Finite Group Databasebeta

← #27158273

Subgroups — #27158273

Order 12416.

496 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
1#20261686C211yesmaximal, characteristic, complemented
2#20261685C211yesmaximal, characteristic
3#20261685C211yesmaximal, characteristic
4#20261685C211yesmaximal, characteristic
5#20261684C211yesmaximal, characteristic, complemented
6#20261684C211yesmaximal, characteristic, complemented
7#20261684C211yesmaximal, characteristic, complemented
8#20261682C211yesmaximal, characteristic
9C4⋊C4×#87604C211yesmaximal, characteristic, complemented
10C4⋊C4×#87604C211yesmaximal, characteristic, complemented
11C4×C4⋊#87604C211yesmaximal, characteristic, complemented
12C22⋊C4×#87604C211yesmaximal, characteristic
13C22⋊C4×#87604C211yesmaximal, characteristic
14#20261671C211yesmaximal, characteristic, complemented
15#126×D97C211yesmaximal, characteristic
16C23×#87604C2211yescharacteristic
17C2×C4⋊#87604C2211yescharacteristic
18C2×C4⋊#87604C2211yescharacteristic
19C2×C4⋊#87604C2211yescharacteristic
20C2×C4⋊#87604C2211yescharacteristic
21C2×C4⋊#87604C2211yescharacteristic
22C2×C22⋊#87604C2211yescharacteristic
23C2×C22⋊#87604C2211yescharacteristic
24C2×C22⋊#87604C2211yescharacteristic
25C2×C4×#87604C2211yescharacteristic
26C2×C4×#87604C2211yescharacteristic
27C2×C4×#87604C2211yescharacteristic
28C2×C4×#87604C2211yescharacteristic
29C2×C4×#87604C2211yescharacteristic
30C2×C4×#87604C2211yescharacteristic
31C2×C4×#87604C2211yescharacteristic
32(C2×#87604)⋊C4C2211yescharacteristic
33(C2×#87604)⋊C4C2211yescharacteristic
34(C2×#87604)⋊C4C2211yescharacteristic
35(C2×#87604)⋊C4C2211yescharacteristic
36(C2×#87604)⋊C4C2211yescharacteristic
37(C2×#87604)⋊C4C2211yescharacteristic
38(C2×#87604)⋊C4C2211yescharacteristic
39(C2×#87604)⋊C4C2211yescharacteristic
40(C2×#87604)⋊C4C2211yescharacteristic
41(C2×#87604)⋊C4C2211yescharacteristic
42(C2×#87604)⋊C4C2211yescharacteristic
43(C2×#87604)⋊C4C2211yescharacteristic
44#15690805C411yescharacteristic, complemented
45#15690805C411yescharacteristic, complemented
46#15690805C411yescharacteristic, complemented
47#15690804C411yescharacteristic, complemented
48#15690803C411yescharacteristic, complemented
49#15690803C411yescharacteristic, complemented
50#15690803C411yescharacteristic, complemented
51C4⋊C4×D97C2211yescharacteristic
52C4⋊C4×D97C2211yescharacteristic
53C4⋊C4×D97C2211yescharacteristic
54C22⋊C4×D97C2211yescharacteristic
55C22⋊C4×D97C2211yescharacteristic
56C22⋊C4×D97C2211yescharacteristic
57C42×D97C2211yescharacteristic
58#126×C97C411yescharacteristic, complemented
59C22×#87604—22no—
60C22×#87604—22no—
61C22×#87604—41no—
62C22×#87604C2311yescharacteristic
63C22×#87604C2311yescharacteristic
64C22×#87604C2311yescharacteristic
65C22×#87604C2311yescharacteristic
66C22×#87604C2311yescharacteristic
67C22×#87604C2311yescharacteristic
68C22×#87604C2311yescharacteristic
69C22×#87604C2311yescharacteristic
70C4⋊#87604—22no—
71C4⋊#87604—22no—
72C4⋊#87604—22no—
73C4⋊#87604—22no—
74C4⋊#87604—41no—
75C22⋊#87604—22no—
