Finite Group Databasebeta

← #27144451

Subgroups — #27144451

Order 12416.

516 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690809C211yesmaximal, characteristic
2C2×C4×Dic194C211yesmaximal, characteristic
3C2×#15690775C211yesmaximal, characteristic
4#20260639C211yesmaximal, characteristic
5#20260639C211yesmaximal, characteristic
6#20260622C211yesmaximal, characteristic
7#20260621C211yesmaximal, characteristic
8#20260620C211yesmaximal, characteristic
9#20260619C211yesmaximal, characteristic
10#20260618C211yesmaximal, characteristic
11#20260617C211yesmaximal, characteristic
12#20260616C211yesmaximal, characteristic
13#20260599C211yesmaximal, characteristic
14#20260580C211yesmaximal, characteristic
15#387×C97C211yesmaximal, characteristic
16C22×Dic194C2211yescharacteristic
17C22×Dic194C2211yescharacteristic
18#15690809C418yescomplemented
19C4×Dic194—24no—
20#15690775—24no—
21C2×C4⋊Dic97C2211yescharacteristic
22C2×C4⋊Dic97C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×#13095490C2211yescharacteristic
29C2×C4×Dic97C2211yescharacteristic
30C2×C4×Dic97C2211yescharacteristic
31C2×C4×Dic97C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35#15690717C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C2×C4⋊C4×C97C2211yescharacteristic
48C2×C4⋊C4×C97C2211yescharacteristic
49C2×C42×C97C2211yescharacteristic
50#95×C97C2211yescharacteristic
51#95×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53#95×C97C2211yescharacteristic
54C2×Dic194—22no—
55C2×Dic194—22no—
56C2×Dic194—22no—
57C2×Dic194—22no—
58C2×Dic194C2×C414yes—
59C2×Dic194C2×C414yes—
60C22×Dic97C2311yescharacteristic
61C22×Dic97C2311yescharacteristic
62C22×Dic97C2311yescharacteristic
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C4⋊Dic97—22no—
69C4⋊Dic97C2×C414yes—
70#13095490—22no—
71#13095490—22no—
72#13095490—22no—
73#13095490—22no—
74#13095490C2×C414yes—
75#13095490C2×C414yes—
76C4×Dic97—22no—
77C4×Dic97—22no—
78C4×Dic97—22no—
79C4×Dic97—22no—
80C4×Dic97C2×C414yes—
81C4×Dic97D414yes—
82C22×C4×C97C2311yescharacteristic
83C22×C4×C97C2311yescharacteristic
84C22×C4×C97C2311yescharacteristic
85C22×C4×C97C2311yescharacteristic
86C22×C4×C97C2311yescharacteristic
87C22×C4×C97C2311yescharacteristic
88C22×C4×C97C2311yescharacteristic
89C4⋊C4×C97—22no—
90C4⋊C4×C97C2×C414yes—
91C42×C97—22no—
92C23×C97C2411yescharacteristic
93C2×C4×C97—21no—
94C2×C4×C97—21no—
95C2×C4×C97—21no—
96C2×C4×C97—21no—
97C2×C4×C97—21no—
98C2×C4×C97—21no—
99C2×C4×C97—21no—
100C2×C4×C97—21no—
101C2×C4×C97—21no—
102C2×C4×C97—21no—
103C2×C4×C97—21no—
104C2×C4×C97—21no—
105C2×C4×C97—21no—
106C2×C4×C97—21no—
107C2×C4×C97—21no—
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97C22×C412yes—
111C2×C4×C97C22×C412yes—
112C2×C4×C97C22×C412yes—
113C2×C4×C97C2×D412yes—
114C2×Dic97—21no—
115C2×Dic97—21no—
116C2×Dic97—21no—
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97—21no—
122C2×Dic97—21no—
123C2×Dic97—21no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97C22×C412yes—
132C2×Dic97C22×C412yes—
133C2×Dic97C22×C412yes—
134C2×Dic97C22×C412yes—
135C2×Dic97C2×D412yes—
136C2×Dic97C2×D412yes—
137C2×Dic97#4112yes—
138Dic194—28no—
139Dic194—28no—
140C22×C97C23×C411yescharacteristic
141C22×C97C22×D411yescharacteristic
142C22×C97C2×#4111yescharacteristic
143C22×C972−1+411yescharacteristic
144C22×C972−1+411yescharacteristic
145C22×C972−1+411yescharacteristic
146C22×C972−1+411yescharacteristic
147C4×C97—22no—
148C4×C97—22no—
149C4×C97—22no—
150C4×C97—22no—
151C4×C97—41no—
152C4×C97—41no—
153C4×C97—41no—
154Dic97—22no—
155Dic97—22no—
156Dic97—22no—
157Dic97—22no—
158Dic97—41no—
159Dic97—41no—
160Dic97—41no—
161Dic97C4×D414yes—
162C2×C97C2×C4×D411yescharacteristic
163C2×C97#51911yescharacteristic
164C2×C97#51911yescharacteristic
