Finite Group Databasebeta

← #27158253

Subgroups — #27158253

Order 12416.

588 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261685C212yesmaximal, complemented
2#20261684C212yesmaximal, complemented
3#20261683C212yesmaximal
4#20261680C211yesmaximal, characteristic, complemented
5#20261678C211yesmaximal, characteristic, complemented
6#20261677C211yesmaximal, characteristic, complemented
7C22⋊C4×#87604C212yesmaximal, complemented
8C4×C22⋊#87604C211yesmaximal, characteristic, complemented
9#20261671C211yesmaximal, characteristic, complemented
10C2×(C2×#87604)⋊C4C211yesmaximal, characteristic, complemented
11C4×D4×D97C211yesmaximal, characteristic
12C23×#87604C2212yes—
13C2×C4⋊#87604C2212yes—
14C2×C22⋊#87604C2212yes—
15C2×C22⋊#87604C2212yes—
16C2×C22⋊#87604C2212yes—
17C2×C4×#87604C2212yes—
18C2×C4×#87604C2212yes—
19(C2×#87604)⋊C4C2211yescharacteristic
20(C2×#87604)⋊C4C2211yescharacteristic
21(C2×#87604)⋊C4C2212yes—
22(C2×#87604)⋊C4C2211yescharacteristic
23(C2×#87604)⋊C4C2212yes—
24(C2×#87604)⋊C4C2211yescharacteristic
25(C2×#87604)⋊C4C2212yes—
26(C2×#87604)⋊C4C2212yes—
27(C2×#87604)⋊C4C2212yes—
28C2×D4×D97C2211yescharacteristic
29C22×C4×D97C2211yescharacteristic
30C22×C4×D97C2211yescharacteristic
31C4×D388C411yescharacteristic, complemented
32D4×Dic97C411yescharacteristic, complemented
33#15690772C411yescharacteristic, complemented
34C4×C22⋊D97C411yescharacteristic, complemented
35C4×C22⋊D97C411yescharacteristic, complemented
36#15690770C411yescharacteristic, complemented
37#15690770C411yescharacteristic, complemented
38C4⋊C4×D97C2211yescharacteristic
39C22⋊C4×D97C2211yescharacteristic
40C22⋊C4×D97C2211yescharacteristic
41C42×D97C2211yescharacteristic
42C4×D4×C97C411yescharacteristic, complemented
43C22×#87604—24no—
44C22×#87604—24no—
45C22×#87604—42no—
46C22×#87604C2312yes—
47C22×#87604C2312yes—
48C22×#87604C2312yes—
49C22×#87604C2312yes—
50C4⋊#87604—42no—
51C22⋊#87604—24no—
52C22⋊#87604—24no—
53C22⋊#87604—42no—
54C4×#87604—24no—
55C4×#87604—42no—
56C23×D97C2311yescharacteristic
57C23×D97C2311yescharacteristic
58C2×D388C2×C411yescharacteristic
59C2×C22⋊D97C2×C411yescharacteristic
60C2×C22⋊D97C2×C411yescharacteristic
61D4×D97—22no—
62D4×D97—22no—
63C22×Dic97C2×C411yescharacteristic
64C22×Dic97C2×C411yescharacteristic, complemented
65C2×C4×D97—42no—
66C2×C4×D97—42no—
67C2×C4×D97C2×C411yescharacteristic
68C2×C4×D97C2×C411yescharacteristic
69C2×C4×D97C2×C411yescharacteristic, complemented
70C2×C4×D97C2×C411yescharacteristic, complemented
71C2×C4×D97C2311yescharacteristic
72C2×C4×D97C2311yescharacteristic
73C2×C4×D97C2311yescharacteristic
74C2×C4×D97C2311yescharacteristic
75C2×C4×D97C2311yescharacteristic
