Finite Group Databasebeta

← #38745269

Subgroups — #38745269

Order 25216.

484 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263316C211yesmaximal, characteristic
2#27160976C211yesmaximal, characteristic
3#27160974C211yesmaximal, characteristic
4#27160966C211yesmaximal, characteristic
5#27160963C211yesmaximal, characteristic
6#27160962C211yesmaximal, characteristic
7#27160943C211yesmaximal, characteristic
8C4⋊C4×Dic197C211yesmaximal, characteristic
9C4⋊C4×Dic197C211yesmaximal, characteristic
10#27160930C211yesmaximal, characteristic
11#27160930C211yesmaximal, characteristic
12#27160930C211yesmaximal, characteristic
13#27160930C211yesmaximal, characteristic
14C42×Dic197C211yesmaximal, characteristic
15#387×C197C211yesmaximal, characteristic
16#20263316C418yescomplemented
17C2×C4⋊Dic197C2211yescharacteristic
18C2×C4⋊Dic197C2211yescharacteristic
19C2×#15691337C2211yescharacteristic
20C2×#15691337C2211yescharacteristic
21C2×#15691337C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×#15691337C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
27C2×C4×Dic197C2211yescharacteristic
28C2×C4×Dic197C2211yescharacteristic
29C2×C4×Dic197C2211yescharacteristic
30C2×C4×Dic197C2211yescharacteristic
31C2×C4×Dic197C2211yescharacteristic
32C2×C4×Dic197C2211yescharacteristic
33C2×C4×Dic197C2211yescharacteristic
34C2×C4×Dic197C2211yescharacteristic
35C2×C4×Dic197C2211yescharacteristic
36C2×C4×Dic197C2211yescharacteristic
37#20263233C2211yescharacteristic
38#20263233C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45C2×C4⋊C4×C197C2211yescharacteristic
46C2×C4⋊C4×C197C2211yescharacteristic
47C2×C42×C197C2211yescharacteristic
48#95×C197C2211yescharacteristic
49#95×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52C22×Dic197C2311yescharacteristic
53C22×Dic197C2311yescharacteristic
54C22×Dic197C2311yescharacteristic
55C22×Dic197C2311yescharacteristic
56C22×Dic197C2311yescharacteristic
57C22×Dic197C2311yescharacteristic
58C22×Dic197C2311yescharacteristic
59C22×Dic197C2311yescharacteristic
60C4⋊Dic197—22no—
61C4⋊Dic197C2×C414yes—
62#15691337—22no—
63#15691337—22no—
64#15691337—22no—
65#15691337—22no—
66#15691337C2×C414yes—
67#15691337C2×C414yes—
68#15691337C2×C414yes—
69#15691337C2×C414yes—
70C4×Dic197—22no—
71C4×Dic197—22no—
72C4×Dic197—22no—
73C4×Dic197—22no—
74C4×Dic197—22no—
75C4×Dic197—22no—
76C4×Dic197—22no—
77C4×Dic197—22no—
78C4×Dic197C2×C414yes—
79C4×Dic197Q814yes—
80C22×C4×C197C2311yescharacteristic
81C22×C4×C197C2311yescharacteristic
82C22×C4×C197C2311yescharacteristic
83C22×C4×C197C2311yescharacteristic
84C22×C4×C197C2311yescharacteristic
85C22×C4×C197C2311yescharacteristic
86C22×C4×C197C2311yescharacteristic
87C4⋊C4×C197—22no—
88C4⋊C4×C197C2×C414yes—
89C42×C197—22no—
