Finite Group Databasebeta

← #30598712

Subgroups — #30598712

Order 15616.

690 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 92 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#21461720C211yesmaximal, characteristic
2#21461716C211yesmaximal, characteristic
3#21461229C212yesmaximal
4#21461227C212yesmaximal
5C2×#16840370C211yesmaximal, characteristic
6#16840734C2212yes—
7#16840705C2212yes—
8#16840642—21no—
9#16840642—21no—
10#16840640—21no—
11#16840640—21no—
12#16840616—24no—
13#16840615—24no—
14#16840370—24no—
15#16840370—24no—
16C2×#14519775C411yescharacteristic, complemented
17C2×#14519772—21no—
18C2×#14519771—21no—
19C2×#14519769—21no—
20C2×#14519768C411yescharacteristic, complemented
21C2×#14519767—21no—
22C2×#14519757C2211yescharacteristic
23C2×#14519753C2211yescharacteristic
24C2×C4×Dic122C411yescharacteristic, complemented
25C2×#14519734C2211yescharacteristic
26C2×C4×Q8×C61C411yescharacteristic, complemented
27C2×#11758217—22no—
28C2×C22⋊#6923—22no—
29#14519834—22no—
30#14519834—22no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4—22no—
33C22×Dic122C2×C411yescharacteristic
34C22×C4×D61C2311yescharacteristic
35#14519775—22no—
36#14519775—22no—
37#14519772—28no—
38#14519771—28no—
39#14519769—28no—
40#14519768—22no—
41#14519768—22no—
42#14519767—28no—
43#14519757—22no—
44#14519757D414yes—
45#14519753—22no—
46#14519753D414yes—
47C4×Dic122—22no—
48C4×Dic122—22no—
49#14519734—22no—
50#14519734—22no—
51C2×C4⋊Dic61—21no—
52C2×C4⋊Dic61—21no—
53C2×C4⋊Dic61C2×C411yescharacteristic
54C2×#11758196—21no—
55C2×#11758196—21no—
56C2×#11758196—21no—
57C2×#11758196—21no—
58C2×#11758196—21no—
59C2×#11758196D411yescharacteristic
60C2×#11758196D411yescharacteristic
61C2×#11758198—21no—
62C2×#11758198—21no—
63C2×#11758198D411yescharacteristic
64C2×#11758198D411yescharacteristic
65C2×C4×Dic61—21no—
66C2×C4×Dic61C2×C411yescharacteristic
67C22×Q8×C61C2×C411yescharacteristic
68C4×Q8×C61—22no—
69C4×Q8×C61—22no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61C2×C411yescharacteristic
72C2×C42×C61—21no—
73C2×C42×C61C2×C411yescharacteristic
74C23×D61C2×D411yescharacteristic
75C22×#6923—42no—
76C2×Q8×C61—21no—
77C2×Q8×C61—42no—
78C2×Q8×C61C22⋊C412yes—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61C22⋊C411yescharacteristic
82C22×C4×C61C22⋊C411yescharacteristic
83C22×C4×C61C22×C411yescharacteristic
84C22×Dic61—21no—
85C22×Dic61—21no—
86C22×Dic61C22⋊C411yescharacteristic
87C22×Dic61C22⋊C411yescharacteristic
88C22×Dic61C2×D411yescharacteristic
89C2×C4×D61—42no—
90C2×C4×D61C2×D412yes—
91C2×C4×D61C2×D412yes—
92C2×Dic122—21no—
93C2×Dic122—42no—
94C2×Dic122C22⋊C412yes—
95C22⋊#6923—44no—
96#11758217—44no—
97C2×#92420—42no—
98C4⋊C4×C61—24no—
99C4⋊C4×C61—21no—
100C4⋊C4×C61C22⋊C412yes—