76C4⋊Dic97C2×C411yescharacteristic, complemented
77#13095490C2×C411yescharacteristic, complemented
78#13095490C2×C411yescharacteristic, complemented
79C22⋊Dic97C2×C411yescharacteristic
80C22⋊Dic97C2×C411yescharacteristic, complemented
81#13095488C2×C411yescharacteristic
82#13095488C2×C411yescharacteristic
83#13095488C2×C411yescharacteristic, complemented
84#13095488C2×C411yescharacteristic, complemented
85C4×Dic97C2×C411yescharacteristic, complemented
86C4×Dic97C2×C411yescharacteristic, complemented
87C4×Dic97C2×C411yescharacteristic, complemented
88C2×D4×C97C2×C411yescharacteristic
89C22×C4×C97C2×C411yescharacteristic
90C22×C4×C97C2×C411yescharacteristic, complemented
91C4⋊C4×C97C2×C411yescharacteristic, complemented
92C22⋊C4×C97C2×C411yescharacteristic
93C22⋊C4×C97C2×C411yescharacteristic, complemented
94C42×C97C2×C411yescharacteristic, complemented
95C23×C97C22×C411yescharacteristic
96C23×C97C22×C411yescharacteristic
97C22×D97—22no—
98C22×D97—22no—
99C22×D97—21no—
100C22×D97—21no—
101C22×D97—22no—
102C22×D97—22no—
103C22×D97C22×C411yescharacteristic
104C22×D97C22×C411yescharacteristic
105C22×D97#4112yes—
106C22×D97#4112yes—
107C22×D97C2411yescharacteristic
108C2×#87604—22no—
109C2×#87604—22no—
110C2×#87604—22no—
111C2×#87604—22no—
112C2×#87604—22no—
113C2×#87604—22no—
114C2×#87604—22no—
115C2×#87604—22no—
116C2×#87604—22no—
117C2×#87604—22no—
118C2×#87604—22no—
119C2×#87604—48no—
120C2×#87604#4114yes—
121D4×C97—22no—
122C2×C4×C97—41no—
123C2×C4×C97—41no—
124C2×C4×C97C22×C411yescharacteristic
125C2×C4×C97C22×C411yescharacteristic
126C2×C4×C97C22×C411yescharacteristic
127C2×C4×C97C22×C411yescharacteristic
128C2×C4×C97C22×C411yescharacteristic
129C22⋊D97—22no—
130C22⋊D97—22no—
131C2×Dic97—41no—
132C2×Dic97—41no—
133C2×Dic97C22×C411yescharacteristic
134C2×Dic97C22×C411yescharacteristic
135C2×Dic97C22×C411yescharacteristic
136C2×Dic97C22×C411yescharacteristic
137C2×Dic97C22×C411yescharacteristic
138D388—22no—
139C4×D97—21no—
140C4×D97—21no—
141C4×D97—22no—
142C4×D97—22no—
143C4×D97—22no—
144C4×D97—22no—
145C4×D97—41no—
146C4×D97—41no—
147C4×D97—41no—
148C4×D97—41no—
149C22×C97—21no—
150C22×C97—21no—
151C22×C97—21no—
152C22×C97—21no—
153C22×C97—21no—
154C22×C97#11712yes—
155C22×C97C23×C411yescharacteristic
156C2×D97—21no—
157C2×D97—21no—
158C2×D97—24no—
159C2×D97—24no—
160C2×D97—21no—
161C2×D97—21no—
162C2×D97—21no—
163C2×D97—42no—
164C2×D97—42no—
165C2×D97#11712yes—
166C2×D97C2×#4111yescharacteristic
167C2×D97C2×#4112yes—
168C2×D97C2×#4111yescharacteristic
169C2×D97H(5,𝔽2)12yes—
170#87604—24no—
171#87604—42no—
172#87604—42no—
173#87604—42no—
174C4×C97—21no—
175C4×C97—21no—
176C4×C97—21no—
177C4×C97—21no—
178C4×C97—21no—
