Finite Group Databasebeta

← #27144781

Subgroups — #27144781

Order 12416.

716 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690782C211yesmaximal, characteristic
2C2×#15690770C211yesmaximal, characteristic, complemented
3#20260629C211yesmaximal, characteristic, complemented
4#20260625C211yesmaximal, characteristic, complemented
5#20260619C211yesmaximal, characteristic, complemented
6#20260616C211yesmaximal, characteristic, complemented
7#20260615C211yesmaximal, characteristic, complemented
8#20260614C211yesmaximal, characteristic, complemented
9#20260614C211yesmaximal, characteristic, complemented
10#20260611C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
12C4×#13095488C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic97C211yesmaximal, characteristic, complemented
14C2×#15690717C211yesmaximal, characteristic, complemented
15C2×C4×D4×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C23×Dic97C2211yescharacteristic
19C23×Dic97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21#15690782C418yescomplemented
22#15690770—24no—
23C2×#13095490C2211yescharacteristic
24C2×#13095490C2211yescharacteristic
25C2×C22⋊Dic97C2211yescharacteristic
26C2×C22⋊Dic97C2211yescharacteristic
27C2×C22⋊Dic97C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×#13095488C2211yescharacteristic
30C2×#13095488C2211yescharacteristic
31C2×#13095488C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
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
48C23×C4×C97C2211yescharacteristic
49C4×D4×C97—24no—
50C2×C4⋊C4×C97C2211yescharacteristic
51C2×C22⋊C4×C97C2211yescharacteristic
52C2×C22⋊C4×C97C2211yescharacteristic
53C2×C42×C97C2211yescharacteristic
54C23×D97C2311yescharacteristic
55C2×C22⋊D97—22no—
56C2×C22⋊D97—22no—
57C2×C22⋊D97—41no—
58C2×C22⋊D97—41no—
59C2×C22⋊D97C2×C414yescomplemented
60C2×C22⋊D97C2×C414yescomplemented
61C22×Dic97—22no—
62C22×Dic97—22no—
63C22×Dic97—41no—
64C22×Dic97—41no—
65C22×Dic97—41no—
66C22×Dic97C2×C414yescomplemented
67C22×Dic97C2311yescharacteristic
68C22×Dic97C2311yescharacteristic
69C22×Dic97C2311yescharacteristic
70C22×Dic97C2311yescharacteristic
71C22×Dic97C2311yescharacteristic
72C22×Dic97C2311yescharacteristic
73C22×Dic97C2311yescharacteristic
74C2×C4×D97—22no—
75C2×C4×D97—22no—
76C2×C4×D97—41no—
77#13095490—22no—
78#13095490C2×C414yescomplemented
79C22⋊Dic97—22no—
80C22⋊Dic97—22no—
81C22⋊Dic97—41no—
82C22⋊Dic97C2×C414yescomplemented
83#13095488—22no—
84#13095488—22no—
85#13095488—41no—
86#13095488C2×C414yescomplemented
87C4×Dic97—22no—
88C4×Dic97—41no—
89C4×Dic97—41no—
90C24×C97C2311yescharacteristic
91C24×C97C2311yescharacteristic
92C2×D4×C97—22no—
93C2×D4×C97—22no—
94C2×D4×C97C2×C414yescomplemented
95C22×C4×C97—22no—
96C22×C4×C97—22no—
97C22×C4×C97—22no—
98C22×C4×C97—22no—
99C22×C4×C97—41no—
100C22×C4×C97—41no—
101C22×C4×C97C2311yescharacteristic
102C22×C4×C97C2311yescharacteristic
103C22×C4×C97C2311yescharacteristic
104C22×C4×C97C2311yescharacteristic
105C22×C4×C97C2311yescharacteristic
106C4⋊C4×C97—22no—
107C22⋊C4×C97—22no—
108C22⋊C4×C97D414yescomplemented
109C42×C97—22no—
110C23×C97—21no—
111C23×C97—21no—
112C23×C97—21no—
113C23×C97—21no—
114C23×C97—21no—
115C23×C97—21no—
116C23×C97—21no—
117C23×C97—21no—
118C23×C97—21no—
119C23×C97—21no—
120C23×C97C22×C412yes—
121C23×C97C22×C412yes—
122C23×C97C2×D412yes—
123C23×C97#4112yes—
124C23×C97C2411yescharacteristic
125C22×D97—21no—
126C22×D97—21no—
127C22×D97—21no—
128C22×D97—21no—
129C22×D97—21no—
130C22×D97—21no—
131C22×D97C22×C412yes—
132D4×C97—28no—
