Finite Group Databasebeta

← #27144771

Subgroups — #27144771

Order 12416.

688 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690782C211yesmaximal, characteristic
2C2×#15690770C211yesmaximal, characteristic, complemented
3C2×#15690766C211yesmaximal, characteristic, complemented
4#20260635C211yesmaximal, characteristic, complemented
5#20260635C211yesmaximal, characteristic, complemented
6#20260630C211yesmaximal, characteristic, complemented
7#20260625C211yesmaximal, characteristic, complemented
8#20260623C211yesmaximal, characteristic, complemented
9#20260615C211yesmaximal, characteristic, complemented
10#20260611C211yesmaximal, characteristic, complemented
11C4×#13095490C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic97C211yesmaximal, characteristic, complemented
14C4×#13095488C211yesmaximal, characteristic, complemented
15#390×C97C211yesmaximal, characteristic, complemented
16C22×C22⋊D97C2211yescharacteristic
17C22×C22⋊D97C2211yescharacteristic
18C23×Dic97C2211yescharacteristic
19C23×Dic97C2211yescharacteristic
20C22×C4×D97C2211yescharacteristic
21#15690782C418yescomplemented
22#15690770—24no—
23#15690766—24no—
24C2×#13095490C2211yescharacteristic
25C2×#13095490C2211yescharacteristic
26C2×#13095490C2211yescharacteristic
27C2×#13095490C2211yescharacteristic
28C2×C22⋊Dic97C2211yescharacteristic
29C2×C22⋊Dic97C2211yescharacteristic
30C2×C22⋊Dic97C2211yescharacteristic
31C2×C22⋊Dic97C2211yescharacteristic
32C2×#13095488C2211yescharacteristic
33C2×#13095488C2211yescharacteristic
34C2×#13095488C2211yescharacteristic
35C2×#13095488C2211yescharacteristic
36C2×C4×Dic97C2211yescharacteristic
37C2×C4×Dic97C2211yescharacteristic
38C2×C4×Dic97C2211yescharacteristic
39C2×C4×Dic97C2211yescharacteristic
40C2×C4×Dic97C2211yescharacteristic
41#15690717C2211yescharacteristic
42#15690717C2211yescharacteristic
43#15690717C2211yescharacteristic
44#15690717C2211yescharacteristic
45#15690717C2211yescharacteristic
46#15690717C2211yescharacteristic
47C22×D4×C97C2211yescharacteristic
48C2×C4⋊C4×C97C2211yescharacteristic
49C2×C22⋊C4×C97C2211yescharacteristic
50C2×C22⋊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×C414yes—
60C2×C22⋊D97C2×C414yescomplemented
61C22×Dic97—22no—
62C22×Dic97—22no—
63C22×Dic97—22no—
64C22×Dic97—22no—
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#13095490—41no—
79#13095490C2×C414yescomplemented
80#13095490D414yescomplemented
81C22⋊Dic97—22no—
82C22⋊Dic97—22no—
83C22⋊Dic97—41no—
84C22⋊Dic97C2×C414yescomplemented
85#13095488—22no—
86#13095488—22no—
87#13095488—41no—
88#13095488C2×C414yescomplemented
89C4×Dic97—22no—
90C4×Dic97—22no—
91C4×Dic97—22no—
92C4×Dic97—22no—
93C4×Dic97—41no—
94C24×C97C2311yescharacteristic
95C24×C97C2311yescharacteristic
96C2×D4×C97—41no—
97C2×D4×C97—41no—
98C2×D4×C97C2×C414yescomplemented
99C22×C4×C97C2311yescharacteristic
100C22×C4×C97C2311yescharacteristic
101C22×C4×C97C2311yescharacteristic
102C22×C4×C97C2311yescharacteristic
103C22×C4×C97C2311yescharacteristic
104C4⋊C4×C97—41no—
105C22⋊C4×C97—22no—
106C22⋊C4×C97—22no—
107C22⋊C4×C97—22no—
108C22⋊C4×C97—22no—
109C42×C97—41no—
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×C97—21no—
121C23×C97—21no—
