Finite Group Databasebeta

← #30599756

Subgroups — #30599756

Order 15616.

354 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 74 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#21462777C211yesmaximal, characteristic
2#21461761C211yesmaximal, characteristic, complemented
3#21461385C211yesmaximal, characteristic, complemented
4#21461177C211yesmaximal, characteristic
5#21460713C211yesmaximal, characteristic, complemented
6#21460704C211yesmaximal, characteristic, complemented
7#21444406C211yesmaximal, characteristic
8#16840814—21no—
9#16840796—21no—
10#16840792—21no—
11#16840792—21no—
12#16840787—21no—
13#16840784C2211yescharacteristic
14#16840780C2211yescharacteristic
15C2×#14519848—21no—
16C2×#14519847—21no—
17#16840769—21no—
18#16840767C2211yescharacteristic
19#16840766C2211yescharacteristic
20#16840626—21no—
21#16840624—21no—
22#16840612—21no—
23#16840610—21no—
24#16840297C2211yescharacteristic
25#16839714C2211yescharacteristic
26#97×D61C2211yescharacteristic
27#16839504C411yescharacteristic
28#16839501C411yescharacteristic
29#16839501C411yescharacteristic
30C61×#330C411yescharacteristic
31C2×#11758213—21no—
32C2×#11758213—21no—
33C2×#11758213D411yescharacteristic
34C2×#11758213D411yescharacteristic
35C2×D61⋊C8—21no—
36#14519848—42no—
37#14519847—42no—
38#14519844—21no—
39#14519844—21no—
40#14519844—21no—
41#14519844—21no—
42C4⋊#92420—21no—
43C4⋊#92420—21no—
44C4⋊#92420—21no—
45C4⋊#92420D411yescharacteristic
46C4⋊#92420D411yescharacteristic
47#14519834—21no—
48#14519834—21no—
49#14519834—21no—
50#14519834—21no—
51#14519832—21no—
52#14519832—21no—
53C4×#92420—21no—
54C4×#92420—21no—
55C2×#11758189—21no—
56C2×C8×D61—21no—
57#14519772—21no—
58#14519765C2×C411yescharacteristic
59#14519762—21no—
60#14519757Q811yescharacteristic
61#14519752Q811yescharacteristic
62#14519742D411yescharacteristic
63#14519739—21no—
64#14519739D411yescharacteristic
65#14519735—21no—
66C42×D61C2311yescharacteristic
67#14519688C2×C411yescharacteristic
68#14519688C2×C411yescharacteristic
69#14519687—21no—
70C8×Dic61—21no—
71#125×C61C2×C411yescharacteristic
72#97×C61C2×C411yescharacteristic
73#97×C61C2×C411yescharacteristic
74C2×D244—21no—
75C2×D244C4⋊C411yescharacteristic
76C2×C4×D61—21no—
77C2×C4×D61C2×D411yescharacteristic
78C2×C4×D61C2×Q811yescharacteristic
79C2×C4×D61#4111yescharacteristic
80C2×Dic122C4⋊C411yescharacteristic
81C2×#92420—21no—
82C2×#92420—21no—
83C2×#92420—21no—
84C2×#92420—21no—
85C2×#92420—41no—
86C2×#92420—41no—
87#11758213—41no—
88#11758213—41no—
89#11758213—41no—
90#11758213—41no—
91#11758213—81no—
92D61⋊C8—81no—
93C2×C8×C61—21no—
94C2×C8×C61—21no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61C4⋊C411yescharacteristic
98C4⋊C4×C61C4⋊C411yescharacteristic
99C42×C61C22×C411yescharacteristic
100#11758198—21no—
101#11758198—21no—
102#11758198—21no—
103#11758198—21no—
104#11758198—21no—
105C4⋊Dic61—21no—
106#11758196—21no—
107C4×Dic61—21no—
108C4×Dic61C2×D411yescharacteristic
109C4×Dic61C2×D411yescharacteristic
110C4×Dic61#4111yescharacteristic
111C2×#92418—21no—
112C2×#92418—21no—
113#11758189—42no—
114C8×D61—22no—
115C8×D61—22no—
116C22×D61—21no—
117C22×D61C22⋊Q811yescharacteristic
118C2×C4×C61—21no—
119C2×C4×C61—21no—
120C2×C4×C61C2×C4⋊C411yescharacteristic
121C2×C4×C61C4×D411yescharacteristic
122C2×C4×C61C4×D411yescharacteristic
123C2×Dic61—21no—
124C2×Dic61#12011yescharacteristic
125C2×Dic61C22⋊Q811yescharacteristic
126C2×Dic61#12311yescharacteristic
127D244—41no—
128D244—81no—
129C4×D61—21no—
130C4×D61—21no—
131C4×D61—41no—
132C4×D61—41no—
133C4×D61—41no—
134C4×D61—41no—
135Dic122—41no—
136#92420—41no—
137#92420—41no—
138#92420—41no—
139#92420—41no—
140#92420—81no—
141#92420—81no—
142C8×C61—22no—
143C8×C61—41no—
144#92418—22no—
145#92418—22no—
146#8916—611nomaximal, Sylow
147C22×C61#38511yescharacteristic
148C2×D61—21no—
149C2×D61—21no—
150C2×D61—21no—
