Finite Group Databasebeta

← #29576067

Subgroups — #29576067

Order 15616.

612 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 119 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#21448417C211yesmaximal, characteristic
2#21444391C212yesmaximal, complemented
3C2×#16839486C211yesmaximal, characteristic
4#21443686C212yesmaximal, complemented
5C61×#2626C211yesmaximal, characteristic
6C2×C22⋊#92418C2212yes—
7#16839655—24no—
8#16839622C2212yes—
9#16839618—21no—
10C4×C22⋊Dic61—21no—
11#16839499—24no—
12#16839486—22no—
13C2×C22⋊Q8×C61C2211yescharacteristic
14C22×C22⋊C4×C61C2211yescharacteristic
15C61×#399C412yescomplemented
16C61×#396—22no—
17C61×#393C412yescomplemented
18C61×#385—21no—
19C61×#380C2211yescharacteristic
20C61×#379—21no—
21C22×#92418—22no—
22C2×C4⋊Dic61—22no—
23C2×C22⋊Dic61—22no—
24C2×C22⋊Dic61—22no—
25C2×C22⋊Dic61—22no—
26C2×C4×Dic61—22no—
27C22⋊#92418—22no—
28C22⋊#92418—22no—
29#14519684—22no—
30#14519684D414yes—
31C25×C61D411yescharacteristic
32C22×Q8×C61C2×C411yescharacteristic
33C23×C4×C61D411yescharacteristic
34C23×C4×C61C2311yescharacteristic
35C22⋊Q8×C61—22no—
36C22⋊Q8×C61—22no—
37C2×C4⋊C4×C61—21no—
38C2×C4⋊C4×C61—22no—
39C2×C4⋊C4×C61—21no—
40C2×C4⋊C4×C61C2×C411yescharacteristic
41C2×C22⋊C4×C61—22no—
42C2×C22⋊C4×C61—22no—
43C2×C22⋊C4×C61—21no—
44C2×C22⋊C4×C61—21no—
45C2×C22⋊C4×C61—22no—
46C2×C22⋊C4×C61—22no—
47C2×C22⋊C4×C61C2×C412yes—
48C2×C22⋊C4×C61D412yes—
49#95×C61—22no—
50#95×C61C2×C412yes—
51C24×C61—21no—
52C24×C61—21no—
53C24×C61—22no—
54C24×C61—22no—
55C24×C61—81no—
56C24×C61—81no—
57C24×C61C22⋊C412yes—
58C24×C61C2×D411yescharacteristic
59C2×Q8×C61—22no—
60C2×Q8×C61—81no—
61C22×C4×C61—21no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—22no—
65C22×C4×C61—21no—
66C22×C4×C61—81no—
67C22×C4×C61—81no—
68C22×C4×C61C22⋊C412yes—
69C22×C4×C61C22×C411yescharacteristic
70C22×C4×C61C2×D412yes—
71C22×C4×C61C2×D411yescharacteristic
72C22×C4×C61#4112yes—
73C22×Dic61—24no—
74C4⋊C4×C61—22no—
75C4⋊C4×C61—42no—
76C4⋊C4×C61—81no—
77C4⋊C4×C61—82no—
78C22⋊C4×C61—24no—
79C22⋊C4×C61—24no—
80C22⋊C4×C61—41no—
81C22⋊C4×C61—41no—
82C22⋊C4×C61—42no—
83C22⋊C4×C61—82no—
84C22⋊C4×C61—82no—
85C22⋊Dic61—42no—
86C22⋊Dic61—42no—
87C22⋊Dic61—82no—
88C22⋊Dic61—82no—
89C4⋊Dic61—82no—
90C4×Dic61—82no—
91C2×#92418—42no—
92C2×#92418—82no—
93C23×C61—22no—
94C23×C61—22no—
95C23×C61—22no—
96C23×C61—41no—
97C23×C61—41no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61—42no—
101C23×C61—42no—
102C23×C61—42no—
103C23×C61—42no—
104C23×C61—81no—
105C23×C61—81no—
106C23×C61—84no—
107C23×C61—84no—
108C23×C61C2×C22⋊C411yescharacteristic
109C23×C61#12011yescharacteristic
110C23×C61#12311yescharacteristic
111C23×C61#13512yes—
112C23×C61#13712yes—
113Q8×C61—82no—
114C2×C4×C61—22no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61—42no—
122C2×C4×C61—42no—
123C2×C4×C61—42no—
124C2×C4×C61—42no—
125C2×C4×C61—41no—
126C2×C4×C61—41no—
127C2×C4×C61—42no—
128C2×C4×C61—81no—
129C2×C4×C61C4×D412yes—
130C2×C4×C61#12112yes—
131C2×Dic61—44no—
132C2×Dic61—44no—
133C2×Dic61—44no—
134#92418—82no—
135#9201—611nomaximal, Sylow
136C22×C61—21no—
137C22×C61—21no—
138C22×C61—21no—
139C22×C61—21no—
140C22×C61—21no—
141C22×C61—21no—
142C22×C61—42no—
143C22×C61—42no—
144C22×C61—44no—
145C22×C61—42no—
146C22×C61—42no—
147C22×C61—42no—
148C22×C61—82no—
149C22×C61—84no—
150C22×C61—84no—
151C22×C61#38611yescharacteristic
152C22×C61#41611yescharacteristic
153C22×C61#41911yescharacteristic
154C22×C61#45212yes—
155C22×C61#48512yes—