122C23×C97C22×C412yes—
123C23×C97C22×C412yes—
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—44no—
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—21no—
145C2×C4×C97C22×C412yes—
146C2×C4×C97C2×D412yes—
147C2×C4×C97#4112yes—
148C22⋊D97—28no—
149C22⋊D97—44no—
150C2×Dic97—21no—
151C2×Dic97—21no—
152C2×Dic97—21no—
153C2×Dic97—21no—
154C2×Dic97—21no—
155C2×Dic97—21no—
156C2×Dic97—21no—
157C2×Dic97—21no—
158C2×Dic97—28no—
159C2×Dic97—21no—
160C2×Dic97—21no—
161C2×Dic97—21no—
162C2×Dic97—21no—
163C2×Dic97—21no—
164C2×Dic97—21no—
165C2×Dic97—44no—
166C2×Dic97C22×C412yes—
167C2×Dic97C22×C412yes—
168C2×Dic97C22×C412yes—
169C2×Dic97C2×D412yes—
170C2×Dic97C2×D412yes—
171C2×Dic97#4112yes—
172C2×Dic97#4112yes—
173C4×D97—44no—
174C22×C97—22no—
175C22×C97—22no—
176C22×C97—22no—
177C22×C97—22no—
178C22×C97—22no—
179C22×C97—22no—
180C22×C97—41no—
181C22×C97—41no—
182C22×C97—41no—
183C22×C97—41no—
184C22×C97—41no—
185C22×C97—41no—
186C22×C97—41no—
187C22×C97—41no—
188C22×C97C23×C411yescharacteristic
189C22×C97C22×D411yescharacteristic
190C22×C97C2×#4111yescharacteristic
191C22×C97C2×#4111yescharacteristic
192C22×C97C2×#4111yescharacteristic
193C22×C97H(5,𝔽2)11yescharacteristic
194C22×C97H(5,𝔽2)11yescharacteristic
195C2×D97—22no—
196C2×D97—22no—
197C2×D97—22no—
198C2×D97—41no—
199C2×D97—41no—
200C2×D97—41no—
201C2×D97—41no—
202C4×C97—22no—
203C4×C97—22no—
204C4×C97—22no—
205C4×C97—41no—
206C4×C97—41no—
207Dic97—22no—
208Dic97—22no—
209Dic97—22no—
210Dic97—22no—
211Dic97—41no—
212Dic97—41no—
213Dic97C4×D414yes—
214C2×C97—42no—
215C2×C97—42no—
216C2×C97C2×C4×D411yescharacteristic
217C2×C97C4×#4111yescharacteristic
218C2×C97#51811yescharacteristic
219C2×C97#54611yescharacteristic
220C2×C97#55011yescharacteristic
221C2×C97#55311yescharacteristic
222C2×C97#55511yescharacteristic
223D97—42no—
224#2363—971nomaximal, Sylow
225C97#236311yescharacteristic, complemented, Sylow
226C2×#121—971no—
227C2×C4×D4—971no—
228C2×#117—971no—
229#390—971no—
230#390—971no—
231#390D9711yescharacteristic, complemented
232#389—971no—
233#388—971no—
234#388—971no—
235#386—971no—
236#386—971no—
237C4×C4⋊C4—971no—
238C4×C22⋊C4—971no—
239C4×C22⋊C4—971no—
240C4×C22⋊C4—971no—
241C22×D4—971no—
242C22×D4—971no—
243C22×D4C2×D9711yescharacteristic
244C23×C4—971no—
245C23×C4—971no—
246C23×C4—971no—
247#121—978no—
248C4×D4—1944no—
249#117—1944no—
250C2×C4⋊C4—971no—
251C2×C4⋊C4—971no—
252C2×C4⋊C4—971no—
253C2×C4⋊C4—971no—
254C2×C4⋊C4C2×D9711yescharacteristic
255C2×C22⋊C4—971no—
256C2×C22⋊C4—971no—
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⋊C4C2×D9711yescharacteristic
264C2×C22⋊C4C2×D9711yescharacteristic
265C2×C22⋊C4C2×D9711yescharacteristic
266C2×C22⋊C4C2×D9711yescharacteristic
267C2×C42—971no—
268C2×C42—971no—
269C2×C42—971no—
270C2×C42—971no—
271C2×C42—971no—
272C2×C42C2×D9711yescharacteristic
273#95—971no—
274#95—971no—
275#95—971no—
276#95—971no—
277#95—971no—
278#95—971no—
279C24—971no—
280C24C22×D9711yescharacteristic
281C24C22×D9711yescharacteristic
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—974no—
285C2×D4—974no—
286C2×D4—3881no—
287C2×D4—3881no—
288C2×D4—1942no—
289C2×D4—1942no—
290C2×D4C4×D9714yescomplemented
291C22×C4—971no—
