Finite Group Databasebeta

← #27144732

Subgroups — #27144732

Order 12416.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic97C211yesmaximal, characteristic
2C2×#15690769C211yesmaximal, characteristic, complemented
3#20260634C211yesmaximal, characteristic, complemented
4#20260628C211yesmaximal, characteristic, complemented
5#20260628C211yesmaximal, characteristic, complemented
6#20260621C211yesmaximal, characteristic, complemented
7#20260621C211yesmaximal, characteristic, complemented
8#20260615C211yesmaximal, characteristic
9#20260610C211yesmaximal, characteristic, complemented
10#20260610C211yesmaximal, characteristic, complemented
11C4⋊C4×Dic97C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
14C2×#15690717C211yesmaximal, characteristic, complemented
15C2×#123×C97C211yesmaximal, characteristic
16C23×Dic97C2211yescharacteristic
17C23×Dic97C2211yescharacteristic
18C23×Dic97C2211yescharacteristic
19D4×Dic97—24no—
20#15690769—24no—
21C2×C4⋊Dic97C2211yescharacteristic
22C2×C4⋊Dic97C2211yescharacteristic
23C2×C4⋊Dic97C2211yescharacteristic
24C2×C4⋊Dic97C2211yescharacteristic
25C2×C4⋊Dic97C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×C22⋊Dic97C2211yescharacteristic
31C2×C4×Dic97C2211yescharacteristic
32C2×C4×Dic97C2211yescharacteristic
33C2×C4×Dic97C2211yescharacteristic
34C2×C4×Dic97C2211yescharacteristic
35C2×C4×Dic97C2211yescharacteristic
36#15690717C2211yescharacteristic
37#15690717C2211yescharacteristic
38#15690717C2211yescharacteristic
39#15690717C2211yescharacteristic
40#15690717C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46C22×D4×C97C2211yescharacteristic
47C23×C4×C97C2211yescharacteristic
48#123×C97C418yescomplemented
49C2×C4⋊C4×C97C2211yescharacteristic
50C2×C4⋊C4×C97C2211yescharacteristic
51C2×C22⋊C4×C97C2211yescharacteristic
52C2×C22⋊C4×C97C2211yescharacteristic
53C2×C22⋊C4×C97C2211yescharacteristic
54C22×Dic97—22no—
55C22×Dic97—22no—
56C22×Dic97—22no—
57C22×Dic97—22no—
58C22×Dic97—22no—
59C22×Dic97—22no—
60C22×Dic97—41no—
61C22×Dic97—41no—
62C22×Dic97—41no—
63C22×Dic97C2311yescharacteristic
64C22×Dic97C2311yescharacteristic
65C22×Dic97C2311yescharacteristic
66C22×Dic97C2311yescharacteristic
67C22×Dic97C2311yescharacteristic
68C22×Dic97C2311yescharacteristic
69C22×Dic97C2311yescharacteristic
70C22×Dic97C2311yescharacteristic
71C4⋊Dic97—22no—
72C4⋊Dic97—22no—
73C4⋊Dic97—22no—
74C4⋊Dic97—22no—
75C4⋊Dic97—41no—
76C22⋊Dic97—22no—
77C22⋊Dic97—22no—
78C22⋊Dic97—22no—
79C22⋊Dic97—41no—
80C22⋊Dic97D414yes—
81C4×Dic97—22no—
82C4×Dic97—22no—
83C4×Dic97—22no—
84C4×Dic97—22no—
85C4×Dic97—41no—
86C24×C97C2311yescharacteristic
87C24×C97C2311yescharacteristic
88C2×D4×C97—22no—
89C2×D4×C97—22no—
90C2×D4×C97C2×C414yes—
91C22×C4×C97—22no—
92C22×C4×C97—22no—
93C22×C4×C97C2×C414yescomplemented
94C22×C4×C97C2311yescharacteristic
95C22×C4×C97C2311yescharacteristic
96C22×C4×C97C2311yescharacteristic
97C22×C4×C97C2311yescharacteristic
98C22×C4×C97C2311yescharacteristic
99C4⋊C4×C97C2×C414yescomplemented
100C4⋊C4×C97C2×C414yescomplemented
101C22⋊C4×C97C2×C414yescomplemented
102C22⋊C4×C97C2×C414yescomplemented
103C22⋊C4×C97C2×C414yescomplemented
104C23×C97—21no—
105C23×C97—21no—
106C23×C97—21no—
107C23×C97—21no—
108C23×C97—21no—
109C23×C97—21no—
110C23×C97—21no—
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—21no—
114C23×C97C22×C412yes—
115C23×C97C22×C412yes—
116C23×C97C2×D412yes—
