Finite Group Databasebeta

← #30605220

Subgroups — #30605220

Order 15616.

416 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 70 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#21461685C211yesmaximal, characteristic, complemented
2#21461081C211yesmaximal, characteristic, complemented
3#21460818C212yesmaximal, complemented
4#21460801C212yesmaximal
5#21449103C211yesmaximal, characteristic
6#16840783—21no—
7#16840765—21no—
8C2×#14519838—21no—
9#16840642C2212yes—
10#16840608C2212yes—
11#16840607—21no—
12C2×C4×Dic122C2211yescharacteristic
13#16839700C411yescharacteristic
14#16839700C2211yescharacteristic
15#16839687—21no—
16#16839676C411yescharacteristic
17#16839676C2211yescharacteristic
18#16839674—21no—
19C4×C22⋊Dic61—21no—
20C4×C22⋊Dic61—21no—
21#16839608C411yescharacteristic
22C2×C4×D4×C61C411yescharacteristic
23C2×#11758217—22no—
24#14519844—41no—
25C4⋊#92420—41no—
26C4⋊#92420—41no—
27#14519838—42no—
28C22⋊#92420—22no—
29#14519832—41no—
30C22×Dic122D411yescharacteristic
31C4×Dic122—41no—
32C4×Dic122—41no—
33C2×C4⋊Dic61D411yescharacteristic
34C2×#11758196C2×C411yescharacteristic
35C2×#11758196C2311yescharacteristic
36C2×C22⋊Dic61—21no—
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—21no—
39C2×C22⋊Dic61—21no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61D411yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684C2×C411yescharacteristic
49#14519684C2×C411yescharacteristic
50C22×D4×C61C2×C411yescharacteristic
51C23×C4×C61—21no—
52C4×D4×C61—41no—
53C4×D4×C61—41no—
54C2×C4⋊C4×C61C2×C411yescharacteristic
55C2×C22⋊C4×C61—21no—
56C2×C42×C61C2×C411yescharacteristic
57C24×C61—21no—
58C2×D4×C61—41no—
59C2×D4×C61—42no—
60C22×C4×C61—21no—
61C22×C4×C61—42no—
62C22×C4×C61—42no—
63C22×C4×C61—41no—
64C22×C4×C61—41no—
65C22×C4×C61C22⋊C411yescharacteristic
66C22×C4×C61C22⋊C411yescharacteristic
67C22×C4×C61C22×C411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61C22⋊C411yescharacteristic
71C22×Dic61C22⋊C411yescharacteristic
72C22×Dic61C2×D411yescharacteristic
73C22×Dic61C2×D411yescharacteristic
74C2×Dic122—21no—
75C2×Dic122—21no—
76C2×Dic122—41no—
77C2×Dic122—41no—
78#11758217—42no—
79#11758217—42no—
80C2×#92420—42no—
81C2×#92420—42no—
82C4⋊C4×C61—41no—
83C22⋊C4×C61—42no—
84C42×C61—21no—
85C42×C61—21no—
86C22⋊Dic61—42no—
87C22⋊Dic61—42no—
88C22⋊Dic61—42no—
89C22⋊Dic61—41no—
90C22⋊Dic61—41no—
91C4⋊Dic61—41no—
92#11758196—41no—
93#11758196—41no—
94C4×Dic61—21no—
95C4×Dic61—21no—
96C4×Dic61—21no—
97C4×Dic61—21no—
98C4×Dic61—42no—
99C23×C61—22no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61—41no—
104C23×C61—41no—
105C23×C61—41no—
106C23×C61C2×C22⋊C411yescharacteristic
107D4×C61—42no—
108D4×C61—42no—
109C2×C4×C61—22no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—82no—
119C2×C4×C61—82no—
120C2×C4×C61#9911yescharacteristic
121C2×C4×C61#9911yescharacteristic
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61AGL(1,ℤ/8)11yescharacteristic
138C2×Dic61#13711yescharacteristic
139Dic122—42no—
140Dic122—42no—
141#92420—82no—
142#92420—82no—
143#11829—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—42no—
147C22×C61—42no—
148C22×C61—42no—
149C22×C61—81no—
150C22×C61—81no—
151C22×C61—81no—
152C22×C61—81no—
153C22×C61C2×#9911yescharacteristic
154C22×C61#41711yescharacteristic
155C22×C61#42111yescharacteristic
156C4×C61—21no—
157C4×C61—41no—
158C4×C61—41no—
159C4×C61—81no—
160C4×C61#35512yes—
161Dic61—21no—
162Dic61—21no—
