Finite Group Databasebeta

← #30605208

Subgroups — #30605208

Order 15616.

414 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 66 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#21461679C211yesmaximal, characteristic
2#21461078C211yesmaximal, characteristic
3#21460814C212yesmaximal, complemented
4#21460799C212yesmaximal, complemented
5#21451737C211yesmaximal, characteristic
6#16840783—21no—
7C2×#14519835—21no—
8#16840704—21no—
9#16840640C2212yes—
10#16840606C2212yes—
11#16840605—21no—
12C2×C4×C22⋊D61C2211yescharacteristic
13#16839693C411yescharacteristic, complemented
14#16839690—21no—
15#16839688C2211yescharacteristic
16#16839681C411yescharacteristic, complemented
17#16839677—21no—
18#16839673C2211yescharacteristic
19#16839643—21no—
20#16839633—21no—
21#16839625C411yescharacteristic, complemented
22C2×C22⋊Q8×C61C411yescharacteristic, complemented
23C2×C4⋊#6923—41no—
24C2×C22⋊#6923—22no—
25C4⋊#92420—41no—
26#14519835—42no—
27C22⋊#92420—22no—
28#14519832—41no—
29(C2×#6923)⋊C4—41no—
30C22×C22⋊D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C4×C22⋊D61—41no—
33C4×C22⋊D61—41no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196C2×C411yescharacteristic
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61D411yescharacteristic
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198C2311yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684C2×C411yescharacteristic
49#14519684C2×C411yescharacteristic
50C22×Q8×C61C2×C411yescharacteristic
51C23×C4×C61C2×C411yescharacteristic
52C22⋊Q8×C61—41no—
53C22⋊Q8×C61—41no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C22⋊C4×C61—21no—
57C24×C61C22⋊C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C2×Q8×C61—41no—
61C2×Q8×C61—81no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—81no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22×C411yescharacteristic
69C2×C22⋊D61—21no—
70C2×C22⋊D61—21no—
71C2×C22⋊D61—41no—
72C2×C22⋊D61—41no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C2×D411yescharacteristic
78C2×C4×D61—21no—
79C2×C4×D61—21no—
80C2×C4×D61—81no—
81C22⋊#6923—42no—
82C22⋊#6923—42no—
83C2×#92420—42no—
84C4⋊#6923—82no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C4⋊C4×C61—41no—
88C22⋊C4×C61—42no—
89C22⋊Dic61—41no—
90C22⋊Dic61—81no—
91#11758198—41no—
92#11758198—41no—
93#11758198—41no—
94#11758198—81no—
95#11758196—41no—
96#11758196—42no—
97#11758196—81no—
98C4×Dic61—21no—
99C4×Dic61—21no—
100C23×C61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61C2×C22⋊C411yescharacteristic
106C22×D61—21no—
107C22×D61—21no—
108C22×D61—41no—
109C22×D61—41no—
110C22×D61AGL(1,ℤ/8)11yescharacteristic
111C22×D61#13711yescharacteristic
112C2×#6923—82no—
113C2×#6923—82no—
114C2×#6923—82no—
115Q8×C61—82no—
116C2×C4×C61—21no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—81no—
126C2×C4×C61—81no—
127C2×C4×C61#9911yescharacteristic
128C2×C4×C61#9911yescharacteristic
129C22⋊D61—42no—
130C22⋊D61—42no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—21no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C4×D61—42no—
145C4×D61—81no—
146#92420—82no—
147#11826—611nomaximal, Sylow
148C22×C61—21no—
149C22×C61—21no—
150C22×C61—41no—
151C22×C61—41no—
152C22×C61—41no—
153C22×C61—81no—
154C22×C61—81no—
155C22×C61C2×#9911yescharacteristic
156C22×C61#41711yescharacteristic
157C22×C61#42111yescharacteristic
158C2×D61—21no—
159C2×D61—21no—
160C2×D61—41no—
161C2×D61—41no—
162C2×D61—41no—
163C2×D61—41no—
164C2×D61—81no—
165#6923—162no—
166C4×C61—21no—
167C4×C61—21no—
168C4×C61—41no—
169C4×C61—81no—
170C4×C61—81no—
171Dic61—41no—
172Dic61—41no—
173Dic61—41no—
174Dic61—81no—
175Dic61—81no—
176#2638—611no—
177#1799—611no—
178#1466—611no—
179#1351—612no—
180#1344—612no—
181C2×C61—21no—
182C2×C61—21no—