117C23×C97#4112yes—
118C23×C97C2411yescharacteristic
119D4×C97—28no—
120C2×C4×C97—28no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—21no—
124C2×C4×C97—21no—
125C2×C4×C97—21no—
126C2×C4×C97—21no—
127C2×C4×C97—21no—
128C2×C4×C97—21no—
129C2×C4×C97—21no—
130C2×C4×C97—21no—
131C2×C4×C97C22×C412yes—
132C2×C4×C97C22×C412yes—
133C2×C4×C97C22×C412yes—
134C2×C4×C97C22×C412yes—
135C2×C4×C97C22×C412yes—
136C2×Dic97—21no—
137C2×Dic97—21no—
138C2×Dic97—21no—
139C2×Dic97—21no—
140C2×Dic97—21no—
141C2×Dic97—21no—
142C2×Dic97—21no—
143C2×Dic97—21no—
144C2×Dic97—21no—
145C2×Dic97—21no—
146C2×Dic97—21no—
147C2×Dic97—21no—
148C2×Dic97—21no—
149C2×Dic97—21no—
150C2×Dic97—21no—
151C2×Dic97—21no—
152C2×Dic97—21no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—44no—
157C2×Dic97—44no—
158C2×Dic97—44no—
159C2×Dic97C2×D412yes—
160C2×Dic97C2×D412yes—
161C2×Dic97#4112yes—
162C2×Dic97#4112yes—
163C22×C97—22no—
164C22×C97—22no—
165C22×C97—22no—
166C22×C97—22no—
167C22×C97—22no—
168C22×C97—22no—
169C22×C97—22no—
170C22×C97—22no—
171C22×C97—22no—
172C22×C97—41no—
173C22×C97—41no—
174C22×C97—41no—
175C22×C97—41no—
176C22×C97C4×D414yes—
177C22×C97C23×C411yescharacteristic
178C22×C97C22×D411yescharacteristic
179C22×C97C2×#4111yescharacteristic
180C22×C97C2×#4111yescharacteristic
181C22×C97C2×#4111yescharacteristic
182C22×C97H(5,𝔽2)11yescharacteristic
183C22×C972−1+411yescharacteristic
184C4×C97—22no—
185C4×C97—22no—
186C4×C97—22no—
187C4×C97—22no—
188C4×C97—22no—
189Dic97—22no—
190Dic97—22no—
191Dic97—22no—
192Dic97—22no—
193Dic97—41no—
194Dic97—41no—
195Dic97—41no—
196Dic97—41no—
197C2×C97—24no—
198C2×C97—42no—
199C2×C97C2×C4×D411yescharacteristic
200C2×C97C4×#4111yescharacteristic
201C2×C97#52011yescharacteristic
202C2×C97#54611yescharacteristic
203C2×C97#54711yescharacteristic
204C2×C97#55111yescharacteristic
205C2×C97#55211yescharacteristic
206#2359—971nomaximal, Sylow
207C97#235911yescharacteristic, complemented, Sylow
208C2×#123D9711yescharacteristic
209C2×C4×D4—971no—
210C2×#117—971no—
211#389—971no—
212#388—971no—
213#388—971no—
214#387—971no—
215#387—971no—
216#386—971no—
217#385—971no—
218#385—971no—
219C4×C4⋊C4—971no—
220C4×C22⋊C4—971no—
221C4×C22⋊C4—971no—
222C2×#95—971no—
223C22×D4C2×D9711yescharacteristic
224C23×C4—971no—
225C23×C4—971no—
226C23×C4—971no—
227C23×C4C2×D9711yescharacteristic
228#123Dic9718yescomplemented
229C4×D4—1944no—
230#117—1944no—
231C2×C4⋊C4—971no—
232C2×C4⋊C4—971no—
233C2×C4⋊C4—971no—
234C2×C4⋊C4—971no—
235C2×C4⋊C4—971no—
236C2×C4⋊C4C2×D9711yescharacteristic
237C2×C4⋊C4C2×D9711yescharacteristic
238C2×C22⋊C4—971no—
239C2×C22⋊C4—971no—
240C2×C22⋊C4—971no—
241C2×C22⋊C4—971no—
242C2×C22⋊C4—971no—
243C2×C22⋊C4C2×D9711yescharacteristic
244C2×C22⋊C4C2×D9711yescharacteristic
245C2×C22⋊C4C2×D9711yescharacteristic
246C2×C42—971no—
247C2×C42—971no—
248C2×C42—971no—
249C2×C42—971no—
250C2×C42—971no—
251#95—971no—
252#95—971no—
253#95—971no—
254#95—971no—
255#95—971no—
256#95—971no—
257#95—971no—
258#95—971no—
259#95—971no—
260#95—971no—
261C24C22×D9711yescharacteristic
262C24C22×D9711yescharacteristic
263C2×D4—22no—
264C2×D4—22no—
265C2×D4C2×Dic9714yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—971no—
269C22×C4—971no—
270C22×C4—971no—
271C22×C4—971no—
272C22×C4—971no—
273C22×C4—971no—
274C22×C4—971no—
275C22×C4—971no—
276C22×C4—3881no—
277C22×C4—3881no—
278C22×C4—3881no—
279C22×C4—1942no—
280C22×C4—1942no—
