Finite Group Databasebeta

← #27144415

Subgroups — #27144415

Order 12416.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690787C211yesmaximal, characteristic
2C2×C4×Dic194C211yesmaximal, characteristic, complemented
3C2×#15690766C211yesmaximal, characteristic
4#20260639C211yesmaximal, characteristic, complemented
5#20260632C211yesmaximal, characteristic, complemented
6#20260631C211yesmaximal, characteristic, complemented
7#20260630C211yesmaximal, characteristic, complemented
8#20260628C211yesmaximal, characteristic
9#20260623C211yesmaximal, characteristic
10#20260623C211yesmaximal, characteristic
11#20260620C211yesmaximal, characteristic, complemented
12#20260619C211yesmaximal, characteristic, complemented
13#20260593C211yesmaximal, characteristic
14#20260586C211yesmaximal, characteristic, complemented
15#388×C97C211yesmaximal, characteristic
16C22×Dic194C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18#15690787C418yescomplemented
19C4×Dic194—24no—
20#15690766—24no—
21C2×C4⋊Dic97C2211yescharacteristic
22C2×C4⋊Dic97C2211yescharacteristic
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×#13095490C2211yescharacteristic
29C2×#13095490C2211yescharacteristic
30C2×#13095490C2211yescharacteristic
31C2×C22⋊Dic97C2211yescharacteristic
32C2×C22⋊Dic97C2211yescharacteristic
33C2×C22⋊Dic97C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35C2×C4×Dic97C2211yescharacteristic
36C2×C4×Dic97C2211yescharacteristic
37C2×C4×Dic97C2211yescharacteristic
38C2×C4×Dic97C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C2×C4⋊C4×C97C2211yescharacteristic
48C2×C22⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C22⋊C4×C97C2211yescharacteristic
51C2×C42×C97C2211yescharacteristic
52#95×C97C2211yescharacteristic
53#95×C97C2211yescharacteristic
54C2×Dic194—22no—
55C2×Dic194—22no—
56C2×Dic194C2×C414yes—
57C22×Dic97—22no—
58C22×Dic97—22no—
59C22×Dic97C2×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#13095490—22no—
75#13095490—41no—
76#13095490C2×C414yes—
77#13095490C2×C414yes—
78C22⋊Dic97—22no—
79C22⋊Dic97—41no—
80C22⋊Dic97C2×C414yes—
81C4×Dic97—22no—
82C4×Dic97—22no—
83C4×Dic97—41no—
84C4×Dic97—41no—
85C4×Dic97D414yes—
86C24×C97C2311yescharacteristic
87C22×C4×C97C2311yescharacteristic
88C22×C4×C97C2311yescharacteristic
89C22×C4×C97C2311yescharacteristic
90C22×C4×C97C2311yescharacteristic
91C22×C4×C97C2311yescharacteristic
92C22×C4×C97C2311yescharacteristic
93C4⋊C4×C97—41no—
94C22⋊C4×C97—22no—
95C22⋊C4×C97—41no—
96C22⋊C4×C97C2×C414yes—
97C42×C97—22no—
98C23×C97—21no—
99C23×C97—21no—
100C23×C97—21no—
101C23×C97—21no—
102C23×C97—21no—
103C23×C97—21no—
104C23×C97C22×C412yes—
105C23×C97C2411yescharacteristic
106C2×C4×C97—21no—
107C2×C4×C97—21no—
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97—21no—
115C2×C4×C97—21no—
116C2×C4×C97—21no—
117C2×C4×C97—21no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97C22×C412yes—
122C2×C4×C97C22×C412yes—
123C2×C4×C97C2×D412yes—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—21no—
127C2×Dic97—21no—
128C2×Dic97—21no—
129C2×Dic97—21no—
130C2×Dic97—21no—
131C2×Dic97—21no—
132C2×Dic97—28no—
133C2×Dic97—21no—
134C2×Dic97—21no—
135C2×Dic97—21no—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—21no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142C2×Dic97C22×C412yes—
143C2×Dic97C22×C412yes—
144C2×Dic97C22×C412yes—
145C2×Dic97C22×C412yes—
146C2×Dic97C2×D412yes—
147C2×Dic97C2×D412yes—
148C2×Dic97#4112yes—
149Dic194—28no—
150C22×C97—22no—
151C22×C97—22no—
152C22×C97—22no—
153C22×C97—41no—
154C22×C97—41no—
155C22×C97—41no—
156C22×C97—41no—
157C22×C97C23×C411yescharacteristic
158C22×C97C22×D411yescharacteristic
159C22×C97C2×#4111yescharacteristic
160C22×C97H(5,𝔽2)11yescharacteristic
161C22×C972−1+411yescharacteristic
162C22×C972−1+411yescharacteristic
163C22×C972−1+411yescharacteristic
164C4×C97—22no—
165C4×C97—22no—
166C4×C97—22no—
167C4×C97—41no—
168C4×C97—41no—
169C4×C97—41no—
170Dic97—22no—
171Dic97—22no—
172Dic97—22no—
173Dic97—22no—
174Dic97—41no—
175Dic97—41no—
176Dic97—41no—
177Dic97C4×D414yes—
178C2×C97—42no—
179C2×C97C2×C4×D411yescharacteristic
180C2×C97#51911yescharacteristic
181C2×C97#52011yescharacteristic