101C42×C61—24no—
102C42×C61—21no—
103C42×C61—21no—
104#11758198—24no—
105#11758198—24no—
106#11758198—22no—
107#11758198—22no—
108C4⋊Dic61—24no—
109C4⋊Dic61—21no—
110C4⋊Dic61—24no—
111C4⋊Dic61C22⋊C412yes—
112#11758196—24no—
113#11758196—24no—
114#11758196—24no—
115#11758196—24no—
116#11758196—24no—
117#11758196—22no—
118#11758196—22no—
119C4×Dic61—24no—
120C4×Dic61—21no—
121C4×Dic61—21no—
122C23×C61C2×C22⋊C411yescharacteristic
123C22×D61—22no—
124C22×D61—41no—
125C22×D61—41no—
126C22×D61#12012yes—
127C2×#6923—82no—
128C2×#6923—82no—
129C2×#6923—82no—
130Q8×C61—42no—
131Q8×C61—42no—
132C2×C4×C61—22no—
133C2×C4×C61—21no—
134C2×C4×C61—21no—
135C2×C4×C61—22no—
136C2×C4×C61—22no—
137C2×C4×C61—42no—
138C2×C4×C61—42no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61C2×C22⋊C411yescharacteristic
142C2×C4×C61C2×C22⋊C411yescharacteristic
143C2×Dic61—22no—
144C2×Dic61—21no—
145C2×Dic61—21no—
146C2×Dic61—22no—
147C2×Dic61—22no—
148C2×Dic61—42no—
149C2×Dic61—42no—
150C2×Dic61—41no—
151C2×Dic61—41no—
152C2×Dic61#12012yes—
153C4×D61—42no—
154C4×D61—42no—
155Dic122—42no—
156Dic122—42no—
157#92420—82no—
158#8521—611nomaximal, Sylow
159C22×C61—21no—
160C22×C61—21no—
161C22×C61#37911yescharacteristic
162C22×C61#41012yes—
163C2×D61—22no—
164C2×D61—22no—
165C2×D61—22no—
166C2×D61—82no—
167#6923—162no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—41no—
171C4×C61—41no—
172C4×C61—42no—
173C4×C61—42no—
174Dic61—22no—
175Dic61—41no—
176Dic61—41no—
177Dic61—42no—
178Dic61—42no—
179C2×#541—611no—
180#1822—611no—
181#1820—611no—
182#1511—612no—
183#1511—612no—
184C2×C61—22no—
185C2×C61#178211yescharacteristic
186C2×C61#179611yescharacteristic
187C2×C61#179611yescharacteristic
188D61—42no—
189#541—1224no—
190#541—1224no—
191C2×C4⋊Q8—611no—
192C2×#126—1221no—
193C2×#126—1221no—
194C2×#125—611no—
195C2×#125—1221no—
196C2×#125—1221no—
197C2×C22⋊Q8—611no—
198C2×C22⋊Q8—611no—
199C2×C4×Q8—611no—
200C2×C4×Q8#692311yescharacteristic, complemented
201C2×#117—611no—
202#407—612no—
203#381—612no—
204#343—1221no—
205#343—1221no—
206#342—1221no—
207#342—1221no—
208#328—1224no—
209#328—1224no—
210C61#852111yescharacteristic, complemented, Sylow
211C22×Q8—611no—
212C22×Q8C2×#692311yescharacteristic
213C23×C4—611no—
214C2×M16—1222no—
215C4⋊Q8—1222no—
216C4⋊Q8—1222no—
217#126—1228no—
218#126—1228no—
219#125—1228no—
220#125—1222no—
221#125—1228no—
222#125—1222no—
223C22⋊Q8—614no—
224C22⋊Q8—614no—
225C22⋊Q8—1222no—
226C22⋊Q8—1222no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—1222no—
230C4×Q8—1222no—
231#117—1222no—
232#117—1222no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—611no—
235C2×C4⋊C4—611no—
236C2×C4⋊C4—611no—
237C2×C4⋊C4—1221no—
238C2×C4⋊C4—1221no—
239C2×C4⋊C4—1221no—
240C2×C4⋊C4—1221no—
241C2×C4⋊C4—1221no—