90C2×Dic197—21no—
91C2×Dic197—21no—
92C2×Dic197—21no—
93C2×Dic197—21no—
94C2×Dic197—21no—
95C2×Dic197—21no—
96C2×Dic197—21no—
97C2×Dic197—21no—
98C2×Dic197—21no—
99C2×Dic197—21no—
100C2×Dic197—21no—
101C2×Dic197—21no—
102C2×Dic197—21no—
103C2×Dic197—21no—
104C2×Dic197C22×C412yes—
105C2×Dic197C22×C412yes—
106C2×Dic197C22×C412yes—
107C2×Dic197C22×C412yes—
108C2×Dic197C2×Q812yes—
109C2×Dic197C2×Q812yes—
110C2×Dic197#4112yes—
111C2×Dic197#4112yes—
112C2×Dic197#4112yes—
113C2×Dic197#4112yes—
114C23×C197C2411yescharacteristic
115C2×C4×C197—21no—
116C2×C4×C197—21no—
117C2×C4×C197—21no—
118C2×C4×C197—21no—
119C2×C4×C197—21no—
120C2×C4×C197—21no—
121C2×C4×C197—21no—
122C2×C4×C197—21no—
123C2×C4×C197—21no—
124C2×C4×C197—21no—
125C2×C4×C197—21no—
126C2×C4×C197—21no—
127C2×C4×C197—21no—
128C2×C4×C197—21no—
129C2×C4×C197—21no—
130C2×C4×C197—21no—
131C2×C4×C197C22×C412yes—
132C2×C4×C197C22×C412yes—
133C2×C4×C197C22×C412yes—
134C2×C4×C197C2×Q812yes—
135C2×C4×C197#4112yes—
136C22×C197C23×C411yescharacteristic
137C22×C197C22×Q811yescharacteristic
138C22×C197C2×#4111yescharacteristic
139C22×C197C2×#4111yescharacteristic
140C22×C197C2×#4111yescharacteristic
141C22×C197C2×#4111yescharacteristic
142C22×C197C2×#4111yescharacteristic
143C4×C197—22no—
144C4×C197—22no—
145C4×C197—22no—
146C4×C197—22no—
147C4×C197—22no—
148C4×C197—41no—
149C4×C197—41no—
150Dic197—22no—
151Dic197—22no—
152Dic197—22no—
153Dic197—22no—
154Dic197—41no—
155Dic197—41no—
156Dic197C4×Q814yes—
157Dic197#12514yescomplemented
158C2×C197C2×C4×Q811yescharacteristic
159C2×C197C4×#4111yescharacteristic
160C2×C197C4×#4111yescharacteristic
161C2×C197C2×#12511yescharacteristic
162C2×C197#52911yescharacteristic
163C2×C197#52911yescharacteristic
164C2×C197#53311yescharacteristic
165C197C4×#12511yescharacteristic, complemented, Sylow
166C4×#125—1971nomaximal, Sylow
167C2×#125—1971no—
168#389—1971no—
169#389—1971no—
170#387—1971no—
171#387—1971no—
172#387—1971no—
173#387D19711yescharacteristic
174#382—1971no—
175C4×C4⋊C4—1971no—
176C4×C4⋊C4—1971no—
177C4×C4⋊C4—1971no—
178C4×C4⋊C4—1971no—
179C4×C4⋊C4—1971no—
180C4×C4⋊C4—1971no—
181C43—1971no—
182#125—1978no—
183C2×C4⋊C4—1971no—
184C2×C4⋊C4—1971no—
185C2×C4⋊C4—1971no—
186C2×C4⋊C4—1971no—
187C2×C4⋊C4—1971no—
188C2×C4⋊C4—1971no—
189C2×C4⋊C4—1971no—
190C2×C4⋊C4—1971no—
191C2×C4⋊C4—1971no—
192C2×C4⋊C4—1971no—
193C2×C4⋊C4C2×D19711yescharacteristic
194C2×C4⋊C4C2×D19711yescharacteristic
195C2×C42—1971no—
196C2×C42—1971no—
197C2×C42—1971no—
198C2×C42—1971no—
199C2×C42—1971no—
200C2×C42—1971no—
201C2×C42—1971no—
202C2×C42—1971no—
203C2×C42—1971no—
204C2×C42—1971no—
205C2×C42C2×D19711yescharacteristic
206#95—1971no—
207#95—1971no—
208#95—1971no—
209#95—1971no—
210#95—1971no—
211#95—1971no—