76C22⋊#87604—22no—
77C22⋊#87604—41no—
78C4×#87604—22no—
79C4×#87604—22no—
80C4×#87604—22no—
81C4×#87604—22no—
82C4×#87604—41no—
83C4×#87604—41no—
84C4×#87604—41no—
85C23×D97C2311yescharacteristic
86C2×C4×D97C2311yescharacteristic
87C2×C4×D97C2311yescharacteristic
88C2×C4×D97C2311yescharacteristic
89C2×C4×D97C2311yescharacteristic
90C2×C4×D97C2311yescharacteristic
91C2×C4×D97C2311yescharacteristic
92C4⋊Dic97C2×C411yescharacteristic, complemented
93C4⋊Dic97C2×C411yescharacteristic, complemented
94C4⋊Dic97C2×C411yescharacteristic, complemented
95#13095490C2×C411yescharacteristic, complemented
96#13095490C2×C411yescharacteristic, complemented
97#13095490C2×C411yescharacteristic, complemented
98#13095490C2×C411yescharacteristic, complemented
99#13095490C2×C411yescharacteristic, complemented
100#13095490C2×C411yescharacteristic, complemented
101C22⋊Dic97C2×C411yescharacteristic
102C22⋊Dic97C2×C411yescharacteristic
103C22⋊Dic97C2×C411yescharacteristic
104#13095488C2×C411yescharacteristic
105#13095488C2×C411yescharacteristic
106#13095488C2×C411yescharacteristic
107#13095488C2×C411yescharacteristic
108#13095488C2×C411yescharacteristic
109#13095488C2×C411yescharacteristic
110C4×Dic97C2×C411yescharacteristic, complemented
111C4×Dic97C2×C411yescharacteristic, complemented
112C4×Dic97C2×C411yescharacteristic, complemented
113C4⋊C4×C97C2×C411yescharacteristic, complemented
114C4⋊C4×C97C2×C411yescharacteristic, complemented
115C4⋊C4×C97C2×C411yescharacteristic, complemented
116C22⋊C4×C97C2×C411yescharacteristic
117C22⋊C4×C97C2×C411yescharacteristic
118C22⋊C4×C97C2×C411yescharacteristic
119C42×C97C2×C411yescharacteristic, complemented
120C23×C97C22×C411yescharacteristic
121C22×D97—21no—
122C22×D97—21no—
123C22×D97—21no—
124C22×D97—21no—
125C22×D97—21no—
126C22×D97—21no—
127C22×D97C22×C411yescharacteristic
128C22×D97C2411yescharacteristic
129C2×#87604—21no—
130C2×#87604—21no—
131C2×#87604—21no—
132C2×#87604—21no—
133C2×#87604—21no—
134C2×#87604—21no—
135C2×#87604—21no—
136C2×#87604—21no—
137C2×#87604—21no—
138C2×#87604—21no—
139C2×#87604—21no—
140C2×#87604—21no—
141C2×#87604—21no—
142C2×#87604—21no—
143C2×#87604—21no—
144C2×#87604—21no—
145C2×#87604—21no—
146C2×#87604—21no—
147C2×#87604—21no—
148C2×#87604—21no—
149C2×#87604—44no—
150C2×#87604#4112yes—
151C2×#87604#4112yes—
152C2×#87604#4112yes—
153C2×#87604#4112yes—
154C2×C4×C97C22×C411yescharacteristic
155C2×C4×C97C22×C411yescharacteristic
156C2×C4×C97C22×C411yescharacteristic
157C2×C4×C97C22×C411yescharacteristic
158C2×C4×C97C22×C411yescharacteristic
159C2×C4×C97C22×C411yescharacteristic
160C2×Dic97C22×C411yescharacteristic
161C2×Dic97C22×C411yescharacteristic
162C2×Dic97C22×C411yescharacteristic
163C2×Dic97C22×C411yescharacteristic
164C2×Dic97C22×C411yescharacteristic
165C2×Dic97C22×C411yescharacteristic
166C4×D97—21no—
167C4×D97—21no—
168C4×D97—21no—
169C4×D97—21no—
170C4×D97—21no—
171C4×D97—21no—
172C4×D97—21no—
173C4×D97—21no—
174C4×D97—21no—
175C4×D97—21no—
176C4×D97—21no—
177C4×D97—21no—
178C22×C97—21no—
179C22×C97—21no—
