Finite Group Databasebeta

← #27144325

Subgroups — #27144325

Order 12416.

880 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095511C211yesmaximal, characteristic
2C2×#15690768C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
4#20260639C212yesmaximal, complemented
5#20260635C212yesmaximal, complemented
6#20260613C212yesmaximal, complemented
7#20260611C212yesmaximal, complemented
8#20260595C211yesmaximal, characteristic, complemented
9#20260592C211yesmaximal, characteristic, complemented
10#20260592C211yesmaximal, characteristic, complemented
11#380×C97C211yesmaximal, characteristic, complemented
12C2×#13095511—24no—
13C2×#13095511—24no—
14C2×#13095511C418yescomplemented
15C22×Dic194C2211yescharacteristic
16C22×D388C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C22×C4×D97C2211yescharacteristic
20C22×C4×D97C2212yes—
21C22×C4×D97C2211yescharacteristic
22#15690768—24no—
23C2×C4⋊Dic97C2211yescharacteristic
24C2×C4⋊Dic97C2211yescharacteristic
25C2×#13095490C2212yes—
26C4⋊C4×D97—24no—
27C2×C22⋊Dic97C2212yes—
28C2×#13095488C2212yes—
29C2×#13095488C2212yes—
30C2×#13095488C2212yes—
31C2×C4×Dic97C2212yes—
32#15690717C2212yes—
33#15690717C2212yes—
34#15690717C2212yes—
35C23×C4×C97C2211yescharacteristic
36C2×C4⋊C4×C97C2211yescharacteristic
37C2×C4⋊C4×C97C2211yescharacteristic
38C2×C22⋊C4×C97C2212yes—
39#95×C97C2212yes—
40C23×D97C2311yescharacteristic
41C23×D97C2311yescharacteristic
42#13095511—232no—
43C2×Dic194—22no—
44C2×Dic194—22no—
45C2×Dic194C2×C414yescomplemented
46C2×D388—22no—
47C2×D388—22no—
48C2×D388C2×C414yes—
49C2×C22⋊D97—22no—
50C2×C22⋊D97—22no—
51C2×C22⋊D97—22no—
52C2×C22⋊D97—22no—
53C2×C22⋊D97C2×C414yes—
54C2×C22⋊D97C2×C414yescomplemented
55C22×Dic97C2312yes—
56C22×Dic97C2312yes—
57C22×Dic97C2311yescharacteristic
58C22×Dic97C2311yescharacteristic
59C2×C4×D97—22no—
60C2×C4×D97—22no—
61C2×C4×D97—24no—
62C2×C4×D97—42no—
63C2×C4×D97—42no—
64C2×C4×D97C2×C414yes—
65C2×C4×D97C2×C414yescomplemented
66C2×C4×D97D414yes—
67C2×C4×D97D414yes—
68C4⋊Dic97—22no—
69C4⋊Dic97—22no—
70#13095490—24no—
71C22⋊Dic97—42no—
72#13095488—24no—
73#13095488—42no—
74#13095488—42no—
75C4×Dic97—24no—
76C24×C97C2311yescharacteristic
77C22×C4×C97—22no—
78C22×C4×C97—22no—
79C22×C4×C97C2×C414yescomplemented
80C22×C4×C97C2312yes—
81C22×C4×C97C2312yes—
82C22×C4×C97C2311yescharacteristic
83C22×C4×C97C2311yescharacteristic
84C4⋊C4×C97—22no—
85C4⋊C4×C97—22no—
86C22⋊C4×C97—42no—
87C23×C97—21no—
88C23×C97—21no—
89C23×C97—21no—
90C23×C97—21no—
91C23×C97—21no—
92C23×C97—21no—
93C23×C97C22×C412yes—
94C23×C97C2411yescharacteristic
95C22×D97—21no—
96C22×D97—21no—
97C22×D97—21no—
98C22×D97—21no—
99C22×D97—21no—
100C22×D97—21no—
101C22×D97—21no—
102C22×D97—21no—
103C22×D97—21no—
104C22×D97—21no—
105C22×D97C22×C412yes—
106C22×D97C22×C412yes—
107C22×D97C2×D412yes—
108C22×D97C2×D412yes—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—28no—
112C2×C4×C97—22no—
113C2×C4×C97—22no—