292C22×C4—974no—
293C22×C4—971no—
294C22×C4—971no—
295C22×C4—971no—
296C22×C4—971no—
297C22×C4—971no—
298C22×C4—971no—
299C22×C4—3881no—
300C22×C4—3881no—
301C22×C4—1942no—
302C22×C4—1942no—
303C22×C4—1942no—
304C22×C4—1942no—
305C22×C4—1942no—
306C22×C4—1942no—
307C22×C4C22×D9711yescharacteristic
308C22×C4C22×D9711yescharacteristic
309C22×C4C22×D9711yescharacteristic
310C22×C4C22×D9711yescharacteristic
311C22×C4C22×D9711yescharacteristic
312C4⋊C4—41no—
313C4⋊C4—974no—
314C4⋊C4—974no—
315C4⋊C4—3881no—
316C4⋊C4—1942no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—974no—
322C22⋊C4—974no—
323C22⋊C4—3881no—
324C22⋊C4—3881no—
325C22⋊C4—1942no—
326C22⋊C4—1942no—
327C22⋊C4—1942no—
328C22⋊C4—1942no—
329C42—41no—
330C42—3881no—
331C42—1942no—
332C42—1942no—
333C42—1942no—
334C42—1942no—
335C23—21no—
336C23—21no—
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—
356C23C23×D9711yescharacteristic
357D4—44no—
358D4—3884no—
359D4—1948no—
360C2×C4—21no—
361C2×C4—21no—
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—972no—
373C2×C4—972no—
374C2×C4—972no—
375C2×C4—972no—
376C2×C4—972no—
377C2×C4—972no—
378C2×C4—972no—
379C2×C4—3884no—
380C2×C4—3884no—
381C2×C4—1948no—
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×C4C2×C4×D9712yes—
397C2×C4D4×D9712yes—
398C2×C4#1309551012yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—3881no—
414C22—3881no—
415C22—3881no—
416C22—3881no—
417C22—1942no—
418C22—1942no—
419C22—1942no—
420C22C22×C4×D9711yescharacteristic
421C22C2×D4×D9711yescharacteristic
422C22#41×D9711yescharacteristic
423C22#41×D9711yescharacteristic
424C22C2×#1309551011yescharacteristic
425C22#1569085511yescharacteristic
426C22#1569085511yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—974no—
433C4—3881no—
434C4—3881no—
435C4—1942no—
436C4—1942no—
437C4—1942no—
438C4—1942no—
439C2—42no—
440C2—42no—
441C2—3882no—
442C2C4×D4×D9711yescharacteristic
443C2C4×#1309551011yescharacteristic
444C2#2026120811yescharacteristic
445C2#2026137411yescharacteristic
446C2#2026140511yescharacteristic
447C2#2026142811yescharacteristic
448C2#2026144711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D97
commutatorC22×C97C23×C4
Fitting#390×C97C2
socle, cosocleC23×C97C24
prime radicalC2×#15690782C2
hypercenter#390D97
hypocenterC97#2363
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C2×#15690770C2semidirect
3. C2×#15690766C2semidirect
4. #20260635C2semidirect
5. #20260635C2semidirect
6. #20260630C2semidirect
7. #20260625C2semidirect
8. #20260623C2semidirect
9. #20260615C2semidirect
10. #20260611C2semidirect
11. C4×#13095490C2semidirect
12. C22⋊C4×Dic97C2semidirect
13. C22⋊C4×Dic97C2semidirect
14. C4×#13095488C2semidirect
15. #390×C97C2semidirect
21. #15690782C4semidirect
60. C2×C22⋊D97C2×C4semidirect
66. C22×Dic97C2×C4semidirect
79. #13095490C2×C4semidirect
80. #13095490D4semidirect
84. C22⋊Dic97C2×C4semidirect
88. #13095488C2×C4semidirect
98. C2×D4×C97C2×C4semidirect
225. C97#2363semidirect
231. #390D97semidirect
290. C2×D4C4×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
C2×#156907662
#202606352
#202606302
#202606252
#202606232
#202606152
#202606112
C4×#130954902
C22⋊C4×Dic972
C4×#130954882
#390×C972
#236397