Finite Group Databasebeta

← #30595556

Subgroups — #30595556

Order 15616.

506 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 98 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#21461710C211yesmaximal, characteristic
2#21461545C211yesmaximal, characteristic
3#21460639C212yesmaximal, complemented
4#21460637C212yesmaximal, complemented
5C22⋊Q8×Dic61C211yesmaximal, characteristic
6C2×C22⋊#92420—21no—
7C2×C22⋊#92420—21no—
8C2×#14519832—21no—
9#16840628—21no—
10#16840628C2212yes—
11#16840628C2212yes—
12C2×Q8×Dic61C411yescharacteristic
13C22×C4×Dic61C2211yescharacteristic
14#16839669C2211yescharacteristic
15#16839661C2211yescharacteristic
16#16839648—21no—
17#16839637—21no—
18#16839622C411yescharacteristic
19C4⋊C4×Dic61—21no—
20C4⋊C4×Dic61C411yescharacteristic
21C22⋊C4×Dic61—21no—
22C2×C22⋊Q8×C61C411yescharacteristic
23C22×#92420—22no—
24C22×#92420—22no—
25C22⋊#92420—24no—
26C22⋊#92420—42no—
27#14519832—42no—
28C23×Dic61D411yescharacteristic
29C23×Dic61D411yescharacteristic
30Q8×Dic61—42no—
31C2×C4⋊Dic61—21no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61C2×C411yescharacteristic
34C2×C4⋊Dic61C2×C411yescharacteristic
35C2×C22⋊Dic61—21no—
36C2×C4×Dic61—21no—
37C2×C4×Dic61—21no—
38C2×C4×Dic61—42no—
39C2×C4×Dic61C2×C411yescharacteristic
40C2×C4×Dic61D411yescharacteristic
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61C2311yescharacteristic
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48C22×Q8×C61C2×C411yescharacteristic
49C23×C4×C61C2×C411yescharacteristic
50C22⋊Q8×C61—22no—
51C22⋊Q8×C61—22no—
52C2×C4⋊C4×C61—21no—
53C2×C4⋊C4×C61C2×C411yescharacteristic
54C2×C22⋊C4×C61—21no—
55C24×C61C22⋊C411yescharacteristic
56C2×Q8×C61—21no—
57C2×Q8×C61—41no—
58C2×Q8×C61—41no—
59C2×Q8×C61M1612yes—
60C22×C4×C61—21no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—41no—
65C22×C4×C61—41no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—22no—
69C22×Dic61—22no—
70C22×Dic61—22no—
71C22×Dic61—22no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—41no—
75C22×Dic61—41no—
76C22×Dic61C22⋊C411yescharacteristic
77C22×Dic61C22⋊C411yescharacteristic
78C22×Dic61C2×D411yescharacteristic
79C22×Dic61C2×D411yescharacteristic
80C2×#92420—24no—
81C2×#92420—42no—
82C2×#92420—42no—
83C2×#92420—42no—
84C2×#92420—42no—
85C2×#92420—42no—
86C4⋊C4×C61—24no—
87C4⋊C4×C61—21no—
88C4⋊C4×C61M1612yes—
89C22⋊C4×C61—24no—
90C22⋊Dic61—42no—
91C4⋊Dic61—42no—
92C4⋊Dic61—42no—
93C4⋊Dic61—41no—
94C4⋊Dic61—41no—
95C4×Dic61—22no—
96C4×Dic61—22no—
97C4×Dic61—41no—
98C4×Dic61—41no—
99C4×Dic61—42no—
100C4×Dic61—42no—
101C23×C61—21no—
102C23×C61—21no—
103C23×C61—21no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61#10412yes—
108C23×C61C2×C22⋊C411yescharacteristic
109Q8×C61—42no—
110Q8×C61—42no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—22no—
114C2×C4×C61—22no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61C2×M1611yescharacteristic
125C2×C4×C61C2×M1611yescharacteristic
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—21no—
135C2×Dic61—44no—
136C2×Dic61—44no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61—41no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61#9912yes—
144C2×Dic61#13512yes—
145C2×Dic61#13712yes—
146#92420—44no—
147#92420—82no—
148#7623—611nomaximal, Sylow
149C22×C61—22no—
150C22×C61—22no—
151C22×C61—22no—
152C22×C61—41no—
153C22×C61—41no—
154C22×C61—41no—
155C22×C61—41no—
156C22×C61#40711yescharacteristic
157C22×C61C2×#9911yescharacteristic
158C22×C61#41311yescharacteristic
159C22×C61#41611yescharacteristic
160C22×C61#41911yescharacteristic
161C22×C61C2×#10411yescharacteristic
162C22×C61#42111yescharacteristic
163C4×C61—21no—
164C4×C61—21no—
165C4×C61—41no—
166C4×C61—42no—
167C4×C61—42no—
168Dic61—22no—
169Dic61—22no—
170Dic61—22no—
171Dic61—41no—
172Dic61—81no—
173Dic61—81no—
