Finite Group Databasebeta

← #30598724

Subgroups — #30598724

Order 15616.

910 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 96 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#21461352C212yesmaximal, complemented
2#21461298C212yesmaximal, complemented
3#21461072C212yesmaximal, complemented
4C2×#16840332C211yesmaximal, characteristic
5#16840782C2214yes—
6#16840622—24no—
7#16840617—24no—
8#16840608—22no—
9#16840606—22no—
10C22×#11758224C411yescharacteristic
11C22×Q8×D61C411yescharacteristic
12#16840332—24no—
13#16840332—24no—
14C2×#14519777C411yescharacteristic
15C2×#14519775C2211yescharacteristic
16C2×#14519764C2211yescharacteristic
17C2×#14519759—22no—
18C2×#14519754—22no—
19C2×#14519734C2211yescharacteristic
20C2×#124×C61C411yescharacteristic
21#14519844—24no—
22C22⋊#92420—24no—
23#14519832—24no—
24C2×#11758224—22no—
25C2×#11758224—22no—
26C2×#11758224—42no—
27C2×#11758224—42no—
28C22×Dic122—21no—
29C22×Dic122C2×C411yescharacteristic
30C22×Dic122C2×C411yescharacteristic
31C2×Q8×D61—22no—
32C2×Q8×D61—22no—
33C2×Q8×D61—42no—
34C2×Q8×D61—42no—
35C22×C22⋊D61—21no—
36C23×Dic61—21no—
37C22×C4×D61—21no—
38C22×C4×D61C2×C411yescharacteristic
39#14519777—22no—
40#14519777—22no—
41#14519775—22no—
42#14519775D414yes—
43#14519764—22no—
44#14519764D414yes—
45#14519759—216no—
46#14519754—216no—
47#14519734—22no—
48#14519734—22no—
49C2×C4⋊Dic61—22no—
50C2×#11758196—22no—
51C2×#11758196D412yes—
52C2×C22⋊Dic61—22no—
53C2×#11758198D412yes—
54C2×C4×Dic61C2311yescharacteristic
55C22×Q8×C61C2×C411yescharacteristic
56C22×D4×C61C2×C411yescharacteristic
57#124×C61—22no—
58#124×C61—22no—
59C2×C22⋊C4×C61—22no—
60C2×C42×C61C2×C411yescharacteristic
61C24×C61—21no—
62C23×D61C22⋊C411yescharacteristic
63C2×Q8×C61—21no—
64C2×Q8×C61—41no—
65C2×Q8×C61—41no—
66C2×Q8×C61C22⋊C412yes—
67C2×D4×C61—21no—
68C2×D4×C61—41no—
69C2×D4×C61—41no—
70C2×D4×C61C22⋊C412yes—
71C22×C4×C61—21no—
72C22×C4×C61C22⋊C412yes—
73C22×C4×C61C22×C411yescharacteristic
74C2×C22⋊D61—24no—
75C2×C22⋊D61—42no—
76C2×C22⋊D61—42no—
77C22×Dic61—24no—
78C22×Dic61—21no—
79C22×Dic61—21no—
80C22×Dic61—42no—
81C22×Dic61—42no—
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C2×D412yes—
84Q8×D61—48no—
85Q8×D61—48no—
86#11758224—48no—
87#11758224—48no—
88C2×C4×D61—22no—
89C2×C4×D61—21no—
90C2×C4×D61—21no—
91C2×C4×D61—22no—
92C2×C4×D61—24no—
93C2×C4×D61—42no—
94C2×C4×D61—42no—
95C2×Dic122—24no—
96C2×Dic122—21no—
97C2×Dic122—21no—
98C2×Dic122—42no—
99C2×Dic122—42no—
100C2×Dic122—41no—
101C2×Dic122—41no—
102C2×Dic122—41no—
103C2×Dic122—41no—
104C2×Dic122C22⋊C412yes—
105C2×Dic122C22⋊C412yes—
106C2×#92420—44no—
107C22⋊C4×C61—28no—
108C42×C61—21no—
109C42×C61—21no—