281C22×C4—1942no—
282C22×C4—1942no—
283C22×C4—1942no—
284C22×C4—1942no—
285C22×C4C2×Dic9714yescomplemented
286C22×C4C22×D9711yescharacteristic
287C22×C4C22×D9711yescharacteristic
288C22×C4C22×D9711yescharacteristic
289C22×C4C22×D9711yescharacteristic
290C22×C4C22×D9711yescharacteristic
291C4⋊C4—3881no—
292C4⋊C4—1942no—
293C4⋊C4—1942no—
294C4⋊C4—1942no—
295C4⋊C4—1942no—
296C4⋊C4C2×Dic9714yescomplemented
297C4⋊C4C2×Dic9714yescomplemented
298C22⋊C4—974no—
299C22⋊C4—3881no—
300C22⋊C4—1942no—
301C22⋊C4—1942no—
302C22⋊C4—1942no—
303C22⋊C4C2×Dic9714yescomplemented
304C22⋊C4C2×Dic9714yescomplemented
305C22⋊C4C2×Dic9714yescomplemented
306C42—3881no—
307C42—1942no—
308C42—1942no—
309C42—1942no—
310C42—1942no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23C22×Dic9712yes—
322C23C22×Dic9712yes—
323C23D4×D9712yes—
324C23#1309551012yes—
325C23C23×D9711yescharacteristic
326D4—28no—
327C2×C4—28no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—972no—
339C2×C4—972no—
340C2×C4—972no—
341C2×C4—972no—
342C2×C4—3884no—
343C2×C4—3884no—
344C2×C4—3884no—
345C2×C4—1941no—
346C2×C4—1941no—
347C2×C4—1941no—
348C2×C4—1941no—
349C2×C4—1941no—
350C2×C4—1941no—
351C2×C4—1941no—
352C2×C4—1941no—
353C2×C4—1941no—
354C2×C4—1941no—
355C2×C4—1941no—
356C2×C4—1941no—
357C2×C4—1941no—
358C2×C4—1941no—
359C2×C4—1941no—
360C2×C4—1941no—
361C2×C4—1941no—
362C2×C4—1941no—
363C2×C4—1941no—
364C2×C4—1941no—
365C2×C4C22×Dic9712yes—
366C2×C4C22×Dic9712yes—
367C2×C4C22×Dic9712yes—
368C2×C4C22×Dic9712yes—
369C2×C4C22×Dic9712yes—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22D4×Dic9714yes—
384C22C23×Dic9711yescharacteristic
385C22C2×D4×D9711yescharacteristic
386C22#41×D9711yescharacteristic
387C22#41×D9711yescharacteristic
388C22C2×#1309551011yescharacteristic
389C22#1569085611yescharacteristic
390C22#1569085711yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—3881no—
397C4—3881no—
398C4—3881no—
399C4—3881no—
400C4—1942no—
401C4—1942no—
402C4—1942no—
403C4—1942no—
404C2—24no—
405C2—42no—
406C2C2×D4×Dic9711yescharacteristic
407C2#41×Dic9711yescharacteristic
408C2#2026121811yescharacteristic
409C2#2026137711yescharacteristic
410C2#2026138311yescharacteristic
411C2#2026140811yescharacteristic
412C2#2026142211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC2×#123×C97C2
socle, cosocleC23×C97C24
prime radicalC22×D4×C97C22
hypercenterC2×#123D97
hypocenterC97#2359
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#15690769C2semidirect
3. #20260634C2semidirect
4. #20260628C2semidirect
5. #20260628C2semidirect
6. #20260621C2semidirect
7. #20260621C2semidirect
9. #20260610C2semidirect
10. #20260610C2semidirect
11. C4⋊C4×Dic97C2semidirect
12. C22⋊C4×Dic97C2semidirect
13. C22⋊C4×Dic97C2semidirect
14. C2×#15690717C2semidirect
48. #123×C97C4semidirect
93. C22×C4×C97C2×C4semidirect
99. C4⋊C4×C97C2×C4semidirect
100. C4⋊C4×C97C2×C4semidirect
101. C22⋊C4×C97C2×C4semidirect
102. C22⋊C4×C97C2×C4semidirect
103. C22⋊C4×C97C2×C4semidirect
207. C97#2359semidirect
228. #123Dic97semidirect
285. C22×C4C2×Dic97semidirect
296. C4⋊C4C2×Dic97semidirect
297. C4⋊C4C2×Dic97semidirect
303. C22⋊C4C2×Dic97semidirect
304. C22⋊C4C2×Dic97semidirect
305. C22⋊C4C2×Dic97semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic972
C2×#156907692
#202606342
#202606282
#202606212
#202606152
#202606102
C4⋊C4×Dic972
C22⋊C4×Dic972
C2×#156907172
C2×#123×C972
#235997