Finite Group Databasebeta

← #30600523

Subgroups — #30600523

Order 15616.

478 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 80 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#21462717C211yesmaximal, characteristic, complemented
2#21461572C211yesmaximal, characteristic, complemented
3#21461278C212yesmaximal
4#21460678C212yesmaximal, complemented
5#21451684C211yesmaximal, characteristic
6#16840812—24no—
7#16840783—21no—
8#16840783C2212yes—
9#16840778—21no—
10C2×#14519838—21no—
11C4×#11758217C2212yes—
12#16840642—21no—
13#16840617—24no—
14C2×#14519764C2211yescharacteristic
15C2×#14519742C2211yescharacteristic
16#16839698—21no—
17#16839683C411yescharacteristic
18#16839682C411yescharacteristic
19#16839676C411yescharacteristic
20#16839674—21no—
21#16839656—21no—
22#16839655—21no—
23#16839629C2211yescharacteristic
24C61×#394C411yescharacteristic
25C2×#11758217—22no—
26C4⋊#92420—22no—
27C4⋊#92420—22no—
28C4⋊#92420—22no—
29#14519838—24no—
30C22⋊#92420—22no—
31#14519832—22no—
32#14519832—22no—
33#14519832D414yes—
34C4×#92420—22no—
35C4×#92420—22no—
36C22×Dic122C2×C411yescharacteristic
37C22×Dic122D411yescharacteristic
38#14519764—22no—
39#14519764—22no—
40#14519742—22no—
41#14519742—22no—
42C2×C4⋊Dic61—21no—
43C2×#11758196—21no—
44C2×#11758196C2×C411yescharacteristic
45C2×#11758196D411yescharacteristic
46C2×C22⋊Dic61—21no—
47C2×C22⋊Dic61—21no—
48C2×C22⋊Dic61—21no—
49C2×C22⋊Dic61—21no—
50C2×C4×Dic61D411yescharacteristic
51C2×C4×Dic61D411yescharacteristic
52C2×C4×Dic61C2311yescharacteristic
53#14519684—21no—
54#14519684—21no—
55#14519684—21no—
56#14519684—21no—
57#14519684C2×C411yescharacteristic
58C22×D4×C61C2×C411yescharacteristic
59C2×C4⋊C4×C61C2×C411yescharacteristic
60C2×C22⋊C4×C61—21no—
61C2×C22⋊C4×C61—21no—
62#95×C61C2×C411yescharacteristic
63C24×C61—21no—
64C2×D4×C61—22no—
65C2×D4×C61—81no—
66C22×C4×C61—21no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22⋊C411yescharacteristic
69C22×C4×C61C22×C411yescharacteristic
70C22×Dic61—21no—
71C22×Dic61—21no—
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C22⋊C411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C22×Dic61C2×D411yescharacteristic
76C2×Dic122—22no—
77C2×Dic122—22no—
78C2×Dic122—22no—
79C2×Dic122—41no—
80C2×Dic122—81no—
81#11758217—24no—
82#11758217—42no—
83C2×#92420—24no—
84C2×#92420—42no—
85C4⋊C4×C61—22no—
86C22⋊C4×C61—42no—
87C22⋊C4×C61—81no—
88C22⋊Dic61—42no—
89C22⋊Dic61—41no—
90C22⋊Dic61—42no—
91C22⋊Dic61—41no—
92C22⋊Dic61—81no—
93C4⋊Dic61—81no—
94#11758196—22no—
95#11758196—22no—
96#11758196—81no—
97C4×Dic61—21no—
98C4×Dic61—21no—
99C4×Dic61—21no—
100C4×Dic61—21no—
101C4×Dic61C2×D412yes—
102C4×Dic61#4112yes—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61—41no—
107C23×C61—41no—
108C23×C61—41no—
109C23×C61—41no—
110C23×C61C2×C22⋊C411yescharacteristic
111D4×C61—82no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61#10412yes—
121C2×C4×C61C4×D412yes—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61#12011yescharacteristic
137C2×Dic61#12112yes—
138C2×Dic61#12311yescharacteristic
139Dic122—44no—
140Dic122—82no—
141#92420—44no—
142#92420—44no—
143#9188—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—81no—
147C22×C61—81no—
148C22×C61—81no—
149C22×C61—81no—
150C22×C61—81no—
151C22×C61—81no—
152C22×C61—81no—
153C22×C61#38611yescharacteristic
154C22×C61C2×#10411yescharacteristic
155C22×C61#42111yescharacteristic
