Finite Group Databasebeta

← #30605481

Subgroups — #30605481

Order 15616.

452 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 76 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#21461058C211yesmaximal, characteristic
3#21460816C212yesmaximal, complemented
4#21460814C212yesmaximal, complemented
5#21451001C211yesmaximal, characteristic
6#16840779—21no—
7C2×#14519835—21no—
8#16840704—21no—
9#16840640C2212yes—
10#16840606C2212yes—
11#16840605—21no—
12C2×C4×C22⋊D61C2211yescharacteristic
13#16839700C411yescharacteristic, complemented
14#16839697C2211yescharacteristic
15#16839681C411yescharacteristic, complemented
16#16839673C2211yescharacteristic
17#16839643—21no—
18#16839639—21no—
19#16839633—21no—
20#16839620—21no—
21C2×#14519684C411yescharacteristic, complemented
22C2×C22⋊Q8×C61C411yescharacteristic, complemented
23C2×C4⋊#6923—41no—
24C2×C22⋊#6923—22no—
25#14519844—41no—
26#14519835—42no—
27C22⋊#92420—22no—
28C4×#92420—41no—
29(C2×#6923)⋊C4—41no—
30C22×C22⋊D61D411yescharacteristic
31C23×Dic61—21no—
32C22×C4×D61D411yescharacteristic
33C4×C22⋊D61—41no—
34C4×C22⋊D61—41no—
35C2×#11758196—21no—
36C2×#11758196—21no—
37C2×#11758196C2×C411yescharacteristic
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—41no—
70C2×C22⋊D61—41no—
71C2×C22⋊D61SD1612yes—
72C2×C22⋊D61Q1612yes—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—42no—
76C22×Dic61—42no—
77C22×Dic61—81no—
78C22×Dic61C22⋊C411yescharacteristic
79C22×Dic61C22⋊C411yescharacteristic
80C22×Dic61C2×D411yescharacteristic
81C2×C4×D61—21no—
82C2×C4×D61—21no—
83C2×C4×D61—81no—
84C22⋊#6923—42no—
85C22⋊#6923—42no—
86C2×#92420—42no—
87C4⋊#6923—82no—
88C4⋊C4×C61—41no—
89C4⋊C4×C61—41no—
90C4⋊C4×C61—41no—
91C22⋊C4×C61—42no—
92C22⋊Dic61—41no—
93#11758198—41no—
94#11758198—42no—
95#11758198—81no—
96#11758196—41no—
97#11758196—42no—
98#11758196—81no—
99C4×Dic61—21no—
100C4×Dic61—21no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61#10312yes—
106C23×C61C2×C22⋊C411yescharacteristic
107C22×D61—21no—
108C22×D61—21no—
109C22×D61—41no—
110C22×D61—41no—
111C22×D61C2×SD1611yescharacteristic
112C22×D61C2×Q1611yescharacteristic
113C2×#6923—82no—
114C2×#6923—82no—
115C2×#6923—82no—
116Q8×C61—82no—
117C2×C4×C61—21no—
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—41no—
126C2×C4×C61—81no—
127C2×C4×C61—81no—
128C2×C4×C61#9911yescharacteristic
129C2×C4×C61#9911yescharacteristic
130C22⋊D61—42no—
131C22⋊D61—42no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—22no—
135C2×Dic61—21no—
136C2×Dic61—21no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—41no—
144C2×Dic61—84no—
145C2×Dic61#10312yes—
146C4×D61—42no—
147C4×D61—81no—
148#92420—82no—
149#11902—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—22no—
153C22×C61—42no—
154C22×C61—42no—
155C22×C61—41no—
156C22×C61—41no—
157C22×C61C2×#9911yescharacteristic
158C22×C61C2×#10311yescharacteristic
159C22×C61#42111yescharacteristic
160C2×D61—21no—
161C2×D61—21no—
162C2×D61—41no—
163C2×D61—41no—
164C2×D61—41no—
165C2×D61—41no—
166C2×D61—81no—
167#6923—162no—
168C4×C61—21no—
169C4×C61—21no—
170C4×C61—41no—
171C4×C61—81no—
172C4×C61—81no—
173Dic61—22no—
174Dic61—42no—
175Dic61—41no—
176Dic61—41no—
177Dic61—81no—
178#2564—611no—
179#1799—611no—
180#1460—611no—
181#1352—612no—
182#1351—612no—
183C2×C61—21no—
184C2×C61—21no—
185C2×C61—42no—
186C2×C61#152011yescharacteristic
187C2×C61#212111yescharacteristic
188C2×C61#212611yescharacteristic
189D61—41no—
190D61—41no—
191C2×C22⋊Q8#692311yescharacteristic, complemented
192C2×C4×D4—611no—