133C2×C4×C97—21no—
134C2×C4×C97—21no—
135C2×C4×C97—21no—
136C2×C4×C97—21no—
137C2×C4×C97—21no—
138C2×C4×C97—21no—
139C2×C4×C97—21no—
140C2×C4×C97—21no—
141C2×C4×C97—21no—
142C2×C4×C97—21no—
143C2×C4×C97—21no—
144C2×C4×C97—44no—
145C2×C4×C97—44no—
146C2×C4×C97C22×C412yes—
147C2×C4×C97C2×D412yes—
148C2×C4×C97C2×D412yes—
149C2×C4×C97#4112yes—
150C22⋊D97—28no—
151C22⋊D97—44no—
152C2×Dic97—21no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—21no—
157C2×Dic97—21no—
158C2×Dic97—21no—
159C2×Dic97—21no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162C2×Dic97—21no—
163C2×Dic97—21no—
164C2×Dic97—21no—
165C2×Dic97—21no—
166C2×Dic97—21no—
167C2×Dic97—21no—
168C2×Dic97—21no—
169C2×Dic97—44no—
170C2×Dic97—44no—
171C2×Dic97C22×C412yes—
172C2×Dic97C22×C412yes—
173C2×Dic97C22×C412yes—
174C2×Dic97#4112yes—
175C4×D97—44no—
176C22×C97—22no—
177C22×C97—22no—
178C22×C97—22no—
179C22×C97—22no—
180C22×C97—22no—
181C22×C97—22no—
182C22×C97—22no—
183C22×C97—22no—
184C22×C97—22no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97—41no—
189C22×C97C4×D414yes—
190C22×C97C23×C411yescharacteristic
191C22×C97C22×D411yescharacteristic
192C22×C97C2×#4111yescharacteristic
193C22×C97C2×#4111yescharacteristic
194C22×C97C2×#4111yescharacteristic
195C22×C97H(5,𝔽2)11yescharacteristic
196C22×C97H(5,𝔽2)11yescharacteristic
197C2×D97—22no—
198C2×D97—22no—
199C2×D97—22no—
200C2×D97—41no—
201C2×D97—41no—
202C2×D97—41no—
203C2×D97—41no—
204C4×C97—22no—
205C4×C97—22no—
206C4×C97—22no—
207C4×C97—22no—
208C4×C97—41no—
209Dic97—22no—
210Dic97—22no—
211Dic97—22no—
212Dic97—22no—
213Dic97—41no—
214Dic97—41no—
215Dic97—41no—
216C2×C97—24no—
217C2×C97—42no—
218C2×C97C2×C4×D411yescharacteristic
219C2×C97C4×#4111yescharacteristic
220C2×C97#51811yescharacteristic
221C2×C97#54611yescharacteristic
222C2×C97#54611yescharacteristic
223C2×C97#55311yescharacteristic
224C2×C97#55311yescharacteristic
225D97—42no—
226#2364—971nomaximal, Sylow
227C97#236411yescharacteristic, complemented, Sylow
228C2×#121—971no—
229C2×C4×D4—971no—
230C2×C4×D4D9711yescharacteristic, complemented
231#388—971no—
232#388—971no—
233#387—971no—
234#387—971no—
235#386—971no—
236#386—971no—
237#386—971no—
238#386—971no—
239C4×C22⋊C4—971no—
240C4×C22⋊C4—971no—
241C4×C22⋊C4—971no—
242C2×#95—971no—
243C22×D4—971no—
244C22×D4—971no—
245C22×D4C2×D9711yescharacteristic
246C23×C4—971no—
247C23×C4—971no—
248C23×C4—971no—
249C23×C4C2×D9711yescharacteristic
250C23×C4C2×D9711yescharacteristic
251#121—978no—
252C4×D4—24no—
253C4×D4—1944no—
254C2×C4⋊C4—971no—
255C2×C4⋊C4—971no—
256C2×C4⋊C4C2×D9711yescharacteristic
257C2×C22⋊C4—971no—
258C2×C22⋊C4—971no—
259C2×C22⋊C4—971no—
260C2×C22⋊C4—971no—
261C2×C22⋊C4—971no—
262C2×C22⋊C4—971no—
263C2×C22⋊C4—971no—
264C2×C22⋊C4—971no—
265C2×C22⋊C4C2×D9711yescharacteristic
266C2×C22⋊C4C2×D9711yescharacteristic
267C2×C42—971no—
268C2×C42—971no—
269C2×C42—971no—
270C2×C42C2×D9711yescharacteristic
271#95—971no—
272#95—971no—
273#95—971no—
274#95—971no—
275#95—971no—
276#95—971no—
277#95—971no—
278#95—971no—
279#95—971no—
280#95—971no—
281C24—971no—
282C24C22×D9711yescharacteristic
283C24C22×D9711yescharacteristic
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—974no—
287C2×D4—974no—
288C2×D4—3881no—
289C2×D4—3881no—
290C2×D4—1942no—
291C2×D4—1942no—
292C2×D4C4×D9714yescomplemented
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—974no—
300C22×C4—971no—
301C22×C4—971no—
302C22×C4—971no—
303C22×C4—971no—
304C22×C4—971no—
305C22×C4—971no—
306C22×C4—971no—
307C22×C4—3881no—
