Finite Group Databasebeta

← #30599683

Subgroups — #30599683

Order 15616.

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

normal subgroups only

This group has 364 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#21462610C211yesmaximal, characteristic
2#21461919C211yesmaximal, characteristic
3#21461386C211yesmaximal, characteristic
4#21461127C211yesmaximal, characteristic
5#21460717C211yesmaximal, characteristic
6#21460658C211yesmaximal, characteristic
7#21444316C211yesmaximal, characteristic
8#16840814—21no—
9#16840814—21no—
10#16840813—21no—
11#16840812—21no—
12#16840779C2211yescharacteristic
13#16840768—21no—
14#16840765C2211yescharacteristic
15C2×C4⋊#92420—21no—
16C2×C4⋊#92420C2211yescharacteristic
17#16840726—21no—
18C2×#14519832C2211yescharacteristic
19#16840628—21no—
20#16840624—21no—
21#16840623—21no—
22#16840612—21no—
23Q8⋊#92420—21no—
24#16840314C2211yescharacteristic
25#16839812C2211yescharacteristic
26#16839613C2211yescharacteristic
27#16839500C411yescharacteristic
28#16839499C411yescharacteristic
29#16839498C411yescharacteristic
30C61×#328C411yescharacteristic
31C2×#11758217—21no—
32C22×#92420—21no—
33C22×#92420D411yescharacteristic
34C22×#92420D411yescharacteristic
35#14519844—21no—
36#14519844—21no—
37#14519844—21no—
38#14519844—21no—
39C4⋊#92420—21no—
40C4⋊#92420—21no—
41C4⋊#92420—21no—
42C4⋊#92420—21no—
43C4⋊#92420—42no—
44C4⋊#92420D411yescharacteristic
45C4⋊#92420D411yescharacteristic
46C22⋊#92420—21no—
47#14519832—21no—
48#14519832—21no—
49#14519832—21no—
50#14519832—21no—
51C4×#92420—21no—
52C4×#92420—21no—
53C4×#92420—21no—
54#14519776Q811yescharacteristic
55#14519775Q811yescharacteristic
56#14519767—21no—
57#14519756C2×C411yescharacteristic
58#14519754—21no—
59Q8×Dic61D411yescharacteristic
60#14519742—21no—
61#14519742D411yescharacteristic
62#14519733—21no—
63C2×#11758196—21no—
64C2×C4×Dic61—21no—
65C2×C4×Dic61C2311yescharacteristic
66#14519706—21no—
67C22⋊#92418C2×C411yescharacteristic
68C8×Dic61—21no—
69#14519684C2×C411yescharacteristic
70C22⋊Q8×C61C2×C411yescharacteristic
71C22⋊C8×C61C2×C411yescharacteristic
72#95×C61C2×C411yescharacteristic
73C2×Q8×C61C4⋊C411yescharacteristic
74C22×C4×C61—21no—
75C22×C4×C61C22×C411yescharacteristic
76C22×Dic61—21no—
77C22×Dic61C2×D411yescharacteristic
78C22×Dic61#4111yescharacteristic
79C2×Dic122—21no—
80C2×Dic122C4⋊C411yescharacteristic
81#11758217—41no—
82#11758217—41no—
83C2×#92420—21no—
84C2×#92420—21no—
85C2×#92420—21no—
86C2×#92420—21no—
87C2×#92420—21no—
88C2×#92420—21no—
89C2×#92420—21no—
90C2×#92420—41no—
91C2×#92420—41no—
92C2×#92420—41no—
93C2×#92420—41no—
94C2×#92420SD1612yes—
95C2×C8×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61C4⋊C411yescharacteristic
98C22⋊C4×C61—21no—
99C22⋊Dic61—21no—
100C4⋊Dic61—21no—
101C4⋊Dic61C4⋊C411yescharacteristic
102#11758196—21no—
103#11758196—21no—
104#11758196—21no—
105#11758196—21no—
106#11758196—21no—
107#11758196—42no—
108C4×Dic61—21no—
109C4×Dic61—21no—
110C4×Dic61—41no—
111C4×Dic61—41no—
112C4×Dic61C2×D411yescharacteristic
113C4×Dic61C2×D411yescharacteristic
114C4×Dic61C2×Q811yescharacteristic
115C4×Dic61#4111yescharacteristic
116C2×#92418—21no—
117C23×C61C4×D411yescharacteristic
118Q8×C61—41no—
119C2×C4×C61—22no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—41no—
123C2×C4×C61—41no—
124C2×C4×C61—41no—
125C2×C4×C61C2×C4⋊C411yescharacteristic
126C2×C4×C61C4×D411yescharacteristic
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—41no—
130C2×Dic61—41no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61#12011yescharacteristic
134C2×Dic61C22⋊Q811yescharacteristic
135C2×Dic61C22⋊Q811yescharacteristic
136C2×Dic61#12311yescharacteristic
137C2×Dic61C2×SD1611yescharacteristic
138C2×Dic61#13511yescharacteristic
139C2×Dic61#13711yescharacteristic
140C2×Dic61#13711yescharacteristic
141Dic122—41no—
142Dic122—81no—
143#92420—22no—
144#92420—41no—
145#92420—41no—
146#92420—41no—
147#92420—81no—
148#92420—81no—
149C8×C61—41no—
150#92418—41no—
151#8881—611nomaximal, Sylow
152C22×C61—21no—
153C22×C61—21no—