163Dic61—41no—
164Dic61—41no—
165Dic61—41no—
166Dic61—81no—
167Dic61—81no—
168#2398—611no—
169#1801—611no—
170#1467—611no—
171#1353—612no—
172#1345—612no—
173C2×C61—21no—
174C2×C61—21no—
175C2×C61—82no—
176C2×C61#151311yescharacteristic
177C2×C61#212011yescharacteristic
178C2×C61C2×#35511yescharacteristic
179C2×C4×Q8—611no—
180C2×C4×D4#692311yescharacteristic
181#432—1221no—
182#423—1221no—
183C2×#101—1221no—
184#400—611no—
185#400—611no—
186#397—1221no—
187#394—611no—
188#394—611no—
189#393—1221no—
190C4×C22⋊C4—1221no—
191C4×C22⋊C4—1221no—
192#376—611no—
193#343—612no—
194#324—612no—
195#324—1221no—
196C61#1182911yescharacteristic, complemented, Sylow
197C22×Q8—611no—
198C22×D4C2×#692311yescharacteristic
199C23×C4—21no—
200C2×M16—1222no—
201C4×Q8—2441no—
202C4×Q8—2441no—
203C4×D4—41no—
204C4×D4—41no—
205C2×C4⋊C4—611no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—611no—
208C2×C4⋊C4C2×#692311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—1221no—
211C2×C22⋊C4—1221no—
212C2×C22⋊C4—1221no—
213C2×C22⋊C4—1221no—
214C2×C42—611no—
215C2×C42—611no—
216C2×C42—1221no—
217C2×C42C2×#692311yescharacteristic
218C4⋊C8—2441no—
219C4⋊C8—2441no—
220C4⋊C8—2441no—
221#101—2442no—
222C22⋊C8—1222no—
223#97—2441no—
224#95—611no—
225#95—611no—
226#95—1221no—
227#95—1221no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231C24—21no—
232C2×Q8—1221no—
233C2×Q8—1221no—
234C2×Q8—2441no—
235C2×Q8—2441no—
236C2×D4—41no—
237C2×D4—42no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—611no—
244C22×C4—611no—
245C22×C4—611no—
246C22×C4—611no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4C22⋊#692311yescharacteristic
250C22×C4C22⋊#692311yescharacteristic
251C22×C4C22×#692311yescharacteristic
252M16—2442no—
253M16—2442no—
254C2×C8—2442no—
255C2×C8—2442no—
256C4⋊C4—41no—
257C4⋊C4—2441no—
258C4⋊C4—2441no—
259C4⋊C4—2441no—
260C22⋊C4—42no—
261C22⋊C4—2441no—
262C22⋊C4—2441no—
263C22⋊C4—2442no—
264C22⋊C4—2442no—
265C22⋊C4—2442no—
266C42—21no—
267C42—21no—
268C42—1221no—
269C42—1221no—
270C42—1221no—
271C42—1221no—
272C42—2442no—
273C23—22no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×C22⋊#692311yescharacteristic
281Q8—2442no—
282Q8—2442no—
283D4—42no—
284D4—42no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—611no—
297C2×C4—611no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2441no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2441no—
313C2×C4#1451983511yescharacteristic
314C2×C4#1451983511yescharacteristic
315C8—4882no—
316C8—4882no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22C2×#1451983511yescharacteristic
327C22#1684075311yescharacteristic
328C22#1684075911yescharacteristic
329C4—21no—
330C4—41no—
331C4—41no—
332C4—81no—
333C4—1221no—
334C4—1221no—
335C4—4881no—
336C4—4881no—
337C4—2441no—
338C4—2441no—
339C4—2441no—
340C4#1684067012yes—
341C2—21no—
342C2—21no—
343C2—82no—
344C2#2146123411yescharacteristic
345C2#2146207011yescharacteristic
346C2C2×#1684067011yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4×D4×C61C4
socleC22×C61C2×#99
prime radicalC22×D4×C61C2×C4
cosocleC2×#11758196C23
hypercenterC2×C4×D4#6923
hypocenterC61#11829
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #21461685C2semidirect
2. #21461081C2semidirect
3. #21460818C2semidirect
196. C61#11829semidirect

Maximal subgroups

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

subgroupindex
#214616852
#214610812
#214608182
#214608012
#214491032
#1182961