Finite Group Databasebeta

← #30601024

Subgroups — #30601024

Order 15616.

486 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 86 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#21462718C211yesmaximal, characteristic
2#21461686C211yesmaximal, characteristic
3#21461328C212yesmaximal
4#21460717C212yesmaximal
5#21450803C211yesmaximal, characteristic
6#16840814—24no—
7#16840783—21no—
8#16840782—21no—
9#16840779C2212yes—
10#16840765C2212yes—
11C2×#14519838—21no—
12#16840642—21no—
13#16840621—24no—
14C2×#14519776—21no—
15C2×#14519776C2211yescharacteristic
16C2×#14519742C2211yescharacteristic
17#16839694C411yescharacteristic
18#16839693C411yescharacteristic
19#16839692C411yescharacteristic
20#16839666—21no—
21#16839659—21no—
22#16839657—21no—
23#16839613C2211yescharacteristic
24C61×#398C411yescharacteristic
25C2×#11758217—22no—
26#14519844—22no—
27#14519844—22no—
28#14519844D414yes—
29C4⋊#92420—22no—
30C4⋊#92420—22no—
31#14519838—42no—
32C22⋊#92420—22no—
33#14519832—22no—
34#14519832—22no—
35C4×#92420—22no—
36C22×Dic122—21no—
37C22×Dic122D411yescharacteristic
38#14519776—22no—
39#14519776—22no—
40#14519776—44no—
41#14519742—22no—
42#14519742—22no—
43C2×C4⋊Dic61—21no—
44C2×C4⋊Dic61C2×C411yescharacteristic
45C2×#11758196—21no—
46C2×#11758196—21no—
47C2×#11758196—21no—
48C2×#11758196—21no—
49C2×#11758196C2×C411yescharacteristic
50C2×#11758196D411yescharacteristic
51C2×C4×Dic61—21no—
52C2×C4×Dic61D411yescharacteristic
53C2×C4×Dic61D411yescharacteristic
54C2×C4×Dic61C2311yescharacteristic
55#14519684—21no—
56#14519684—21no—
57#14519684—21no—
58#14519684C2×C411yescharacteristic
59C2×C4⋊C4×C61—21no—
60C2×C4⋊C4×C61C2×C411yescharacteristic
61C2×C4⋊C4×C61C2×C411yescharacteristic
62#95×C61—21no—
63#95×C61C2×C411yescharacteristic
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22×C411yescharacteristic
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C22⋊C411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C22×Dic61C2×D411yescharacteristic
75C2×Dic122—22no—
76C2×Dic122—21no—
77C2×Dic122—22no—
78C2×Dic122—21no—
79C2×Dic122—42no—
80C2×Dic122—41no—
81C2×Dic122—42no—
82C2×Dic122—41no—
83#11758217—42no—
84#11758217—42no—
85C2×#92420—24no—
86C2×#92420—42no—
87C4⋊C4×C61—22no—
88C4⋊C4×C61—22no—
89C4⋊C4×C61—42no—
90C4⋊Dic61—22no—
91C4⋊Dic61—42no—
92#11758196—24no—
93#11758196—22no—
94#11758196—22no—
95#11758196—42no—
96#11758196—42no—
97#11758196—41no—
98#11758196—41no—
99C4×Dic61—21no—
100C4×Dic61—21no—
101C4×Dic61—21no—
102C4×Dic61—21no—
103C4×Dic61—42no—
104C4×Dic61C2×D412yes—
105C4×Dic61#4112yes—
106C23×C61C2×C22⋊C411yescharacteristic
107C2×C4×C61—22no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—41no—
113C2×C4×C61—41no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61C4×D412yes—
120C2×Dic61—22no—
121C2×Dic61—22no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—41no—
127C2×Dic61—41no—
128C2×Dic61—41no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61#12011yescharacteristic
133C2×Dic61#12112yes—
134C2×Dic61#12311yescharacteristic
135C2×Dic61AGL(1,ℤ/8)12yes—
136C2×Dic61AGL(1,ℤ/8)11yescharacteristic
137C2×Dic61#13711yescharacteristic
138Dic122—42no—
139Dic122—42no—
140Dic122—82no—
141Dic122—82no—
142#92420—44no—
143#92420—82no—
144#9402—611nomaximal, Sylow
145C22×C61—21no—
146C22×C61—21no—
147C22×C61#38611yescharacteristic
148C22×C61#41711yescharacteristic
149C22×C61#41811yescharacteristic
150C4×C61—22no—
151C4×C61—42no—
152C4×C61—41no—
153C4×C61—41no—
