
249C2×C4⋊C4—612no—
250C2×C4⋊C4—614no—
251C2×C4⋊C4—1222no—
252C2×C4⋊C4—1222no—
253C2×C4⋊C4—1222no—
254C2×C4⋊C4—1228no—
255C2×C4⋊C4C2×#692312yes—
256C2×C22⋊C4—612no—
257C2×C22⋊C4—1222no—
258C2×C42—612no—
259C2×C42—1222no—
260#107—612no—
261#107—612no—
262#107—614no—
263#107—614no—
264#107—1224no—
265#107C2×#692312yes—
266#107C2×#692312yes—
267#106—1224no—
268#97—1224no—
269C4×C8—1224no—
270#95—1228no—
271C24—611no—
272C22×C4—22no—
273C22×C4—611no—
274C22×C4—612no—
275C22×C4—612no—
276C22×C4—611no—
277C22×C4—611no—
278C22×C4—612no—
279C22×C4—612no—
280C22×C4—611no—
281C22×C4—611no—
282C22×C4—614no—
283C22×C4—1224no—
284C22×C4—1224no—
285C22×C4—1222no—
286C22×C4—1224no—
287C22×C4—4884no—
288C22×C4C22×#692311yescharacteristic
289M16—1224no—
290M16—1224no—
291C2×C8—614no—
292C2×C8—611no—
293C2×C8—611no—
294C2×C8—612no—
295C2×C8—612no—
296C2×C8—614no—
297C2×C8—614no—
298C2×C8—612no—
299C2×C8—612no—
300C2×C8—614no—
301C2×C8—1222no—
302C2×C8—1222no—
303C2×C8C4⋊#692314yescomplemented
304C2×C8C22×#692311yescharacteristic
305C2×C8C22×#692311yescharacteristic
306C4⋊C4—22no—
307C4⋊C4—614no—
308C4⋊C4—1224no—
309C4⋊C4—1224no—
310C4⋊C4—1224no—
311C4⋊C4—1224no—
312C4⋊C4—1224no—
313C4⋊C4—1222no—
314C4⋊C4—4888no—
315C4⋊C4C22×#692314yes—
316C22⋊C4—1224no—
317C22⋊C4—2444no—
318C42—1222no—
319C42—1222no—
320C42—2444no—
321C23—611no—
322C23—612no—
323C23—612no—
324C23—611no—
325C23—1222no—
326C23—1222no—
327C23D4×#692311yescharacteristic
328C2×C4—24no—
329C2×C4—82no—
330C2×C4—612no—
331C2×C4—612no—
332C2×C4—611no—
333C2×C4—611no—
334C2×C4—612no—
335C2×C4—612no—
336C2×C4—612no—
337C2×C4—612no—
338C2×C4—612no—
339C2×C4—611no—
340C2×C4—611no—
341C2×C4—612no—
342C2×C4—611no—
343C2×C4—611no—
344C2×C4—1224no—
345C2×C4—1224no—
346C2×C4—1224no—
347C2×C4—1224no—
348C2×C4—4884no—
349C2×C4—4882no—
350C2×C4—2448no—
351C2×C4—2444no—
352C2×C4—2444no—
353C2×C4—2444no—
354C2×C4C2×C4⋊#692312yes—
355C2×C4D4×#692311yescharacteristic
356C2×C4Q8×#692312yes—
357C2×C4C23×#692311yescharacteristic
358C8—612no—
359C8—612no—
360C8—1224no—
361C8C2×C4⋊#692312yes—
362C8C2×C4⋊#692312yes—
363C22—21no—
364C22—21no—
365C22—611no—
366C22—611no—
367C22—612no—
368C22—612no—
369C22—612no—
370C22—1224no—
371C22—1224no—
372C22—1221no—
373C22—1221no—
374C22#1684071312yes—
375C22C2×D4×#692311yescharacteristic
376C4—44no—
377C4—611no—
378C4—611no—
379C4—612no—
380C4—4884no—
381C4—2444no—
382C4—2444no—
383C4#1684071312yes—
384C4C22×C4⋊#692311yescharacteristic
385C4C2×Q8×#692311yescharacteristic
386C2—22no—
387C2—612no—
388C2—612no—
389C2—1222no—
390C2C2×#1684071311yescharacteristic
391C2#2146257811yescharacteristic
392C2#2146258011yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×#107×C61C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC2×#107#6923
hypocenterC61#21158
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#107

Complements

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

NH ≅ G/Ntype
4. #21461924C2semidirect
5. #21461923C2semidirect
23. C2×#14519711C4semidirect
24. C2×#14519710C4semidirect
25. #107×D61C4semidirect
26. #107×D61C4semidirect
29. C2×#107×C61C4semidirect
106. C2×C8×C61C4⋊C4semidirect
115. C2×#92418C4⋊C4semidirect
118. C8×D61C4⋊C4semidirect
119. C8×D61C4⋊C4semidirect
217. C2×#107#6923semidirect
226. C61#21158semidirect
303. C2×C8C4⋊#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407722
C2×#168407252
#214622992
#214619242
#214619232
#214615062
C2×#107×D612
#2115861