308C22×C4—3881no—
309C22×C4—3881no—
310C22×C4—3881no—
311C22×C4—1942no—
312C22×C4—1942no—
313C22×C4—1942no—
314C22×C4—1942no—
315C22×C4C22×D9711yescharacteristic
316C22×C4C22×D9711yescharacteristic
317C22×C4C22×D9711yescharacteristic
318C22×C4C22×D9711yescharacteristic
319C22×C4C22×D9711yescharacteristic
320C4⋊C4—22no—
321C4⋊C4—974no—
322C4⋊C4—1942no—
323C22⋊C4—22no—
324C22⋊C4—974no—
325C22⋊C4—974no—
326C22⋊C4—3881no—
327C22⋊C4—3881no—
328C22⋊C4—1942no—
329C22⋊C4—1942no—
330C22⋊C4—1942no—
331C22⋊C4—1942no—
332C22⋊C4C22⋊D9714yescomplemented
333C42—22no—
334C42—3881no—
335C42—3881no—
336C42—1942no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—972no—
348C23—1941no—
349C23—1941no—
350C23—1941no—
351C23—1941no—
352C23—1941no—
353C23—1941no—
354C23C2×C4×D9712yes—
355C23C2×C4×D9712yes—
356C23C2×C22⋊D9712yes—
357C23#1309551112yes—
358C23C23×D9711yescharacteristic
359D4—28no—
360D4—3884no—
361D4—1948no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—972no—
376C2×C4—972no—
377C2×C4—972no—
378C2×C4—972no—
379C2×C4—3884no—
380C2×C4—3884no—
381C2×C4—3884no—
382C2×C4—1941no—
383C2×C4—1941no—
384C2×C4—1941no—
385C2×C4—1941no—
386C2×C4—1941no—
387C2×C4—1941no—
388C2×C4—1941no—
389C2×C4—1941no—
390C2×C4—1941no—
391C2×C4—1941no—
392C2×C4—1941no—
393C2×C4—1941no—
394C2×C4—1941no—
395C2×C4—1941no—
396C2×C4—1941no—
397C2×C4—1941no—
398C2×C4—1941no—
399C2×C4C2×C4×D9712yes—
400C2×C4C2×C22⋊D9712yes—
401C2×C4C2×C22⋊D9712yes—
402C2×C4#1309551012yes—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—3881no—
417C22—3881no—
418C22—3881no—
419C22—3881no—
420C22—1942no—
421C22—1942no—
422C22—1942no—
423C22C4×C22⋊D9714yes—
424C22C22×C4×D9711yescharacteristic
425C22C22×C22⋊D9711yescharacteristic
426C22#41×D9711yescharacteristic
427C22C2×#1309551011yescharacteristic
428C22C2×#1309551111yescharacteristic
429C22#1569085511yescharacteristic
430C22#1569085611yescharacteristic
431C4—22no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—41no—
436C4—3881no—
437C4—3881no—
438C4—3881no—
439C4—1942no—
440C4—1942no—
441C4—1942no—
442C4—1942no—
443C2—24no—
444C2—42no—
445C2—3882no—
446C2C2×C4×C22⋊D9711yescharacteristic
447C2C4×#1309551011yescharacteristic
448C2#2026120711yescharacteristic
449C2#2026137311yescharacteristic
450C2#2026137911yescharacteristic
451C2#2026143111yescharacteristic
452C2#2026143411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
FittingC2×C4×D4×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690782C2
hypercenterC2×C4×D4D97
hypocenterC97#2364
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. C2×#15690770C2semidirect
3. #20260629C2semidirect
4. #20260625C2semidirect
5. #20260619C2semidirect
6. #20260616C2semidirect
7. #20260615C2semidirect
8. #20260614C2semidirect
9. #20260614C2semidirect
10. #20260611C2semidirect
11. C22⋊C4×Dic97C2semidirect
12. C4×#13095488C2semidirect
13. C4×C22⋊Dic97C2semidirect
14. C2×#15690717C2semidirect
15. C2×C4×D4×C97C2semidirect
21. #15690782C4semidirect
59. C2×C22⋊D97C2×C4semidirect
60. C2×C22⋊D97C2×C4semidirect
66. C22×Dic97C2×C4semidirect
78. #13095490C2×C4semidirect
82. C22⋊Dic97C2×C4semidirect
86. #13095488C2×C4semidirect
94. C2×D4×C97C2×C4semidirect
108. C22⋊C4×C97D4semidirect
227. C97#2364semidirect
230. C2×C4×D4D97semidirect
292. C2×D4C4×D97semidirect
332. C22⋊C4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156907822
C2×#156907702
#202606292
#202606252
#202606192
#202606162
#202606152
#202606142
#202606112
C22⋊C4×Dic972
C4×#130954882
C4×C22⋊Dic972
C2×#156907172
C2×C4×D4×C972
#236497