114C2×C4×C97—22no—
115C2×C4×C97—22no—
116C2×C4×C97—22no—
117C2×C4×C97—22no—
118C2×C4×C97C22×C412yes—
119C2×C4×C97C22×C412yes—
120C2×C4×C97C2×D412yes—
121C2×C4×C97C2×D412yes—
122C22⋊D97—28no—
123C22⋊D97—28no—
124C2×Dic97—21no—
125C2×Dic97—21no—
126C2×Dic97—22no—
127C2×Dic97—22no—
128C2×Dic97—22no—
129C2×Dic97—22no—
130C2×Dic97—22no—
131C2×Dic97—22no—
132C2×Dic97C22×C412yes—
133C2×Dic97C22×C412yes—
134C2×Dic97C2×D412yes—
135C2×Dic97C2×D412yes—
136D388—28no—
137C4×D97—28no—
138C4×D97—48no—
139C4×D97C22⋊C4116yescomplemented
140Dic194—28no—
141C22×C97—22no—
142C22×C97—22no—
143C22×C97—22no—
144C22×C97—41no—
145C22×C97—41no—
146C22×C97—41no—
147C22×C97—41no—
148C22×C97C23×C411yescharacteristic
149C22×C97C22×D411yescharacteristic
150C22×C97C22×D411yescharacteristic
151C22×C97H(5,𝔽2)11yescharacteristic
152C22×C97H(5,𝔽2)11yescharacteristic
153C22×C972−1+411yescharacteristic
154C22×C972−1+411yescharacteristic
155C2×D97—22no—
156C2×D97—22no—
157C2×D97—22no—
158C2×D97—22no—
159C2×D97—22no—
160C2×D97—22no—
161C2×D97—22no—
162C2×D97—22no—
163C2×D97—22no—
164C2×D97—41no—
165C2×D97—41no—
166C2×D97—41no—
167C2×D97—41no—
168C2×D97C2×C22⋊C414yes—
169C4×C97—22no—
170C4×C97—42no—
171C4×C97—42no—
172C4×C97C2×C22⋊C414yes—
173Dic97—22no—
174Dic97—42no—
175Dic97—42no—
176Dic97C2×C22⋊C414yes—
177C2×C97—42no—
178C2×C97C22×C22⋊C411yescharacteristic
179C2×C97#52011yescharacteristic
180C2×C97#52011yescharacteristic
181C2×C97#53511yescharacteristic
182C2×C97#53611yescharacteristic
183C2×C97#53711yescharacteristic
184C2×C97#53711yescharacteristic
185D97—24no—
186D97—42no—
187#2311—971nomaximal, Sylow
188C97#231111yescharacteristic, complemented, Sylow
189C22×#41—971no—
190C2×#117—971no—
191C22×C4⋊C4—971no—
192#391—972no—
193#390—972no—
194#386—972no—
195#386—972no—
196#380—971no—
197#380—971no—
198#380—971no—
199#380D9711yescharacteristic, complemented
200C2×#41—978no—
201C2×#41—1944no—
202C2×#41—1944no—
203C22×Q8—971no—
204C22×D4—971no—
205C22×D4—971no—
206C22×D4—971no—
207C23×C4—971no—
208C23×C4—971no—
209C23×C4—972no—
210C23×C4C2×D9711yescharacteristic
211#117—1944no—
212C2×C4⋊C4—972no—
213C2×C4⋊C4—971no—
214C2×C4⋊C4—971no—
215C2×C4⋊C4—1944no—
216C2×C4⋊C4C2×D9711yescharacteristic
217C2×C4⋊C4C2×D9711yescharacteristic
218C2×C22⋊C4—972no—
219C2×C22⋊C4—972no—
220C2×C22⋊C4—972no—
221C2×C22⋊C4—972no—
222C2×C22⋊C4C2×D9712yes—
223C2×C42—972no—
224#95—972no—
225#95—972no—
226#95—972no—
227#95C2×D9712yes—
228C24—971no—
229C24—971no—
230C24C22×D9711yescharacteristic
231#41—19432no—
232C2×Q8—974no—
233C2×Q8—1942no—
234C2×Q8—1942no—
235C2×D4—974no—
236C2×D4—974no—
237C2×D4—974no—
238C2×D4—1942no—
239C2×D4—1942no—
240C2×D4—1942no—
241C2×D4—1942no—
242C2×D4—1942no—
243C2×D4—1942no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—971no—
247C22×C4—974no—
248C22×C4—974no—
249C22×C4—974no—
250C22×C4—971no—
251C22×C4—974no—
252C22×C4—972no—
253C22×C4—972no—
254C22×C4—3882no—