165C2×C97#53611yescharacteristic
166C2×C97#53611yescharacteristic
167C2×C97#54111yescharacteristic
168C2×C97#54111yescharacteristic
169#2327—971nomaximal, Sylow
170C97#232711yescharacteristic, complemented, Sylow
171C2×C4⋊Q8—971no—
172C2×C4×Q8—971no—
173C2×C4×Q8—971no—
174#391—971no—
175#391—971no—
176#387—971no—
177#387—971no—
178#387—971no—
179#387—971no—
180#387—971no—
181#387—971no—
182#387—971no—
183#387D9711yescharacteristic
184#382—971no—
185#376—971no—
186C22×Q8—971no—
187C22×Q8—971no—
188C4⋊Q8—978no—
189C4×Q8—1944no—
190C4×Q8—1944no—
191C2×C4⋊C4—971no—
192C2×C4⋊C4—971no—
193C2×C4⋊C4—971no—
194C2×C4⋊C4—971no—
195C2×C4⋊C4—971no—
196C2×C4⋊C4—971no—
197C2×C4⋊C4—971no—
198C2×C4⋊C4—971no—
199C2×C4⋊C4C2×D9711yescharacteristic
200C2×C4⋊C4C2×D9711yescharacteristic
201C2×C42—971no—
202C2×C42—971no—
203C2×C42—971no—
204C2×C42—971no—
205C2×C42—971no—
206C2×C42—971no—
207C2×C42C2×D9711yescharacteristic
208#95—971no—
209#95—971no—
210#95—971no—
211#95—971no—
212#95—971no—
213#95—971no—
214#95—971no—
215#95—971no—
216#95—971no—
217#95—971no—
218#95—971no—
219#95—971no—
220#95C2×D9711yescharacteristic
221#95C2×D9711yescharacteristic
222#95C2×D9711yescharacteristic
223#95C2×D9711yescharacteristic
224C2×Q8—974no—
225C2×Q8—974no—
226C2×Q8—1942no—
227C2×Q8—1942no—
228C2×Q8—1942no—
229C2×Q8—1942no—
230C22×C4—971no—
231C22×C4—971no—
232C22×C4—971no—
233C22×C4—971no—
234C22×C4—971no—
235C22×C4—971no—
236C22×C4—971no—
237C22×C4—971no—
238C22×C4C22×D9711yescharacteristic
239C22×C4C22×D9711yescharacteristic
240C22×C4C22×D9711yescharacteristic
241C22×C4C22×D9711yescharacteristic
242C22×C4C22×D9711yescharacteristic
243C22×C4C22×D9711yescharacteristic
244C22×C4C22×D9711yescharacteristic
245C4⋊C4—22no—
246C4⋊C4—974no—
247C4⋊C4—974no—
248C4⋊C4—974no—
249C4⋊C4—1942no—
250C4⋊C4—1942no—
251C4⋊C4—1942no—
252C4⋊C4—1942no—
253C4⋊C4—1942no—
254C4⋊C4C4×D9714yes—
255C42—22no—
256C42—974no—
257C42—974no—
258C42—1942no—
259C42—1942no—
260C42—1942no—
261C42—1942no—
262C23C23×D9711yescharacteristic
263Q8—1948no—
264Q8—1948no—
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—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—972no—
283C2×C4—972no—
284C2×C4—972no—
285C2×C4—972no—
286C2×C4—972no—
287C2×C4—972no—
288C2×C4—972no—
289C2×C4—1941no—
290C2×C4—1941no—
291C2×C4—1941no—
292C2×C4—1941no—
293C2×C4—1941no—
294C2×C4—1941no—
295C2×C4—1941no—
296C2×C4—1941no—
297C2×C4—1941no—
298C2×C4—1941no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4—1941no—
303C2×C4—1941no—
304C2×C4—1941no—
305C2×C4—1941no—
306C2×C4C2×C4×D9712yes—
307C2×C4C2×C4×D9712yes—
308C2×C4C2×C4×D9712yes—
309C2×C4D4×D9712yes—
310C22C22×C4×D9711yescharacteristic
311C22C2×D4×D9711yescharacteristic
312C22#41×D9711yescharacteristic
313C22#1569085711yescharacteristic
314C22#1569085711yescharacteristic
315C22#1569085711yescharacteristic
316C22#1569085811yescharacteristic
317C4—22no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—974no—
325C4—3881no—
326C4—3881no—
327C4—3881no—
328C4—1942no—
329C4—1942no—
330C4—1942no—
331C4—1942no—
332C2C4×D4×D9711yescharacteristic
333C2#2026121111yescharacteristic
334C2#2026121211yescharacteristic
335C2#2026130711yescharacteristic
336C2#2026130811yescharacteristic
337C2#2026134211yescharacteristic
338C2#2026134411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#387×C97C2
socle, prime radical, cosocleC23×C97C24
hypercenter#387D97
hypocenterC97#2327
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
18. #15690809C4semidirect
170. C97#2327semidirect

Maximal subgroups

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

subgroupindex
C2×#156908092
C2×C4×Dic1942
C2×#156907752
#202606392
#202606222
#202606212
#202606202
#202606192
#202606182
#202606172
#202606162
#202605992
#202605802
#387×C972
#232797