174C4×C22⋊Q8—611no—
175#1817—611no—
176#1749—611no—
177#1272—612no—
178#1271—612no—
179C2×C61—42no—
180C2×C61#146511yescharacteristic
181C2×C61#148811yescharacteristic
182C2×C61#149111yescharacteristic
183C2×C61#150111yescharacteristic
184C2×C61#151111yescharacteristic
185C2×C61#153211yescharacteristic
186C2×C61#155411yescharacteristic
187C2×C22⋊Q8#692311yescharacteristic
188C2×C4×Q8—611no—
189C22×C42—611no—
190C2×C22⋊C8—1221no—
191C2×C22⋊C8—1221no—
192C2×#97—1221no—
193#391—611no—
194#389—611no—
195#387—1221no—
196#385—1221no—
197#380—611no—
198C4×C4⋊C4—611no—
199C4×C4⋊C4—1221no—
200C4×C22⋊C4—1221no—
201#336—612no—
202#336—612no—
203#336—1221no—
204C61#762311yescharacteristic, complemented, Sylow
205C22×Q8C2×#692311yescharacteristic
206C23×C4—611no—
207C23×C4—611no—
208C23×C4C2×#692311yescharacteristic
209C22×C8—1222no—
210C22×C8—1222no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C4×Q8—2442no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—611no—
216C2×C4⋊C4—611no—
217C2×C4⋊C4—1221no—
218C2×C4⋊C4—1221no—
219C2×C4⋊C4C2×#692311yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—1221no—
222C2×C42—611no—
223C2×C42—611no—
224C2×C42—611no—
225C2×C42—611no—
226C2×C42—1221no—
227C2×C42—1221no—
228C2×C42—2442no—
229C22⋊C8—1224no—
230C22⋊C8—2442no—
231#97—2442no—
232#95—1221no—
233#95—1221no—
234#95—1221no—
235#95—1221no—
236#95—1221no—
237C24C22⋊#692311yescharacteristic
238C2×Q8—21no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8#1175821712yes—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—611no—
249C22×C4—611no—
250C22×C4—611no—
251C22×C4—611no—
252C22×C4—1222no—
253C22×C4—1222no—
254C22×C4—1222no—
255C22×C4—1222no—
256C22×C4—1221no—
257C22×C4—1221no—
258C22×C4—2441no—
259C22×C4—2441no—
260C22×C4C22⋊#692311yescharacteristic
261C22×C4C22×#692311yescharacteristic
262C2×C8—1224no—
263C2×C8—2442no—
264C2×C8—2442no—
265C2×C8—2442no—
266C2×C8—2442no—
267C2×C8—2442no—
268C4⋊C4—24no—
269C4⋊C4—21no—
270C4⋊C4—2441no—
271C4⋊C4—2441no—
272C4⋊C4—2442no—
273C4⋊C4—2442no—
274C4⋊C4#1175821712yes—
275C22⋊C4—24no—
276C22⋊C4—2442no—
277C42—1222no—
278C42—1222no—
279C42—2441no—
280C42—2441no—
281C42—2442no—
282C42—2442no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23#1451984212yes—
290C23C2×C22⋊#692311yescharacteristic
291Q8—42no—
292Q8—42no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—612no—
307C2×C4—612no—
308C2×C4—612no—
309C2×C4—1221no—
310C2×C4—1221no—
311C2×C4—1221no—
312C2×C4—1221no—
313C2×C4—1221no—
314C2×C4—1221no—
315C2×C4—1221no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—2444no—
319C2×C4—2444no—
320C2×C4—2441no—
321C2×C4—2441no—
322C2×C4—2441no—
323C2×C4—2441no—
324C2×C4—2441no—
325C2×C4—2441no—
326C2×C4C2×#1175821711yescharacteristic
327C2×C4C2×#1175821711yescharacteristic
328C8—4882no—
329C8—2444no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22#1684073211yescharacteristic
338C22D61⋊#9911yescharacteristic
339C22#1684074911yescharacteristic
340C22#1684075211yescharacteristic
341C22#1684075511yescharacteristic
342C22C2×#1451984211yescharacteristic
343C22#1684075911yescharacteristic
344C4—21no—
345C4—21no—
346C4—41no—
347C4—42no—
348C4—42no—
349C4—1222no—
350C4—1222no—
351C4—1222no—
352C4—4881no—
353C4—4881no—
354C4—2441no—
355C2—42no—
356C2#2146107511yescharacteristic
357C2#2146115011yescharacteristic
358C2#2146116111yescharacteristic
359C2#2146119711yescharacteristic
360C2#2146122811yescharacteristic
361C2#2146129711yescharacteristic
362C2#2146134511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#7623
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21460639C2semidirect
4. #21460637C2semidirect
204. C61#7623semidirect

Maximal subgroups

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

subgroupindex
#214617102
#214615452
#214606392
#214606372
C22⋊Q8×Dic612
#762361