255C22×C4—3882no—
256C22×C4—1944no—
257C22×C4—1942no—
258C22×C4—1942no—
259C22×C4C4×D9714yescomplemented
260C22×C4C22×D9712yes—
261C22×C4C22×D9712yes—
262C22×C4C22×D9711yescharacteristic
263C22×C4C22×D9711yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—1944no—
267C4⋊C4—1942no—
268C4⋊C4—1942no—
269C22⋊C4—42no—
270C22⋊C4—3882no—
271C22⋊C4—3882no—
272C22⋊C4—3882no—
273C22⋊C4—1944no—
274C42—1944no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—972no—
282C23—972no—
283C23—972no—
284C23—972no—
285C23—1941no—
286C23—1941no—
287C23—1941no—
288C23—1941no—
289C23—1941no—
290C23—1941no—
291C23—1941no—
292C23—1941no—
293C23—1941no—
294C23—1941no—
295C23C2×C4×D9712yes—
296C23C23×D9711yescharacteristic
297Q8—1948no—
298D4—1948no—
299D4—1948no—
300D4—1948no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—972no—
311C2×C4—972no—
312C2×C4—972no—
313C2×C4—9716no—
314C2×C4—972no—
315C2×C4—3888no—
316C2×C4—1942no—
317C2×C4—1942no—
318C2×C4—1942no—
319C2×C4—1942no—
320C2×C4—1942no—
321C2×C4—1942no—
322C2×C4—1941no—
323C2×C4—1941no—
324C2×C4—1948no—
325C2×C4C2×C4×D9712yes—
326C2×C4C2×C4×D9712yes—
327C2×C4D4×D9712yes—
328C2×C4D4×D9712yes—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—974no—
337C22—3881no—
338C22—3881no—
339C22—3881no—
340C22—3881no—
341C22—1942no—
342C22—1942no—
343C22—1942no—
344C22—1942no—
345C22—1942no—
346C22—1942no—
347C22—1942no—
348C22—1942no—
349C22—1942no—
350C22C22×C4×D9711yescharacteristic
351C22C2×D4×D9711yescharacteristic
352C22C2×D4×D9711yescharacteristic
353C22#1569085511yescharacteristic
354C22#1569085611yescharacteristic
355C22#1569085711yescharacteristic
356C22#1569085811yescharacteristic
357C4—22no—
358C4—42no—
359C4—42no—
360C4—974no—
361C4—3882no—
362C4—3882no—
363C4—1942no—
364C4C22⋊C4×D9714yes—
365C2—42no—
366C2—3882no—
367C2—1944no—
368C2C2×C22⋊C4×D9711yescharacteristic
369C2#2026121511yescharacteristic
370C2#2026121611yescharacteristic
371C2#2026130111yescharacteristic
372C2#2026130811yescharacteristic
373C2#2026131411yescharacteristic
374C2#2026131511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#380×C97C2
socle, cosocleC23×C97C24
prime radicalC22×#13095511C2
hypercenter#380D97
hypocenterC97#2311
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#15690768C2semidirect
3. C2×C4⋊C4×D97C2semidirect
4. #20260639C2semidirect
5. #20260635C2semidirect
6. #20260613C2semidirect
7. #20260611C2semidirect
8. #20260595C2semidirect
9. #20260592C2semidirect
10. #20260592C2semidirect
11. #380×C97C2semidirect
14. C2×#13095511C4semidirect
45. C2×Dic194C2×C4semidirect
54. C2×C22⋊D97C2×C4semidirect
65. C2×C4×D97C2×C4semidirect
79. C22×C4×C97C2×C4semidirect
139. C4×D97C22⋊C4semidirect
188. C97#2311semidirect
199. #380D97semidirect
259. C22×C4C4×D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955112
C2×#156907682
C2×C4⋊C4×D972
#202606392
#202606352
#202606132
#202606112
#202605952
#202605922
#380×C972
#231197