110C22⋊Dic61—28no—
111#11758198—24no—
112C4⋊Dic61—28no—
113#11758196—28no—
114#11758196—24no—
115C4×Dic61C2×D412yes—
116C4×Dic61C2×D412yes—
117C23×C61—22no—
118C23×C61—44no—
119C23×C61—41no—
120C23×C61—41no—
121C23×C61C2×C22⋊C411yescharacteristic
122C22×D61—21no—
123C22×D61—21no—
124C22×D61—42no—
125C22×D61#10012yes—
126Q8×C61—42no—
127Q8×C61—42no—
128D4×C61—42no—
129D4×C61—42no—
130C2×C4×C61—21no—
131C2×C4×C61—21no—
132C2×C4×C61—22no—
133C2×C4×C61—22no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61C2×C22⋊C411yescharacteristic
138C2×C4×C61C2×C22⋊C411yescharacteristic
139C22⋊D61—48no—
140C2×Dic61—21no—
141C2×Dic61—21no—
142C2×Dic61—22no—
143C2×Dic61—22no—
144C2×Dic61—42no—
145C2×Dic61—48no—
146C2×Dic61—41no—
147C2×Dic61—41no—
148C2×Dic61—41no—
149C2×Dic61—41no—
150C2×Dic61#10012yes—
151C2×Dic61#12014yes—
152C4×D61—24no—
153C4×D61—24no—
154C4×D61—48no—
155Dic122—48no—
156Dic122—42no—
157Dic122—42no—
158Dic122—42no—
159Dic122—42no—
160#92420—84no—
161#8524—611nomaximal, Sylow
162C22×C61—22no—
163C22×C61—44no—
164C22×C61—42no—
165C22×C61—82no—
166C22×C61—82no—
167C22×C61#37911yescharacteristic
168C22×C61C2×#10011yescharacteristic
169C22×C61C2×#10011yescharacteristic
170C2×D61—22no—
171C2×D61—42no—
172C2×D61—42no—
173C2×D61—42no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—42no—
177C4×C61—42no—
178Dic61—24no—
179Dic61—22no—
180Dic61—42no—
181Dic61—42no—
182C2×#536—611no—
183#1558—612no—
184#1533—612no—
185#1464—612no—
186C2×C61—21no—
187C2×C61—21no—
188C2×C61—82no—
189C2×C61#178311yescharacteristic
190C2×C61#179612yes—
191D61—42no—
192C22×#41—611no—
193C23×Q8—611no—
194#536—1224no—
195#536—1224no—
196C2×C4⋊Q8—611no—
197C2×C4⋊Q8—611no—
198C2×#124—611no—
199C2×#124#692311yescharacteristic
200C2×#123—1222no—
201C2×C22⋊Q8—1222no—
202C2×#117—611no—
203#432—614no—
204#333—1224no—
205#329—1224no—
206#324—1222no—
207#323—1222no—
208C61#852411yescharacteristic, complemented, Sylow
209C2×#41—1222no—
210C2×#41—1222no—
211C2×#41—2442no—
212C2×#41—2442no—
213C22×Q8—611no—
214C22×Q8—611no—
215C22×Q8—1221no—
216C22×Q8—1222no—
217C22×Q8—1222no—
218C22×Q8—2442no—
219C22×Q8—2442no—
220C22×Q8C2×#692311yescharacteristic
221C22×D4—1221no—
222C22×D4C2×#692311yescharacteristic
223C23×C4—611no—
224C23×C4—1221no—
225C23×C4—1221no—
226C4⋊Q8—614no—
227C4⋊Q8—1222no—
228C4⋊Q8—1222no—
229C4⋊Q8—1222no—
230#124—22no—
231#124—22no—
232#124—614no—
233#124—1222no—
234#123—12216no—
235C22⋊Q8—12216no—
236#117—1222no—
237#117—1222no—
238C2×C4⋊C4—612no—
239C2×C4⋊C4—1222no—
240C2×C4⋊C4—1222no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—612no—
243C2×C22⋊C4—1222no—
244C2×C42—611no—
245C2×C42C2×#692311yescharacteristic