212#95—1971no—
213#95—1971no—
214#95C2×D19711yescharacteristic
215#95C2×D19711yescharacteristic
216#95C2×D19711yescharacteristic
217#95C2×D19711yescharacteristic
218C22×C4—1971no—
219C22×C4—1971no—
220C22×C4—1971no—
221C22×C4—1971no—
222C22×C4—1971no—
223C22×C4—1971no—
224C22×C4—1971no—
225C22×C4—1971no—
226C22×C4C22×D19711yescharacteristic
227C22×C4C22×D19711yescharacteristic
228C22×C4C22×D19711yescharacteristic
229C22×C4C22×D19711yescharacteristic
230C22×C4C22×D19711yescharacteristic
231C22×C4C22×D19711yescharacteristic
232C22×C4C22×D19711yescharacteristic
233C4⋊C4—22no—
234C4⋊C4—3942no—
235C4⋊C4—3942no—
236C4⋊C4—3942no—
237C4⋊C4—3942no—
238C4⋊C4—3942no—
239C4⋊C4—1974no—
240C4⋊C4—1974no—
241C4⋊C4—1974no—
242C4⋊C4—1974no—
243C4⋊C4—1974no—
244C4⋊C4C4×D19714yes—
245C42—22no—
246C42—3942no—
247C42—3942no—
248C42—3942no—
249C42—3942no—
250C42—3942no—
251C42—3942no—
252C42—3942no—
253C42—3942no—
254C42—1974no—
255C42—1974no—
256C23C23×D19711yescharacteristic
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—3941no—
274C2×C4—3941no—
275C2×C4—3941no—
276C2×C4—3941no—
277C2×C4—3941no—
278C2×C4—3941no—
279C2×C4—3941no—
280C2×C4—3941no—
281C2×C4—3941no—
282C2×C4—3941no—
283C2×C4—3941no—
284C2×C4—3941no—
285C2×C4—3941no—
286C2×C4—3941no—
287C2×C4—1972no—
288C2×C4—1972no—
289C2×C4—1972no—
290C2×C4—1972no—
291C2×C4—1972no—
292C2×C4—1972no—
293C2×C4—1972no—
294C2×C4—1972no—
295C2×C4—1972no—
296C2×C4—1972no—
297C2×C4C2×C4×D19712yes—
298C2×C4C2×C4×D19712yes—
299C2×C4C2×C4×D19712yes—
300C2×C4Q8×D19712yes—
301C2×C4#1569135712yes—
302C22C22×C4×D19711yescharacteristic
303C22C2×Q8×D19711yescharacteristic
304C22#41×D19711yescharacteristic
305C22#41×D19711yescharacteristic
306C22#41×D19711yescharacteristic
307C22#41×D19711yescharacteristic
308C22C2×#1569135711yescharacteristic
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—22no—
314C4—41no—
315C4—41no—
316C4—7881no—
317C4—7881no—
318C4—3942no—
319C4—3942no—
320C4—3942no—
321C4—3942no—
322C4—1974no—
323C4—1974no—
324C2C4×Q8×D19711yescharacteristic
325C2C4×#1569135711yescharacteristic
326C2#2716154711yescharacteristic
327C2#125×D19711yescharacteristic
328C2#2716160611yescharacteristic
329C2#2716161211yescharacteristic
330C2#2716163111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#387×C197C2
socle, prime radical, cosocleC23×C197C24
hypercenter#387D197
hypocenterC197C4×#125
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
16. #20263316C4semidirect
157. Dic197#125semidirect
165. C197C4×#125semidirect

Maximal subgroups

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

subgroupindex
C2×#202633162
#271609762
#271609742
#271609662
#271609632
#271609622
#271609432
C4⋊C4×Dic1972
#271609302
C42×Dic1972
#387×C1972
C4×#125197