180C22×C97—21no—
181C22×C97C23×C411yescharacteristic
182C2×D97—21no—
183C2×D97—21no—
184C2×D97—21no—
185C2×D97—41no—
186C2×D97—41no—
187C2×D97—41no—
188C2×D97—41no—
189C2×D97C2×#4111yescharacteristic
190C2×D97C2×#4111yescharacteristic
191C2×D97C2×#4111yescharacteristic
192C2×D97C2×#4111yescharacteristic
193C2×D97H(5,𝔽2)11yescharacteristic
194C2×D972−1+411yescharacteristic
195#87604—22no—
196#87604—22no—
197#87604—22no—
198#87604—22no—
199#87604—41no—
200#87604—41no—
201#87604—41no—
202#87604—41no—
203C4×C97—21no—
204C4×C97—21no—
205C4×C97—21no—
206C4×C97—21no—
207C4×C97—21no—
208C4×C97—21no—
209Dic97—21no—
210Dic97—21no—
211Dic97—21no—
212Dic97—21no—
213Dic97—21no—
214Dic97—21no—
215C2×C97—41no—
216C2×C97C4×#4111yescharacteristic
217C2×C97C4×#4111yescharacteristic
218C2×C97#52011yescharacteristic
219D97—41no—
220D97#55111yescharacteristic
221D97#55211yescharacteristic
222D97#55311yescharacteristic
223D97#55611yescharacteristic
224#2373—971nomaximal, Sylow
225C97#237311yescharacteristic, complemented, Sylow
226C2×#126—971no—
227#389—971no—
228#388—971no—
229#388—971no—
230#388—971no—
231#387—971no—
232#387—971no—
233#387—971no—
234#385—971no—
235C4×C4⋊C4—971no—
236C4×C4⋊C4—971no—
237C4×C4⋊C4—971no—
238C4×C22⋊C4—971no—
239C4×C22⋊C4—971no—
240#376—971no—
241C23×C4—971no—
242#126—971no—
243#126—971no—
244#126—971no—
245#126—971no—
246#126—971no—
247#126—971no—
248#126—971no—
249#126#8760411yescharacteristic, complemented
250C2×C4⋊C4—971no—
251C2×C4⋊C4—971no—
252C2×C4⋊C4—971no—
253C2×C4⋊C4—971no—
254C2×C4⋊C4—971no—
255C2×C4⋊C4—971no—
256C2×C4⋊C4—971no—
257C2×C4⋊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×C42—971no—
265C2×C42—971no—
266C2×C42—971no—
267C2×C42—971no—
268C2×C42—971no—
269C2×C42—971no—
270C2×C42—971no—
271C2×C42—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—
281#95—971no—
282#95—971no—
283#95—971no—
284C24—971no—
285C22×C4—971no—
286C22×C4—971no—
287C22×C4—971no—
288C22×C4—971no—
289C22×C4—971no—
290C22×C4—971no—
291C22×C4—971no—
292C22×C4—971no—
293C22×C4—971no—
294C22×C4—971no—
295C22×C4—971no—
296C22×C4—971no—
297C22×C4—971no—
298C22×C4—971no—
299C22×C4—3881no—
300C22×C4—1942no—
301C22×C4—1942no—
302C4⋊C4—971no—
303C4⋊C4—971no—
304C4⋊C4—971no—
305C4⋊C4—971no—
306C4⋊C4—971no—
307C4⋊C4—971no—
308C4⋊C4—971no—
309C4⋊C4—971no—
310C4⋊C4—971no—
311C4⋊C4—3881no—
312C4⋊C4—1942no—
313C4⋊C4—1942no—
314C4⋊C4—1942no—
315C4⋊C4—1942no—
316C4⋊C4C2×#8760411yescharacteristic, complemented
317C4⋊C4C2×#8760411yescharacteristic, complemented
318C4⋊C4C2×#8760411yescharacteristic, complemented
319C22⋊C4—971no—
320C22⋊C4—971no—
321C22⋊C4—971no—
322C22⋊C4—971no—
323C22⋊C4—971no—
324C22⋊C4—971no—
325C22⋊C4—971no—
326C22⋊C4—971no—
327C22⋊C4—971no—
328C22⋊C4—3881no—
329C22⋊C4—1942no—
330C22⋊C4—1942no—
331C22⋊C4C2×#8760411yescharacteristic
332C22⋊C4C2×#8760411yescharacteristic
333C22⋊C4C2×#8760411yescharacteristic
334C42—971no—
335C42—971no—
336C42—971no—
337C42—3881no—
338C42—3881no—
339C42—3881no—
340C42—1942no—
341C42—1942no—
342C42—1942no—