246C4⋊C8—1224no—
247C22⋊C8—1224no—
248#97—1224no—
249C24—21no—
250C24—611no—
251#41—2448no—
252#41—2448no—
253C2×Q8—21no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—612no—
257C2×Q8—612no—
258C2×Q8—1224no—
259C2×Q8—1221no—
260C2×Q8—1221no—
261C2×Q8—2441no—
262C2×Q8—2441no—
263C2×Q8—2441no—
264C2×Q8—2441no—
265C2×Q8—2442no—
266C2×Q8—2442no—
267C2×Q8—2448no—
268C2×Q8—2448no—
269C2×Q8C22⋊#692312yes—
270C2×D4—21no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—1224no—
274C2×D4—2442no—
275C2×D4—2442no—
276C2×D4C22⋊#692312yes—
277C22×C4—21no—
278C22×C4—611no—
279C22×C4—612no—
280C22×C4—1222no—
281C22×C4—1222no—
282C22×C4—1224no—
283C22×C4—1221no—
284C22×C4—1221no—
285C22×C4—1224no—
286C22×C4—1221no—
287C22×C4—1221no—
288C22×C4—2442no—
289C22×C4—2442no—
290C22×C4—2442no—
291C22×C4—2442no—
292C22×C4C22⋊#692312yes—
293C22×C4C22×#692311yescharacteristic
294C2×C8—2444no—
295C4⋊C4—1224no—
296C4⋊C4—1228no—
297C4⋊C4—1228no—
298C22⋊C4—28no—
299C22⋊C4—1224no—
300C22⋊C4—1228no—
301C42—21no—
302C42—21no—
303C42—612no—
304C42—612no—
305C23—22no—
306C23—44no—
307C23—41no—
308C23—41no—
309C23—612no—
310C23—1221no—
311C23—1221no—
312C23—2442no—
313C23C2×C22⋊#692311yescharacteristic
314Q8—42no—
315Q8—42no—
316Q8—2442no—
317Q8—2442no—
318Q8—2442no—
319Q8—2442no—
320Q8—2448no—
321D4—42no—
322D4—42no—
323D4—2448no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—612no—
332C2×C4—614no—
333C2×C4—1224no—
334C2×C4—1224no—
335C2×C4—1221no—
336C2×C4—1221no—
337C2×C4—1222no—
338C2×C4—1222no—
339C2×C4—2448no—
340C2×C4—2442no—
341C2×C4—2448no—
342C2×C4—2441no—
343C2×C4—2441no—
344C2×C4—2441no—
345C2×C4—2441no—
346C2×C4C2×C22⋊#692311yescharacteristic
347C2×C4C2×C22⋊#692311yescharacteristic
348C8—4884no—
349C22—22no—
350C22—42no—
351C22—44no—
352C22—82no—
353C22—82no—
354C22—1222no—
355C22—2442no—
356C22—2442no—
357C22—2442no—
358C22#1684070311yescharacteristic
359C22D61⋊#10011yescharacteristic
360C22D61⋊#10011yescharacteristic
361C4—21no—
362C4—21no—
363C4—42no—
364C4—42no—
365C4—1222no—
366C4—1224no—
367C4—2442no—
368C4—2442no—
369C2—21no—
370C2—21no—
371C2—82no—
372C2—2442no—
373C2#2146165211yescharacteristic
374C2#2146167212yes—
subgroupG/H
centerC22#16840703
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#124×C61C4
socleC22×C61#379
prime radicalC22×#11758224C4
cosocleC2×C4×Dic61C23
hypercenterC2×#124#6923
hypocenterC61#8524
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. #21461352C2semidirect
2. #21461298C2semidirect
3. #21461072C2semidirect
208. C61#8524semidirect

Maximal subgroups

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

subgroupindex
#214613522
#214612982
#214610722
C2×#168403322
#852461