Finite Group Databasebeta

← #30604165

Subgroups — #30604165

Order 15616.

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

normal subgroups only

This group has 470 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#21461079C211yesmaximal, characteristic
2#21461075C211yesmaximal, characteristic
3#21460763C212yesmaximal, complemented
4#21460745C212yesmaximal, complemented
5#123×Dic61C211yesmaximal, characteristic
6#16840782—21no—
7#16840732—21no—
8#16840607—21no—
9#16840605—21no—
10#16840605C2212yes—
11#16840605C2212yes—
12C2×D4×Dic61C2211yescharacteristic
13C22×C4×Dic61C411yescharacteristic
14#16839655—21no—
15#16839648—21no—
16#16839642C411yescharacteristic
17#16839637C2211yescharacteristic
18#16839625C2211yescharacteristic
19C4⋊C4×Dic61—21no—
20C22⋊C4×Dic61—21no—
21C22⋊C4×Dic61C411yescharacteristic
22C2×#123×C61C411yescharacteristic
23C2×#11758217—41no—
24#14519844—41no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C22⋊#92420—41no—
28#14519832—41no—
29C23×Dic61C2×C411yescharacteristic
30C23×Dic61C2×C411yescharacteristic
31C23×Dic61D411yescharacteristic
32D4×Dic61—41no—
33D4×Dic61—41no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61D411yescharacteristic
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61C2×C411yescharacteristic
38C2×C22⋊Dic61D411yescharacteristic
39C2×C22⋊Dic61C2311yescharacteristic
40C2×C4×Dic61—21no—
41C2×C4×Dic61—21no—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—41no—
44C2×C4×Dic61—41no—
45C2×C4×Dic61D411yescharacteristic
46#14519684—21no—
47#14519684—21no—
48#14519684—21no—
49#14519684—21no—
50#14519684—21no—
51C22×D4×C61C2×C411yescharacteristic
52C23×C4×C61C2×C411yescharacteristic
53#123×C61—22no—
54#123×C61—22no—
55C2×C4⋊C4×C61—21no—
56C2×C22⋊C4×C61—21no—
57C2×C22⋊C4×C61C2×C411yescharacteristic
58C24×C61C22⋊C411yescharacteristic
59C24×C61C22×C411yescharacteristic
60C2×D4×C61—21no—
61C2×D4×C61—21no—
62C2×D4×C61—41no—
63C2×D4×C61—41no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—21no—
67C22×C4×C61—41no—
68C22×C4×C61—41no—
69C22×C4×C61C22⋊C411yescharacteristic
70C22×C4×C61M1612yescomplemented
71C22×Dic61—21no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61—21no—
75C22×Dic61—21no—
76C22×Dic61—21no—
77C22×Dic61—42no—
78C22×Dic61—42no—
79C22×Dic61—42no—
80C22×Dic61—41no—
81C22×Dic61C22⋊C411yescharacteristic
82C22×Dic61C22⋊C411yescharacteristic
83C22×Dic61C2×D411yescharacteristic
84C22×Dic61C2×D411yescharacteristic
85#11758217—82no—
86C2×#92420—42no—
87C2×#92420—42no—
88C4⋊C4×C61—24no—
89C22⋊C4×C61—24no—
90C22⋊C4×C61—21no—
91C22⋊C4×C61M1612yescomplemented
92C22⋊Dic61—41no—
93C22⋊Dic61—41no—
94C22⋊Dic61—42no—
95C22⋊Dic61—41no—
96C4⋊Dic61—41no—
97C4⋊Dic61—42no—
98C4×Dic61—21no—
99C4×Dic61—21no—
100C4×Dic61—42no—
101C4×Dic61—42no—
102C4×Dic61—41no—
103C4×Dic61—41no—
104C23×C61—21no—
105C23×C61—21no—
106C23×C61—21no—
107C23×C61—21no—
108C23×C61—21no—
109C23×C61—41no—
110C23×C61—41no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61C2×C22⋊C411yescharacteristic
114C23×C61C2×M1611yescharacteristic
115C23×C61C2×M1611yescharacteristic
116D4×C61—42no—
117D4×C61—42no—
118C2×C4×C61—22no—
119C2×C4×C61—22no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61—42no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—42no—
129C2×Dic61—22no—
130C2×Dic61—22no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—21no—
134C2×Dic61—42no—
135C2×Dic61—41no—
136C2×Dic61—42no—
137C2×Dic61—41no—
138C2×Dic61—42no—
139C2×Dic61—42no—
140C2×Dic61—41no—
141C2×Dic61—41no—
142C2×Dic61—41no—
143C2×Dic61—42no—
144C2×Dic61—41no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—81no—
148C2×Dic61#9911yescharacteristic
149C2×Dic61#9911yescharacteristic
150#92420—82no—
151#92420—82no—
152#11526—611nomaximal, Sylow
153C22×C61—21no—
154C22×C61—21no—
155C22×C61—21no—
156C22×C61—21no—
157C22×C61—41no—
158C22×C61—41no—
159C22×C61—42no—
160C22×C61—42no—
161C22×C61—41no—
162C22×C61—41no—
163C22×C61—41no—
164C22×C61—41no—
165C22×C61—41no—
166C22×C61—81no—
167C22×C61#40711yescharacteristic
168C22×C61C2×#9911yescharacteristic