182C2×C97#53611yescharacteristic
183C2×C97#53711yescharacteristic
184C2×C97#54111yescharacteristic
185C2×C97#54211yescharacteristic
186#2323—971nomaximal, Sylow
187C97#232311yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—971no—
189C2×C4×Q8—971no—
190C2×#117—971no—
191#391—971no—
192#389—971no—
193#389—971no—
194#389—971no—
195#388—971no—
196#388—971no—
197#388—971no—
198#388D9711yescharacteristic
199#387—971no—
200#387—971no—
201#380—971no—
202C4×C4⋊C4—971no—
203C22×Q8—971no—
204C23×C4—971no—
205C22⋊Q8—978no—
206C4×Q8—1944no—
207#117—1944no—
208C2×C4⋊C4—971no—
209C2×C4⋊C4—971no—
210C2×C4⋊C4—971no—
211C2×C4⋊C4—971no—
212C2×C4⋊C4—971no—
213C2×C4⋊C4—971no—
214C2×C4⋊C4—971no—
215C2×C4⋊C4—971no—
216C2×C4⋊C4—971no—
217C2×C4⋊C4—971no—
218C2×C4⋊C4C2×D9711yescharacteristic
219C2×C22⋊C4—971no—
220C2×C22⋊C4—971no—
221C2×C22⋊C4—971no—
222C2×C22⋊C4C2×D9711yescharacteristic
223C2×C22⋊C4C2×D9711yescharacteristic
224C2×C22⋊C4C2×D9711yescharacteristic
225C2×C42—971no—
226C2×C42—971no—
227C2×C42—971no—
228C2×C42—971no—
229C2×C42—971no—
230C2×C42C2×D9711yescharacteristic
231#95—971no—
232#95—971no—
233#95—971no—
234#95—971no—
235#95—971no—
236#95—971no—
237#95—971no—
238#95—971no—
239#95C2×D9711yescharacteristic
240#95C2×D9711yescharacteristic
241C24C22×D9711yescharacteristic
242C2×Q8—974no—
243C2×Q8—1942no—
244C2×Q8—1942no—
245C22×C4—971no—
246C22×C4—974no—
247C22×C4—971no—
248C22×C4—971no—
249C22×C4—971no—
250C22×C4—971no—
251C22×C4—971no—
252C22×C4—971no—
253C22×C4—971no—
254C22×C4—1942no—
255C22×C4—1942no—
256C22×C4C22×D9711yescharacteristic
257C22×C4C22×D9711yescharacteristic
258C22×C4C22×D9711yescharacteristic
259C22×C4C22×D9711yescharacteristic
260C22×C4C22×D9711yescharacteristic
261C22×C4C22×D9711yescharacteristic
262C4⋊C4—41no—
263C4⋊C4—974no—
264C4⋊C4—974no—
265C4⋊C4—974no—
266C4⋊C4—3881no—
267C4⋊C4—1942no—
268C4⋊C4—1942no—
269C4⋊C4—1942no—
270C4⋊C4—1942no—
271C4⋊C4—1942no—
272C4⋊C4—1942no—
273C22⋊C4—22no—
274C22⋊C4—41no—
275C22⋊C4—974no—
276C22⋊C4—3881no—
277C22⋊C4—1942no—
278C22⋊C4C4×D9714yes—
279C42—22no—
280C42—974no—
281C42—3881no—
282C42—3881no—
283C42—1942no—
284C42—1942no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23C2×C4×D9712yes—
292C23C23×D9711yescharacteristic
293Q8—1948no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—972no—
310C2×C4—972no—
311C2×C4—972no—
312C2×C4—972no—
313C2×C4—972no—
314C2×C4—972no—
315C2×C4—972no—
316C2×C4—1948no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1941no—
321C2×C4—1941no—
322C2×C4—1941no—
323C2×C4—1941no—
324C2×C4—1941no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1941no—
328C2×C4—1941no—
329C2×C4—1941no—
330C2×C4—1941no—
331C2×C4—1941no—
332C2×C4—1941no—
333C2×C4—1941no—
334C2×C4C2×C4×D9712yes—
335C2×C4C2×C4×D9712yes—
336C2×C4D4×D9712yes—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22C22×C4×D9711yescharacteristic
345C22C2×D4×D9711yescharacteristic
346C22#41×D9711yescharacteristic
347C22#1569085511yescharacteristic
348C22#1569085711yescharacteristic
349C22#1569085711yescharacteristic
350C22#1569085711yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—974no—
358C4—3881no—
359C4—3881no—
360C4—3881no—
361C4—1942no—
362C4—1942no—
363C4—1942no—
364C4—1942no—
365C2—42no—
366C2C4×D4×D9711yescharacteristic
367C2#2026121211yescharacteristic
368C2#2026121411yescharacteristic
369C2#2026130711yescharacteristic
370C2#2026131411yescharacteristic
371C2#2026134211yescharacteristic
372C2#2026134911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#388×C97C2
socle, cosocleC23×C97C24
prime radicalC24×C97C23
hypercenter#388D97
hypocenterC97#2323
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic194C2semidirect
4. #20260639C2semidirect
5. #20260632C2semidirect
6. #20260631C2semidirect
7. #20260630C2semidirect
11. #20260620C2semidirect
12. #20260619C2semidirect
14. #20260586C2semidirect
18. #15690787C4semidirect
187. C97#2323semidirect

Maximal subgroups

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

subgroupindex
C2×#156907872
C2×C4×Dic1942
C2×#156907662
#202606392
#202606322
#202606312
#202606302
#202606282
#202606232
#202606202
#202606192
#202605932
#202605862
#388×C972
#232397