343C42—1942no—
344C42C2×#8760411yescharacteristic, complemented
345C23—971no—
346C23—971no—
347C23—1941no—
348C23—1941no—
349C23—1941no—
350C23—1941no—
351C23—1941no—
352C23—1941no—
353C23C22×#8760411yescharacteristic
354C2×C4—972no—
355C2×C4—972no—
356C2×C4—972no—
357C2×C4—972no—
358C2×C4—971no—
359C2×C4—971no—
360C2×C4—971no—
361C2×C4—971no—
362C2×C4—971no—
363C2×C4—971no—
364C2×C4—3884no—
365C2×C4—1941no—
366C2×C4—1941no—
367C2×C4—1941no—
368C2×C4—1941no—
369C2×C4—1941no—
370C2×C4—1941no—
371C2×C4—1941no—
372C2×C4—1941no—
373C2×C4—1941no—
374C2×C4—1941no—
375C2×C4—1941no—
376C2×C4—1941no—
377C2×C4—1941no—
378C2×C4—1941no—
379C2×C4—1941no—
380C2×C4—1941no—
381C2×C4—1941no—
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×C4C22×#8760411yescharacteristic
398C2×C4C22×#8760411yescharacteristic
399C2×C4C22×#8760411yescharacteristic
400C2×C4C22×#8760411yescharacteristic
401C2×C4C22×#8760411yescharacteristic
402C2×C4C22×#8760411yescharacteristic
403C22—21no—
404C22—21no—
405C22—21no—
406C22—971no—
407C22—971no—
408C22—971no—
409C22—971no—
410C22—971no—
411C22—971no—
412C22—3881no—
413C22—3881no—
414C22—3881no—
415C22—3881no—
416C22—1941no—
417C22—1941no—
418C22—1941no—
419C22C23×#8760411yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—3881no—
427C4—3881no—
428C4—3881no—
429C4—3881no—
430C4—1942no—
431C4—1942no—
432C4—1942no—
433C4—1942no—
434C4—1941no—
435C4—1941no—
436C4—1941no—
437C4—1941no—
438C4—1941no—
439C4—1941no—
440C2—41no—
441C2—971no—
442C2—971no—
443C2—971no—
444C2—971no—
445C2—3881no—
446C2#41×#8760411yescharacteristic
447C2#41×#8760411yescharacteristic
448C2#2026180411yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#126×C97C4
prime radicalC23×D97C23
cosocleC22×D97C24
hypercenter#126#87604
hypocenterC97#2373
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. #20261686C2semidirect
5. #20261684C2semidirect
6. #20261684C2semidirect
7. #20261684C2semidirect
9. C4⋊C4×#87604C2semidirect
10. C4⋊C4×#87604C2semidirect
11. C4×C4⋊#87604C2semidirect
14. #20261671C2semidirect
44. #15690805C4semidirect
45. #15690805C4semidirect
46. #15690805C4semidirect
47. #15690804C4semidirect
48. #15690803C4semidirect
49. #15690803C4semidirect
50. #15690803C4semidirect
58. #126×C97C4semidirect
92. C4⋊Dic97C2×C4semidirect
93. C4⋊Dic97C2×C4semidirect
94. C4⋊Dic97C2×C4semidirect
95. #13095490C2×C4semidirect
96. #13095490C2×C4semidirect
97. #13095490C2×C4semidirect
98. #13095490C2×C4semidirect
99. #13095490C2×C4semidirect
100. #13095490C2×C4semidirect
110. C4×Dic97C2×C4semidirect
111. C4×Dic97C2×C4semidirect
112. C4×Dic97C2×C4semidirect
113. C4⋊C4×C97C2×C4semidirect
114. C4⋊C4×C97C2×C4semidirect
115. C4⋊C4×C97C2×C4semidirect
119. C42×C97C2×C4semidirect
225. C97#2373semidirect
249. #126#87604semidirect
316. C4⋊C4C2×#87604semidirect
317. C4⋊C4C2×#87604semidirect
318. C4⋊C4C2×#87604semidirect
344. C42C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202616862
#202616852
#202616842
#202616822
C4⋊C4×#876042
C4×C4⋊#876042
C22⋊C4×#876042
#202616712
#126×D972
#237397