169C22×C61#41311yescharacteristic
170C4×C61—42no—
171C4×C61—41no—
172C4×C61—42no—
173Dic61—21no—
174Dic61—41no—
175Dic61—41no—
176Dic61—41no—
177Dic61—42no—
178Dic61—81no—
179Dic61#35212yes—
180C4×#123—611no—
181#1466—611no—
182#1465—611no—
183#1326—612no—
184#1317—612no—
185C2×C61—21no—
186C2×C61—21no—
187C2×C61—41no—
188C2×C61—41no—
189C2×C61—81no—
190C2×C61#146511yescharacteristic
191C2×C61C2×#35211yescharacteristic
192C2×C61#212011yescharacteristic
193C2×#123#692311yescharacteristic
194C2×C4×D4—611no—
195C22×C42—611no—
196#432—1221no—
197#407—1221no—
198#388—1221no—
199#387—1221no—
200#386—611no—
201#385—611no—
202#381—611no—
203C4×C4⋊C4—1221no—
204C4×C22⋊C4—611no—
205C4×C22⋊C4—1221no—
206#324—1221no—
207#323—612no—
208#323—612no—
209#323—1221no—
210C61#1152611yescharacteristic, complemented, Sylow
211C22×D4C2×#692311yescharacteristic
212C23×C4—611no—
213C23×C4—611no—
214C23×C4—611no—
215C23×C4C2×#692311yescharacteristic
216C2×M16—2441no—
217#123—22no—
218#123—22no—
219C4×D4—2441no—
220C4×D4—2441no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—611no—
223C2×C4⋊C4—1221no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—611no—
226C2×C22⋊C4—611no—
227C2×C22⋊C4—611no—
228C2×C22⋊C4—1221no—
229C2×C22⋊C4C2×#692311yescharacteristic
230C2×C42—611no—
231C2×C42—1221no—
232C2×C42—1221no—
233C2×C42—1221no—
234C2×C42—2441no—
235C2×C42—2441no—
236C4⋊C8—2441no—
237C22⋊C8—1222no—
238C22⋊C8—1222no—
239C22⋊C8—2441no—
240#97—2441no—
241#95—1221no—
242#95—1221no—
243#95—1221no—
244#95—1221no—
245#95—1221no—
246C24C22⋊#692311yescharacteristic
247C24C22×#692311yescharacteristic
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—611no—
258C22×C4—611no—
259C22×C4—611no—
260C22×C4—611no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1221no—
266C22×C4—1221no—
267C22×C4—2442no—
268C22×C4—2442no—
269C22×C4—2442no—
270C22×C4—2441no—
271C22×C4#1175821712yescomplemented
272C22×C4C22⋊#692311yescharacteristic
273M16—4882no—
274C2×C8—2442no—
275C2×C8—2442no—
276C4⋊C4—24no—
277C4⋊C4—2441no—
278C4⋊C4—2442no—
279C22⋊C4—24no—
280C22⋊C4—21no—
281C22⋊C4—2441no—
282C22⋊C4—2441no—
283C22⋊C4—2442no—
284C22⋊C4—2441no—
285C22⋊C4#1175821712yescomplemented
286C42—1221no—
287C42—1221no—
288C42—2441no—
289C42—2441no—
290C42—2442no—
291C42—2442no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23C2×C22⋊#692311yescharacteristic
302C23C2×#1175821711yescharacteristic
303C23C2×#1175821711yescharacteristic
304D4—42no—
305D4—42no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—611no—
318C2×C4—611no—
319C2×C4—1222no—
320C2×C4—1222no—
321C2×C4—1221no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—4881no—
325C2×C4—2442no—
326C2×C4—2442no—
327C2×C4—2442no—
328C2×C4—2441no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2442no—
333C2×C4—2441no—
334C2×C4—2441no—
335C2×C4—2441no—
336C2×C4—2441no—
337C2×C4—2442no—
338C8—4882no—
339C8—4882no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—41no—
345C22—41no—
346C22—42no—
347C22—42no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22#1684073211yescharacteristic
355C22D61⋊#9911yescharacteristic
356C22#1684074911yescharacteristic
357C4—42no—
358C4—41no—
359C4—42no—
360C4—612no—
361C4—1221no—
362C4—4881no—
363C4—2441no—
364C4—2441no—
365C4—2441no—
366C4—2442no—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—81no—
372C2#2146107511yescharacteristic
373C2D61⋊#35211yescharacteristic
374C2#2146207311yescharacteristic
subgroupG/H
centerC22D61⋊#99
commutatorC24×C61C22×C4
FrattiniC24C22×#6923
FittingC2×#123×C61C4
socleC22×C61C2×#99
prime radicalC22×D4×C61C2×C4
cosocleC2×C22⋊Dic61C23
hypercenterC2×#123#6923
hypocenterC61#11526
LCSG ▷ C24×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. #21460763C2semidirect
4. #21460745C2semidirect
70. C22×C4×C61M16semidirect
91. C22⋊C4×C61M16semidirect
210. C61#11526semidirect
271. C22×C4#11758217semidirect
285. C22⋊C4#11758217semidirect

Maximal subgroups

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

subgroupindex
#214610792
#214610752
#214607632
#214607452
#123×Dic612
#1152661