151C2×D61—41no—
152C2×D61—41no—
153C2×D61—41no—
154C4×C61—21no—
155C4×C61—21no—
156C4×C61—21no—
157C4×C61—41no—
158C4×C61—41no—
159Dic61—21no—
160Dic61—21no—
161Dic61—41no—
162Dic61#45311yescharacteristic
163Dic61#45611yescharacteristic
164#2995—611no—
165#1849—611no—
166#1569—611no—
167#1524—611no—
168#1495—611no—
169#1302—611no—
170#1299—611no—
171C2×C61#185811yescharacteristic
172C2×C61#185911yescharacteristic
173C2×C61#186111yescharacteristic
174D61—41no—
175D61—81no—
176#529—611no—
177#446—1221no—
178#436—1221no—
179#435—1221no—
180#435—1221no—
181#433—1221no—
182#432—611no—
183#431—611no—
184C2×#108—1221no—
185C2×#108—1221no—
186#424—1221no—
187#423—611no—
188#423—611no—
189C4×M16—611no—
190C2×#97—611no—
191#335—1221no—
192#334—1221no—
193#330—611no—
194#330#692311yescharacteristic
195#329—611no—
196#329—611no—
197Q8⋊C8—1221no—
198D4⋊C8—1221no—
199C61#891611yescharacteristic, complemented, Sylow
200C2×M16—611no—
201C2×M16—611no—
202C2×M16—1221no—
203C2×M16—1221no—
204C2×M16—1221no—
205C22×C8—1221no—
206C22×C8—1221no—
207#126—1221no—
208#125C2×#692311yescharacteristic
209#124—611no—
210#123—1221no—
211C22⋊Q8—611no—
212#121—611no—
213C4×Q8—611no—
214C4×D4—611no—
215C4×D4—1221no—
216#117—1221no—
217C2×C42—611no—
218#108—2442no—
219#108—2442no—
220C4⋊C8—611no—
221C4⋊C8—611no—
222C4⋊C8—1221no—
223C4⋊C8—1221no—
224C4⋊C8—1221no—
225C4⋊C8—1221no—
226C4⋊C8—1221no—
227C4⋊C8—1221no—
228C4⋊C8—1221no—
229C22⋊C8—1221no—
230C22⋊C8—1221no—
231C22⋊C8—1221no—
232C22⋊C8—1221no—
233#97—611no—
234#97—611no—
235#97—1221no—
236#97—1221no—
237#97—1221no—
238#97C2×#692311yescharacteristic
239#97C2×#692311yescharacteristic
240C4×C8—1221no—
241C4×C8—1221no—
242C4×C8—1221no—
243C2×Q8—611no—
244C2×D4—611no—
245C2×D4—1221no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—1221no—
250M16—4881no—
251M16—2442no—
252M16—2441no—
253M16—2441no—
254M16—2441no—
255M16—2441no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—1222no—
259C2×C8—1222no—
260C2×C8—1221no—
261C2×C8—1221no—
262C2×C8—1221no—
263C2×C8—1221no—
264C2×C8—1221no—
265C2×C8—1221no—
266C2×C8—4881no—
267C2×C8—2441no—
268C2×C8—2441no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—1221no—
272C4⋊C4—1221no—
273C4⋊C4C4⋊#692311yescharacteristic
274C4⋊C4C4⋊#692311yescharacteristic
275C22⋊C4—1221no—
276C22⋊C4—1221no—
277C22⋊C4—1221no—
278C22⋊C4—1221no—
279C22⋊C4—1221no—
280C42—611no—
281C42—611no—
282C42—611no—
283C42—1221no—
284C42C22×#692311yescharacteristic
285C23—611no—
286C23—1221no—
287Q8—2441no—
288D4—4881no—
289D4—2441no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—611no—
293C2×C4—611no—
294C2×C4—611no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—2441no—
299C2×C4—2441no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4C2×C4⋊#692311yescharacteristic
303C2×C4D4×#692311yescharacteristic
304C2×C4D4×#692311yescharacteristic
305C8—22no—
306C8—41no—
307C8—1222no—
308C8—1222no—
309C8—4881no—
310C8—4881no—
311C8—2441no—
312C8—2441no—
313C8—2441no—
314C8—2441no—
315C22—1221no—
316C22—1221no—
317C22—1221no—
318C22—2441no—
319C22—2441no—
320C22—2441no—
321C22#1684070911yescharacteristic
322C4—21no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—611no—
328C4—611no—
329C4—1221no—
330C4—1221no—
331C4—2441no—
332C2—4881no—
333C2—2441no—
334C2#2146177711yescharacteristic
335C2#2146177811yescharacteristic
336C2#2146178111yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#330C4
socleC22×C61#385
prime radical#14519752Q8
cosocleC42×D61C23
hypercenter#330#6923
hypocenterC61#8916
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
2. #21461761C2semidirect
3. #21461385C2semidirect
5. #21460713C2semidirect
6. #21460704C2semidirect
199. C61#8916semidirect

Maximal subgroups

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

subgroupindex
#214627772
#214617612
#214613852
#214611772
#214607132
#214607042
#214444062
#891661