179Dic97—21no—
180Dic97—21no—
181Dic97—21no—
182Dic97—21no—
183Dic97—21no—
184C2×C97—22no—
185C2×C97—41no—
186C2×C97C2×#11711yescharacteristic
187C2×C97C4×#4111yescharacteristic
188C2×C97#51811yescharacteristic
189D97—22no—
190D97—41no—
191D97#55112yes—
192D97#55312yes—
193#2353—971nomaximal, Sylow
194C97#235311yescharacteristic, complemented, Sylow
195C2×C4×D4—971no—
196#388—972no—
197#387—972no—
198#386—972no—
199#383—971no—
200#381—971no—
201#380—971no—
202C4×C22⋊C4—971no—
203C4×C22⋊C4—972no—
204#376—971no—
205C2×#95—971no—
206C22×D4—971no—
207C23×C4—971no—
208C23×C4—971no—
209C23×C4—972no—
210C4×D4—971no—
211C4×D4—971no—
212C4×D4—971no—
213C4×D4—971no—
214C4×D4—971no—
215C4×D4—971no—
216C4×D4—971no—
217C4×D4#8760411yescharacteristic, complemented
218C2×C4⋊C4—971no—
219C2×C4⋊C4—972no—
220C2×C22⋊C4—971no—
221C2×C22⋊C4—971no—
222C2×C22⋊C4—972no—
223C2×C22⋊C4—972no—
224C2×C22⋊C4—972no—
225C2×C42—971no—
226C2×C42—972no—
227C2×C42—972no—
228#95—971no—
229#95—971no—
230#95—972no—
231#95—971no—
232#95—972no—
233#95—971no—
234#95—972no—
235#95—972no—
236#95—972no—
237C24—971no—
238C24—971no—
239C2×D4—971no—
240C2×D4—971no—
241C2×D4—971no—
242C2×D4—1942no—
243C2×D4—1942no—
244C2×D4C2×#8760411yescharacteristic
245C22×C4—971no—
246C22×C4—971no—
247C22×C4—972no—
248C22×C4—971no—
249C22×C4—971no—
250C22×C4—971no—
251C22×C4—971no—
252C22×C4—971no—
253C22×C4—971no—
254C22×C4—971no—
255C22×C4—971no—
256C22×C4—971no—
257C22×C4—972no—
258C22×C4—972no—
259C22×C4—972no—
260C22×C4—3882no—
261C22×C4—3882no—
262C22×C4—3882no—
263C22×C4—1944no—
264C22×C4—1944no—
265C22×C4C2×#8760411yescharacteristic
266C22×C4C2×#8760411yescharacteristic, complemented
267C4⋊C4—971no—
268C4⋊C4—971no—
269C4⋊C4—971no—
270C4⋊C4—3882no—
271C4⋊C4C2×#8760411yescharacteristic, complemented
272C22⋊C4—971no—
273C22⋊C4—971no—
274C22⋊C4—971no—
275C22⋊C4—971no—
276C22⋊C4—971no—
277C22⋊C4—971no—
278C22⋊C4—3882no—
279C22⋊C4—1944no—
280C22⋊C4—1944no—
281C22⋊C4C2×#8760411yescharacteristic
282C22⋊C4C2×#8760411yescharacteristic, complemented
283C42—971no—
284C42—971no—
285C42—971no—
286C42—3882no—
287C42—1944no—
288C42C2×#8760411yescharacteristic, complemented
289C23—971no—
290C23—972no—
291C23—972no—
292C23—971no—
293C23—971no—
294C23—1942no—
295C23—1942no—
296C23—1941no—
297C23—1941no—
298C23—1942no—
299C23—1942no—
300C23C22×#8760411yescharacteristic
301C23C22×#8760411yescharacteristic
302D4—22no—
303D4—1942no—
304D4—1942no—
305D4—1942no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—974no—
309C2×C4—971no—
310C2×C4—971no—
311C2×C4—971no—
312C2×C4—971no—
313C2×C4—971no—
314C2×C4—3881no—
315C2×C4—3881no—
316C2×C4—3881no—
317C2×C4—3881no—
318C2×C4—3888no—
319C2×C4—3881no—