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—42no—
159C4×C61—82no—
160C4×C61—81no—
161C4×C61—81no—
162Dic61—84no—
163#2626D6111yescharacteristic
164#2312—611no—
165#1520—612no—
166C2×#323—611no—
167#1289—612no—
168C2×C61—22no—
169C2×C61—21no—
170C2×C61—21no—
171C2×C61—22no—
172C2×C61—82no—
173C2×C61#187311yescharacteristic
174C2×C61#191111yescharacteristic
175C2×C61#191311yescharacteristic
176C2×C22⋊Q8C2×D6111yescharacteristic
177C22×C22⋊C4C2×D6111yescharacteristic
178C2×C22⋊C8—612no—
179#399Dic6112yescomplemented
180#396—22no—
181#393Dic6112yescomplemented
182#388—1224no—
183#385—21no—
184#380—612no—
185#380C2×D6111yescharacteristic
186#379—21no—
187#379—1221no—
188C4×C22⋊C4—1221no—
189#328—1224no—
190#323—1222no—
191C61#920111yescharacteristic, complemented, Sylow
192C25C22⋊D6111yescharacteristic
193C22×Q8C2×Dic6111yescharacteristic
194C23×C4C22⋊D6111yescharacteristic
195C23×C4C22×D6111yescharacteristic
196C22×C8—1222no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—1222no—
203C2×C4⋊C4C2×Dic6111yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—1222no—
211C2×C22⋊C4—1222no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4C2×Dic6112yes—
214C2×C22⋊C4C22⋊D6112yes—
215C2×C42—1222no—
216C22⋊C8—1222no—
217C22⋊C8—1222no—
218#95—22no—
219#95—614no—
220#95—1222no—
221#95C2×Dic6112yes—
222C24—21no—
223C24—22no—
224C24—21no—
225C24—22no—
226C24—81no—
227C24—81no—
228C24C22⋊Dic6112yes—
229C24C2×C22⋊D6111yescharacteristic
230C2×Q8—22no—
231C2×Q8—81no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—1224no—
240C22×C4C22⋊Dic6112yes—
241C22×C4D4×D6112yes—
242C22×C4#1175822412yes—
243C22×C4C22×Dic6111yescharacteristic
244C22×C4C2×C22⋊D6111yescharacteristic
245C2×C8—4882no—
246C2×C8—2442no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—81no—
250C4⋊C4—82no—
251C4⋊C4—4882no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—82no—
258C22⋊C4—82no—
259C22⋊C4—4882no—
260C22⋊C4—4882no—
261C22⋊C4—2442no—
262C22⋊C4—2442no—
263C42—4882no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23C2×C22⋊Dic6111yescharacteristic
280C23#1451974511yescharacteristic
281C23#1451975811yescharacteristic
282C23#1451979712yes—
283C23#1451980812yes—
284Q8—82no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—2444no—
301C2×C4—2444no—
302C2×C4—2444no—
303C2×C4D4×Dic6112yes—
304C2×C4#1451974912yes—
305C8—4882no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#1683964211yescharacteristic
322C22#1683975811yescharacteristic
323C22#1683977111yescharacteristic
324C22#1683992212yes—
325C22#1684010012yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—82no—
330C4—81no—
331C4—81no—
332C4—4884no—
333C2—21no—
334C2—22no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#2144613211yescharacteristic
339C2#2144635211yescharacteristic
340C2#2144636211yescharacteristic
subgroupG/H
centerC22#16839642
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#2626C2
socleC22×C61#386
prime radicalC25×C61D4
cosocleC23×C4×C61C23
hypercenter#2626D61
hypocenterC61#9201
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2626

Complements

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

NH ≅ G/Ntype
2. #21444391C2semidirect
4. #21443686C2semidirect
15. C61×#399C4semidirect
17. C61×#393C4semidirect
179. #399Dic61semidirect
181. #393Dic61semidirect
191. C61#9201semidirect

Maximal subgroups

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

subgroupindex
#214484172
#214443912
C2×#168394862
#214436862
C61×#26262
#920161