154C22×C61#38511yescharacteristic
155C22×C61C4×SD1611yescharacteristic
156C22×C61#44111yescharacteristic
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—42no—
160C4×C61—41no—
161C4×C61—41no—
162Dic61—21no—
163Dic61—21no—
164Dic61—41no—
165Dic61—41no—
166Dic61—81no—
167Dic61#44811yescharacteristic
168Dic61#45211yescharacteristic
169Dic61#47511yescharacteristic
170Dic61#47911yescharacteristic
171#2961—611no—
172#1940—611no—
173#1570—611no—
174#1503—611no—
175#1481—611no—
176#1303—611no—
177#1281—611no—
178C2×C61—21no—
179C2×C61—21no—
180C2×C61#186111yescharacteristic
181C2×C61#186511yescharacteristic
182C2×C61#186811yescharacteristic
183#533—611no—
184#446—1221no—
185#446—1221no—
186#446—1221no—
187#446—1221no—
188#431—611no—
189#431—611no—
190#424—1221no—
191#423—611no—
192C2×C4⋊C8—611no—
193C2×C4⋊C8—1221no—
194#405—1221no—
195C2×#97—611no—
196C4×C4⋊C4—611no—
197#336—1221no—
198#334—1221no—
199#334—1221no—
200#328—611no—
201#328—611no—
202#328—611no—
203#328#692311yescharacteristic
204Q8⋊C8—1221no—
205Q8⋊C8—1221no—
206C61#888111yescharacteristic, complemented, Sylow
207C2×M16—1221no—
208C22×C8—611no—
209C22×C8—611no—
210C22×C8—1221no—
211C4⋊Q8—611no—
212C4⋊Q8—611no—
213#125—1221no—
214C22⋊Q8—611no—
215C22⋊Q8—1221no—
216C22⋊Q8C2×#692311yescharacteristic
217C4×Q8—611no—
218C4×Q8—611no—
219C4×Q8—1221no—
220#117—1221no—
221C2×C4⋊C4—1221no—
222C2×C42—611no—
223C2×C42—1221no—
224C4⋊C8—611no—
225C4⋊C8—611no—
226C4⋊C8—1221no—
227C4⋊C8—1221no—
228C4⋊C8—1221no—
229C4⋊C8—1221no—
230C4⋊C8—1221no—
231C4⋊C8—1221no—
232C4⋊C8—1221no—
233C4⋊C8—1221no—
234C4⋊C8—1221no—
235C4⋊C8—2442no—
236C22⋊C8—611no—
237C22⋊C8—1221no—
238C22⋊C8C2×#692311yescharacteristic
239#97—1221no—
240#97—1221no—
241#97—1221no—
242#97—1221no—
243C4×C8—1221no—
244C4×C8—1221no—
245C4×C8—1221no—
246C4×C8—1221no—
247#95—611no—
248#95C2×#692311yescharacteristic
249C2×Q8—611no—
250C2×Q8—1221no—
251C2×Q8C4⋊#692311yescharacteristic
252C22×C4—21no—
253C22×C4—611no—
254C22×C4—611no—
255C22×C4—1221no—
256C22×C4C22×#692311yescharacteristic
257M16—2441no—
258M16—2441no—
259C2×C8—21no—
260C2×C8—612no—
261C2×C8—1221no—
262C2×C8—1221no—
263C2×C8—1221no—
264C2×C8—1221no—
265C2×C8—1221no—
266C2×C8—1221no—
267C2×C8—1221no—
268C2×C8—1221no—
269C2×C8—2441no—
270C2×C8—2441no—
271C2×C8—2441no—
272C2×C8—2441no—
273C4⋊C4—21no—
274C4⋊C4—611no—
275C4⋊C4—1221no—
276C4⋊C4—1221no—
277C4⋊C4—1221no—
278C4⋊C4—1221no—
279C4⋊C4—1221no—
280C4⋊C4—1221no—
281C4⋊C4—2442no—
282C4⋊C4C4⋊#692311yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—1221no—
285C42—611no—
286C42—611no—
287C42—611no—
288C42—611no—
289C42—1221no—
290C42—1221no—
291C42—2441no—
292C42—2441no—
293C23D4×#692311yescharacteristic
294Q8—41no—
295Q8—4881no—
296Q8—2441no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—611no—
304C2×C4—611no—
305C2×C4—611no—
306C2×C4—611no—
307C2×C4—611no—
308C2×C4—611no—
309C2×C4—611no—
310C2×C4—611no—
311C2×C4—1221no—
312C2×C4—1221no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4C2×C4⋊#692311yescharacteristic
318C2×C4D4×#692311yescharacteristic
319C8—41no—
320C8—1222no—
321C8—4881no—
322C8—4881no—
323C8—2441no—
324C8—2441no—
325C8—2441no—
326C8—2441no—
327C22—21no—
328C22—21no—
329C22#1684070911yescharacteristic
330C22SD16×#692311yescharacteristic
331C22#1684080211yescharacteristic
332C4—21no—
333C4—21no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—611no—
338C4—611no—
339C4—611no—
340C4—611no—
341C4—1221no—
342C4—1221no—
343C4—4881no—
344C4—2441no—
345C4—2441no—
346C2—21no—
347C2—21no—
348C2#2146178011yescharacteristic
349C2#2146178511yescharacteristic
350C2#2146179111yescharacteristic
subgroupG/H
centerC22#16840709
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#328C4
socleC22×C61#385
prime radicalC23×C61C4×D4
cosocleC2×C4×Dic61C23
hypercenter#328#6923
hypocenterC61#8881
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
206. C61#8881semidirect

Maximal subgroups

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

subgroupindex
#214626102
#214619192
#214613862
#214611272
#214607172
#214606582
#214443162
#888161