183C2×C61—81no—
184C2×C61#151311yescharacteristic
185C2×C61#212111yescharacteristic
186C2×C61#212911yescharacteristic
187D61—41no—
188D61—41no—
189C2×C22⋊Q8#692311yescharacteristic, complemented
190C2×C4×D4—611no—
191#432—1221no—
192C2×#99—1221no—
193#398—611no—
194#397—611no—
195#397—1221no—
196#395—611no—
197#394—1221no—
198#393—611no—
199#387—1221no—
200#385—1221no—
201#381—611no—
202#380—1221no—
203#342—612no—
204#323—612no—
205#323—1221no—
206C61#1182611yescharacteristic, complemented, Sylow
207C22×Q8C2×#692311yescharacteristic
208C22×D4—611no—
209C23×C4—611no—
210C23×C4C2×#692311yescharacteristic
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C4×D4—2441no—
214C4×D4—2441no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—611no—
217C2×C4⋊C4—1221no—
218C2×C4⋊C4—1221no—
219C2×C4⋊C4—2441no—
220C2×C4⋊C4C2×#692311yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—611no—
223C2×C22⋊C4—611no—
224C2×C22⋊C4—1221no—
225C2×C22⋊C4—1221no—
226C2×C22⋊C4—1221no—
227C2×C22⋊C4—1222no—
228C2×C42—611no—
229C4⋊C8—2441no—
230#99—2442no—
231C22⋊C8—1222no—
232#97—2441no—
233#95—611no—
234#95—611no—
235#95—1221no—
236#95—1221no—
237#95—1221no—
238#95—1221no—
239#95—1221no—
240#95—2441no—
241C24—611no—
242C24C22⋊#692311yescharacteristic
243C2×Q8—41no—
244C2×Q8—81no—
245C2×D4—1221no—
246C2×D4—1221no—
247C2×D4—2441no—
248C2×D4—2441no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—81no—
254C22×C4—611no—
255C22×C4—611no—
256C22×C4—611no—
257C22×C4—1221no—
258C22×C4—1221no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—4881no—
262C22×C4—2442no—
263C22×C4C22⋊#692311yescharacteristic
264C22×C4C22×#692311yescharacteristic
265C2×C8—2442no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—4881no—
270C4⋊C4—4882no—
271C4⋊C4—2441no—
272C4⋊C4—2442no—
273C22⋊C4—42no—
274C22⋊C4—4881no—
275C22⋊C4—4881no—
276C22⋊C4—2441no—
277C22⋊C4—2441no—
278C22⋊C4—2441no—
279C22⋊C4—2441no—
280C22⋊C4—2442no—
281C22⋊C4—2442no—
282C42—1221no—
283C42—1221no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—41no—
288C23—41no—
289C23—611no—
290C23—611no—
291C23—1221no—
292C23—1221no—
293C23—2441no—
294C23—2441no—
295C23C2×C22⋊#692311yescharacteristic
296Q8—82no—
297D4—2442no—
298D4—2442no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—81no—
309C2×C4—81no—
310C2×C4—1221no—
311C2×C4—1221no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—1221no—
315C2×C4—4881no—
316C2×C4—4882no—
317C2×C4—4882no—
318C2×C4—4882no—
319C2×C4—2442no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4#1451983511yescharacteristic
329C2×C4#1451983511yescharacteristic
330C8—4882no—
331C22—21no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—81no—
337C22—81no—
338C22—1221no—
339C22—1221no—
340C22—4881no—
341C22—2441no—
342C22—2441no—
343C22—2441no—
344C22—2441no—
345C22C2×#1451983511yescharacteristic
346C22#1684075311yescharacteristic
347C22#1684075911yescharacteristic
348C4—21no—
349C4—21no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—9762no—
354C4—4881no—
355C4—4881no—
356C4—2441no—
357C4—2441no—
358C4—2441no—
359C2—21no—
360C2—21no—
361C2—81no—
362C2—2441no—
363C2—2441no—
364C2#2146123411yescharacteristic
365C2#2146207411yescharacteristic
366C2#2146210111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×#11758198C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#11826
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21460814C2semidirect
4. #21460799C2semidirect
13. #16839693C4semidirect
16. #16839681C4semidirect
21. #16839625C4semidirect
22. C2×C22⋊Q8×C61C4semidirect
189. C2×C22⋊Q8#6923semidirect
206. C61#11826semidirect

Maximal subgroups

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

subgroupindex
#214616792
#214610782
#214608142
#214607992
#214517372
#1182661