320C2×C4—3881no—
321C2×C4—1942no—
322C2×C4—1942no—
323C2×C4—1942no—
324C2×C4—1942no—
325C2×C4—1941no—
326C2×C4—1941no—
327C2×C4—1942no—
328C2×C4—1942no—
329C2×C4—1942no—
330C2×C4—1942no—
331C2×C4—1942no—
332C2×C4—1942no—
333C2×C4—1942no—
334C2×C4—1942no—
335C2×C4—1942no—
336C2×C4—1942no—
337C2×C4—1942no—
338C2×C4C22×#8760411yescharacteristic
339C2×C4C22×#8760411yescharacteristic
340C2×C4C22×#8760411yescharacteristic
341C2×C4C22×#8760411yescharacteristic
342C2×C4C22×#8760411yescharacteristic
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—971no—
349C22—971no—
350C22—972no—
351C22—972no—
352C22—972no—
353C22—3882no—
354C22—3882no—
355C22—1944no—
356C22—1944no—
357C22—1941no—
358C22—1941no—
359C22—1941no—
360C22—1941no—
361C22—1941no—
362C22#1569088912yes—
363C22C23×#8760411yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—3882no—
370C4—3882no—
371C4—3882no—
372C4—1944no—
373C4—1941no—
374C4—1941no—
375C4—1941no—
376C4—1941no—
377C4—1941no—
378C2—22no—
379C2—41no—
380C2—972no—
381C2—972no—
382C2—3881no—
383C2—1942no—
384C2C2×#1569088911yescharacteristic
385C2#41×#8760411yescharacteristic
386C2#2026180211yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
FittingC4×D4×C97C4
prime radicalC2×D4×D97C22
cosocleC22×D97C24
hypercenterC4×D4#87604
hypocenterC97#2353
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. #20261685C2semidirect
2. #20261684C2semidirect
4. #20261680C2semidirect
5. #20261678C2semidirect
6. #20261677C2semidirect
7. C22⋊C4×#87604C2semidirect
8. C4×C22⋊#87604C2semidirect
9. #20261671C2semidirect
10. C2×(C2×#87604)⋊C4C2semidirect
31. C4×D388C4semidirect
32. D4×Dic97C4semidirect
33. #15690772C4semidirect
34. C4×C22⋊D97C4semidirect
35. C4×C22⋊D97C4semidirect
36. #15690770C4semidirect
37. #15690770C4semidirect
42. C4×D4×C97C4semidirect
64. C22×Dic97C2×C4semidirect
69. C2×C4×D97C2×C4semidirect
70. C2×C4×D97C2×C4semidirect
76. C4⋊Dic97C2×C4semidirect
77. #13095490C2×C4semidirect
78. #13095490C2×C4semidirect
80. C22⋊Dic97C2×C4semidirect
83. #13095488C2×C4semidirect
84. #13095488C2×C4semidirect
85. C4×Dic97C2×C4semidirect
86. C4×Dic97C2×C4semidirect
87. C4×Dic97C2×C4semidirect
90. C22×C4×C97C2×C4semidirect
91. C4⋊C4×C97C2×C4semidirect
93. C22⋊C4×C97C2×C4semidirect
94. C42×C97C2×C4semidirect
194. C97#2353semidirect
217. C4×D4#87604semidirect
266. C22×C4C2×#87604semidirect
271. C4⋊C4C2×#87604semidirect
282. C22⋊C4C2×#87604semidirect
288. C42C2×#87604semidirect

Maximal subgroups

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

subgroupindex
#202616852
#202616842
#202616832
#202616802
#202616782
#202616772
C22⋊C4×#876042
C4×C22⋊#876042
#202616712
C2×(C2×#87604)⋊C42
C4×D4×D972
#235397