154C4×C61—81no—
155Dic61—21no—
156Dic61—42no—
157Dic61—41no—
158Dic61—41no—
159Dic61—81no—
160Dic61#45612yes—
161Dic61#46812yes—
162Dic61#46912yes—
163#2983—611no—
164#2546—611no—
165#1801—611no—
166#1546—612no—
167#1303—612no—
168C2×C61—21no—
169C2×C61—21no—
170C2×C61#188611yescharacteristic
171C2×C61#189511yescharacteristic
172C2×C61#191511yescharacteristic
173C2×C4⋊Q8—611no—
174C2×C4⋊Q8—1221no—
175C2×C4×Q8—611no—
176#446—1224no—
177#432—1221no—
178#432—1221no—
179#431—612no—
180#423—612no—
181C2×#101—1221no—
182#398—611no—
183#398—611no—
184#398—611no—
185#398#692311yescharacteristic
186#391—1221no—
187#389—1221no—
188#389—1221no—
189C4×C4⋊C4—611no—
190#343—1221no—
191#332—1224no—
192C61#940211yescharacteristic, complemented, Sylow
193C22×Q8—611no—
194C22×Q8—1221no—
195C2×M16—1222no—
196C4⋊Q8—1222no—
197C4⋊Q8—1222no—
198C4⋊Q8—2444no—
199C4×Q8—1222no—
200C4×Q8—1222no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—611no—
203C2×C4⋊C4—611no—
204C2×C4⋊C4—611no—
205C2×C4⋊C4—1221no—
206C2×C4⋊C4—1221no—
207C2×C4⋊C4—1221no—
208C2×C4⋊C4—1221no—
209C2×C4⋊C4—1221no—
210C2×C4⋊C4C2×#692311yescharacteristic
211C2×C4⋊C4C2×#692311yescharacteristic
212C2×C42—611no—
213C2×C42—611no—
214C2×C42—611no—
215C2×C42—1221no—
216C4⋊C8—614no—
217C4⋊C8—1222no—
218C4⋊C8—1222no—
219C4⋊C8—1222no—
220C4⋊C8—1222no—
221#101—2442no—
222C22⋊C8—1222no—
223#97—1222no—
224#97—1222no—
225C4×C8—1222no—
226#95—21no—
227#95—611no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231#95C2×#692311yescharacteristic
232C2×Q8—1221no—
233C2×Q8—1222no—
234C2×Q8—1222no—
235C2×Q8—1221no—
236C2×Q8—2442no—
237C2×Q8—2442no—
238C2×Q8—2441no—
239C2×Q8—2441no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—611no—
243C22×C4—611no—
244C22×C4—611no—
245C22×C4—611no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4C22⋊#692311yescharacteristic
249C22×C4C22⋊#692311yescharacteristic
250C22×C4C22×#692311yescharacteristic
251M16—2442no—
252M16—2442no—
253C2×C8—1224no—
254C2×C8—2442no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—1222no—
259C4⋊C4—1224no—
260C4⋊C4—1222no—
261C4⋊C4—1222no—
262C4⋊C4—2442no—
263C4⋊C4—2441no—
264C4⋊C4—2441no—
265C4⋊C4—2442no—
266C4⋊C4—2442no—
267C42—612no—
268C42—612no—
269C42—1221no—
270C42—1221no—
271C42—1221no—
272C42—1221no—
273C42—2442no—
274C23C2×C22⋊#692311yescharacteristic
275Q8—4882no—
276Q8—4882no—
277Q8—2442no—
278Q8—2442no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—611no—
292C2×C4—612no—
293C2×C4—611no—
294C2×C4—611no—
295C2×C4—612no—
296C2×C4—611no—
297C2×C4—1222no—
298C2×C4—1222no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4D4×#692312yes—
310C8—4882no—
311C8—2444no—
312C22—21no—
313C22—21no—
314C22#1684071011yescharacteristic
315C22#1684075311yescharacteristic
316C22#1684075411yescharacteristic
317C4—22no—
318C4—42no—
319C4—41no—
320C4—41no—
321C4—81no—
322C4—612no—
323C4—612no—
324C4—612no—
325C4—1221no—
326C4—4881no—
327C4—2442no—
328C4—2441no—
329C4—2441no—
330C2—21no—
331C2—21no—
332C2#2146182511yescharacteristic
333C2#2146183511yescharacteristic
334C2#2146187111yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#398C4
socleC22×C61#386
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C4×Dic61C23
hypercenter#398#6923
hypocenterC61#9402
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
192. C61#9402semidirect

Maximal subgroups

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

subgroupindex
#214627182
#214616862
#214613282
#214607172
#214508032
#940261