242C2×C4⋊C4—1221no—
243C2×C4⋊C4—1221no—
244C2×C4⋊C4C2×#692311yescharacteristic
245C2×C22⋊C4—611no—
246C2×C22⋊C4—611no—
247C2×C22⋊C4—1221no—
248C2×C22⋊C4—1221no—
249C2×C22⋊C4—1222no—
250C2×C42—21no—
251C2×C42—611no—
252C2×C42—1221no—
253C2×C42C2×#692311yescharacteristic
254C22⋊C8—1222no—
255C22⋊C8—1222no—
256#95—1222no—
257#95—1222no—
258C24—611no—
259C2×Q8—21no—
260C2×Q8—42no—
261C2×Q8—612no—
262C2×Q8—1221no—
263C2×Q8—2442no—
264C2×Q8C22⋊#692312yes—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—611no—
268C22×C4—612no—
269C22×C4—612no—
270C22×C4—611no—
271C22×C4—611no—
272C22×C4—1221no—
273C22×C4—1221no—
274C22×C4—2442no—
275C22×C4—2442no—
276C22×C4C22⋊#692311yescharacteristic
277C22×C4C22⋊#692311yescharacteristic
278C22×C4C22×#692311yescharacteristic
279M16—2444no—
280C2×C8—2442no—
281C4⋊C4—24no—
282C4⋊C4—21no—
283C4⋊C4—612no—
284C4⋊C4—1222no—
285C4⋊C4—1222no—
286C4⋊C4—1224no—
287C4⋊C4—1224no—
288C4⋊C4—1224no—
289C4⋊C4—1224no—
290C4⋊C4—1224no—
291C4⋊C4—1224no—
292C4⋊C4—1224no—
293C4⋊C4—1221no—
294C4⋊C4C22⋊#692312yes—
295C22⋊C4—1222no—
296C22⋊C4—1222no—
297C22⋊C4—1224no—
298C22⋊C4—1224no—
299C22⋊C4—2444no—
300C42—24no—
301C42—21no—
302C42—21no—
303C42—1224no—
304C42—1221no—
305C42—1221no—
306C23—612no—
307C23—1222no—
308C23—2441no—
309C23—2441no—
310C23C2×C22⋊#692311yescharacteristic
311Q8—42no—
312Q8—42no—
313Q8—2442no—
314Q8—2442no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—612no—
325C2×C4—1222no—
326C2×C4—1222no—
327C2×C4—1221no—
328C2×C4—1221no—
329C2×C4—1222no—
330C2×C4—4882no—
331C2×C4—4882no—
332C2×C4—4882no—
333C2×C4—2442no—
334C2×C4—2442no—
335C2×C4—2442no—
336C2×C4—2442no—
337C2×C4—2441no—
338C2×C4—2441no—
339C2×C4C2×C22⋊#692311yescharacteristic
340C2×C4C2×C22⋊#692311yescharacteristic
341C8—4882no—
342C22—21no—
343C22—21no—
344C22—1222no—
345C22—1222no—
346C22—1222no—
347C22—4882no—
348C22#1684070311yescharacteristic
349C22#1684074112yes—
350C4—21no—
351C4—21no—
352C4—42no—
353C4—41no—
354C4—41no—
355C4—42no—
356C4—1222no—
357C4—9762no—
358C4—2442no—
359C4—2441no—
360C4—2442no—
361C4—2441no—
362C2—22no—
363C2—2442no—
364C2#2146164811yescharacteristic
365C2#2146167111yescharacteristic
366C2#2146167211yescharacteristic
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×Q8×C61C4
socleC22×C61#379
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenterC2×C4×Q8#6923
hypocenterC61#8521
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
16. C2×#14519775C4semidirect
20. C2×#14519768C4semidirect
24. C2×C4×Dic122C4semidirect
26. C2×C4×Q8×C61C4semidirect
200. C2×C4×Q8#6923semidirect
210. C61#8521semidirect

Maximal subgroups

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

subgroupindex
#214617202
#214617162
#214612292
#214612272
C2×#168403702
#852161