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—41no—
159C4×C61—81no—
160Dic61—22no—
161Dic61—21no—
162Dic61—21no—
163Dic61—41no—
164Dic61—41no—
165Dic61—81no—
166Dic61—81no—
167#2983—611no—
168#2632—611no—
169C4×#101—611no—
170#1527—612no—
171#1288—612no—
172C2×C61—21no—
173C2×C61—21no—
174C2×C61—161no—
175C2×C61#187111yescharacteristic
176C2×C61#191211yescharacteristic
177C2×C61#191411yescharacteristic
178C2×#124—611no—
179C2×C4×Q8—611no—
180#446—1224no—
181#432—612no—
182#432—1221no—
183#431—1221no—
184C2×#101—1221no—
185C4×M16—612no—
186#399—1221no—
187#395—611no—
188#395—611no—
189#394—611no—
190#394#692311yescharacteristic
191#393—1221no—
192#388—1221no—
193#388—1221no—
194#383—611no—
195#343—1221no—
196#329—1224no—
197C61#918811yescharacteristic, complemented, Sylow
198C22×Q8—611no—
199C22×Q8—611no—
200C22×D4C2×#692311yescharacteristic
201C2×M16—1222no—
202#124—1222no—
203#124—1222no—
204C4×Q8—1222no—
205C4×Q8—1222no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4C2×#692311yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—1221no—
214C2×C22⋊C4—1221no—
215C2×C22⋊C4—1221no—
216C2×C22⋊C4—1221no—
217C2×C42—611no—
218C2×C42—611no—
219C2×C42—611no—
220C4⋊C8—1222no—
221C4⋊C8—1222no—
222C4⋊C8—1222no—
223#101—1224no—
224C22⋊C8—1222no—
225#97—614no—
226#97—1222no—
227#97—1222no—
228C4×C8—1222no—
229C4×C8—1222no—
230#95—611no—
231#95—1221no—
232#95—1221no—
233#95—1221no—
234#95—1221no—
235#95C2×#692311yescharacteristic
236C24—21no—
237C2×Q8—1222no—
238C2×Q8—1222no—
239C2×Q8—1222no—
240C2×Q8—4881no—
241C2×Q8—2441no—
242C2×D4—22no—
243C2×D4—81no—
244C22×C4—21no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4C22⋊#692311yescharacteristic
252C22×C4C22⋊#692311yescharacteristic
253C22×C4C22×#692311yescharacteristic
254M16—1224no—
255M16—2442no—
256C2×C8—1224no—
257C2×C8—2442no—
258C4⋊C4—22no—
259C4⋊C4—1222no—
260C4⋊C4—1222no—
261C4⋊C4—4881no—
262C4⋊C4—4881no—
263C22⋊C4—42no—
264C22⋊C4—81no—
265C22⋊C4—4881no—
266C22⋊C4—2441no—
267C22⋊C4—2442no—
268C22⋊C4—2441no—
269C22⋊C4—2442no—
270C42—612no—
271C42—612no—
272C42—1221no—
273C42—1221no—
274C42—1221no—
275C42—1221no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×C22⋊#692311yescharacteristic
284Q8—4882no—
285Q8—2444no—
286D4—82no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—611no—
296C2×C4—612no—
297C2×C4—611no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4#1451984112yes—
313C2×C4D4×#692312yes—
314C8—2444no—
315C8—2444no—
316C22—21no—
317C22—21no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22#1684071011yescharacteristic
326C22C2×#1451984111yescharacteristic
327C22#1684075911yescharacteristic
328C4—22no—
329C4—22no—
330C4—41no—
331C4—81no—
332C4—1222no—
333C4—1221no—
334C4—1221no—
335C4—4881no—
336C4—4881no—
337C4—2441no—
338C4—2441no—
339C2—21no—
340C2—21no—
341C2—161no—
342C2#2146180211yescharacteristic
343C2#2146186711yescharacteristic
344C2#2146187011yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#394C4
socleC22×C61#386
prime radicalC22×D4×C61C2×C4
cosocleC2×C4×Dic61C23
hypercenter#394#6923
hypocenterC61#9188
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #21462717C2semidirect
2. #21461572C2semidirect
4. #21460678C2semidirect
197. C61#9188semidirect

Maximal subgroups

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

subgroupindex
#214627172
#214615722
#214612782
#214606782
#214516842
#918861