193#431—1221no—
194C2×#99—1221no—
195#400—611no—
196#399—611no—
197#395—611no—
198#393—611no—
199#387—1221no—
200#386—1221no—
201#385—1221no—
202#380—1221no—
203#380—1221no—
204C2×#95—611no—
205#342—612no—
206#323—612no—
207#323—1221no—
208C61#1190211yescharacteristic, complemented, Sylow
209C22×Q8C2×#692311yescharacteristic
210C22×D4—611no—
211C23×C4—611no—
212C23×C4—1221no—
213C23×C4C2×#692311yescharacteristic
214C22⋊Q8—41no—
215C22⋊Q8—41no—
216C4×D4—2441no—
217C4×D4—2441no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—611no—
220C2×C4⋊C4—1221no—
221C2×C4⋊C4—1221no—
222C2×C4⋊C4—2441no—
223C2×C4⋊C4C2×#692311yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—611no—
226C2×C22⋊C4—611no—
227C2×C22⋊C4—1221no—
228C2×C22⋊C4—1221no—
229C2×C22⋊C4—1222no—
230C2×C42—611no—
231C4⋊C8—2441no—
232#99—2442no—
233C22⋊C8—1222no—
234C4×C8—2441no—
235#95—611no—
236#95—611no—
237#95—1221no—
238#95—1221no—
239#95—1221no—
240#95—1221no—
241#95—1221no—
242#95—2441no—
243C24—611no—
244C24C22⋊#692311yescharacteristic
245C2×Q8—41no—
246C2×Q8—81no—
247C2×D4—612no—
248C2×D4—612no—
249C2×D4—2441no—
250C2×D4—2441no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—81no—
256C22×C4—611no—
257C22×C4—611no—
258C22×C4—611no—
259C22×C4—1221no—
260C22×C4—1221no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4—4881no—
264C22×C4—4881no—
265C22×C4—2442no—
266C22×C4—2442no—
267C22×C4—2442no—
268C22×C4C22⋊#692311yescharacteristic
269C22×C4C22×#692311yescharacteristic
270C2×C8—2442no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—4881no—
275C4⋊C4—4882no—
276C4⋊C4—2441no—
277C4⋊C4—2442no—
278C22⋊C4—42no—
279C22⋊C4—4881no—
280C22⋊C4—2441no—
281C22⋊C4—2442no—
282C22⋊C4—2441no—
283C22⋊C4—2442no—
284C22⋊C4—2442no—
285C42—1221no—
286C42—1221no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—611no—
292C23—611no—
293C23—1221no—
294C23—1221no—
295C23—2441no—
296C23—2441no—
297C23#1451984012yes—
298C23C2×C22⋊#692311yescharacteristic
299Q8—82no—
300D4—2442no—
301D4—2442no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—612no—
314C2×C4—1221no—
315C2×C4—1221no—
316C2×C4—1222no—
317C2×C4—1221no—
318C2×C4—1221no—
319C2×C4—4881no—
320C2×C4—4882no—
321C2×C4—4882no—
322C2×C4—4882no—
323C2×C4—4884no—
324C2×C4—2442no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4—2441no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4#1451983511yescharacteristic
333C2×C4#1451983511yescharacteristic
334C8—4882no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—41no—
341C22—41no—
342C22—1221no—
343C22—1221no—
344C22—4881no—
345C22—2441no—
346C22—2441no—
347C22—2441no—
348C22—2441no—
349C22C2×#1451983511yescharacteristic
350C22C2×#1451984011yescharacteristic
351C22#1684076111yescharacteristic
352C4—21no—
353C4—21no—
354C4—41no—
355C4—81no—
356C4—81no—
357C4—1222no—
358C4—9762no—
359C4—4881no—
360C4—2442no—
361C4—2441no—
362C4—2441no—
363C2—21no—
364C2—21no—
365C2—42no—
366C2—2441no—
367C2—2441no—
368C2#2146125911yescharacteristic
369C2#2146207411yescharacteristic
370C2#2146209011yescharacteristic
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#11902
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. #21460816C2semidirect
4. #21460814C2semidirect
13. #16839700C4semidirect
15. #16839681C4semidirect
21. C2×#14519684C4semidirect
22. C2×C22⋊Q8×C61C4semidirect
191. C2×C22⋊Q8#6923semidirect
208. C61#11902semidirect

Maximal subgroups

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

subgroupindex
#214616792
#214610582
#214608162
#214608142
#214510012
#1190261