Finite Group Databasebeta

← #27158190

Subgroups — #27158190

Order 12416.

696 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690889C211yesmaximal, characteristic, complemented
2C22×C4×#87604C211yesmaximal, characteristic, complemented
3#20261685C212yesmaximal, complemented
4#20261684C212yesmaximal, complemented
5#20261683C212yesmaximal
6#20261682C212yesmaximal, complemented
7C4×C4⋊#87604C211yesmaximal, characteristic, complemented
8C4×C22⋊#87604C211yesmaximal, characteristic, complemented
9C4×C22⋊#87604C211yesmaximal, characteristic, complemented
10#20261671C211yesmaximal, characteristic, complemented
11C4×D4×D97C211yesmaximal, characteristic
12C23×#87604C2212yes—
13#15690889—24no—
14C2×C4⋊#87604C2212yes—
15C2×C4⋊#87604C2212yes—
16C2×C22⋊#87604C2212yes—
17C2×C22⋊#87604C2212yes—
18C2×C22⋊#87604C2212yes—
19C2×C4×#87604—24no—
20C2×C4×#87604C2211yescharacteristic
21C2×C4×#87604C2211yescharacteristic
22C2×C4×#87604C2211yescharacteristic
23C2×C4×#87604C2212yes—
24C2×C4×#87604C2211yescharacteristic
25(C2×#87604)⋊C4C2212yes—
26(C2×#87604)⋊C4C2212yes—
27(C2×#87604)⋊C4C2212yes—
28(C2×#87604)⋊C4C2212yes—
29(C2×#87604)⋊C4C2212yes—
30C2×D4×D97C2211yescharacteristic
31C22×C4×D97C2211yescharacteristic
32C22×C4×D97C2211yescharacteristic
33C4×D388C411yescharacteristic, complemented
34D4×Dic97C411yescharacteristic, complemented
35#15690772C411yescharacteristic, complemented
36C4×C22⋊D97C411yescharacteristic, complemented
37C4×C22⋊D97C411yescharacteristic, complemented
38#15690770C411yescharacteristic, complemented
39#15690770C411yescharacteristic, complemented
40C4⋊C4×D97C2211yescharacteristic
41C22⋊C4×D97C2211yescharacteristic
42C22⋊C4×D97C2211yescharacteristic
43C42×D97C2211yescharacteristic
44C4×D4×C97C411yescharacteristic, complemented
45C22×#87604—24no—
46C22×#87604—24no—
47C22×#87604—24no—
48C22×#87604C2312yes—
49C22×#87604C2312yes—
50C22×#87604C2312yes—
51C22×#87604C2312yes—
52C4⋊#87604—24no—
53C4⋊#87604—24no—
54C22⋊#87604—24no—
55C22⋊#87604—24no—
56C22⋊#87604—24no—
57C4×#87604—24no—
58C4×#87604—22no—
59C4×#87604—22no—
60C4×#87604—22no—
61C4×#87604D414yescomplemented
62C23×D97C2311yescharacteristic
63C23×D97C2311yescharacteristic
64C2×D388C2×C411yescharacteristic
65C2×C22⋊D97C2×C411yescharacteristic
66C2×C22⋊D97C2×C411yescharacteristic
67D4×D97—24no—
68C22×Dic97C2×C411yescharacteristic
69C22×Dic97C2×C411yescharacteristic, complemented
70C2×C4×D97—22no—
71C2×C4×D97—22no—
72C2×C4×D97—22no—
73C2×C4×D97—22no—
74C2×C4×D97C2×C411yescharacteristic
75C2×C4×D97C2×C411yescharacteristic
76C2×C4×D97C2×C411yescharacteristic, complemented
77C2×C4×D97C2×C411yescharacteristic, complemented
78C2×C4×D97C2311yescharacteristic
79C2×C4×D97C2311yescharacteristic
80C2×C4×D97C2311yescharacteristic
81C2×C4×D97C2311yescharacteristic
82C2×C4×D97C2311yescharacteristic
83C4⋊Dic97C2×C411yescharacteristic, complemented
84#13095490C2×C411yescharacteristic, complemented
85#13095490C2×C411yescharacteristic, complemented
86C22⋊Dic97C2×C411yescharacteristic
87C22⋊Dic97C2×C411yescharacteristic, complemented
88#13095488C2×C411yescharacteristic
89#13095488C2×C411yescharacteristic
90#13095488C2×C411yescharacteristic, complemented
91#13095488C2×C411yescharacteristic, complemented
92C4×Dic97C2×C411yescharacteristic, complemented
93C4×Dic97C2×C411yescharacteristic, complemented
94C4×Dic97C2×C411yescharacteristic, complemented
95C2×D4×C97C2×C411yescharacteristic
96C22×C4×C97C2×C411yescharacteristic
97C22×C4×C97C2×C411yescharacteristic, complemented
98C4⋊C4×C97C2×C411yescharacteristic, complemented
99C22⋊C4×C97C2×C411yescharacteristic
100C22⋊C4×C97C2×C411yescharacteristic, complemented
101C42×C97C2×C411yescharacteristic, complemented
102C23×C97C22×C411yescharacteristic
103C23×C97C22×C411yescharacteristic
104C22×D97—21no—
105C22×D97—21no—
106C22×D97—22no—
107C22×D97—21no—
108C22×D97—21no—
109C22×D97—22no—
110C22×D97—22no—
111C22×D97C22×C411yescharacteristic
112C22×D97C22×C411yescharacteristic
113C22×D97#4114yes—
114C22×D97C2411yescharacteristic
115C2×#87604—216no—
116C2×#87604—22no—
117C2×#87604—22no—
118C2×#87604—22no—
119C2×#87604—22no—
120C2×#87604—22no—
121C2×#87604—22no—
122C2×#87604—22no—
123C2×#87604—22no—
124C2×#87604—22no—
125C2×#87604—22no—
126C2×#87604C2×D414yes—
127C2×#87604#4114yes—
128D4×C97—22no—
129C2×C4×C97—22no—
130C2×C4×C97—22no—
131C2×C4×C97C22×C411yescharacteristic
132C2×C4×C97C22×C411yescharacteristic
133C2×C4×C97C22×C411yescharacteristic
134C2×C4×C97C22×C411yescharacteristic
135C2×C4×C97C22×C411yescharacteristic
136C22⋊D97—22no—
137C22⋊D97—22no—
138C2×Dic97—22no—
139C2×Dic97—22no—
140C2×Dic97C22×C411yescharacteristic
141C2×Dic97C22×C411yescharacteristic
142C2×Dic97C22×C411yescharacteristic
143C2×Dic97C22×C411yescharacteristic
144C2×Dic97C22×C411yescharacteristic
145D388—22no—
146C4×D97—21no—
147C4×D97—21no—
148C4×D97—21no—
149C4×D97—21no—
150C4×D97—22no—
151C4×D97—22no—
152C4×D97—22no—
153C4×D97—22no—
154C4×D97—22no—
155C4×D97—22no—
156C4×D97C2×D412yes—
157C4×D97#4112yes—
158C22×C97—21no—
159C22×C97—21no—
160C22×C97—21no—
161C22×C97—21no—
162C22×C97—21no—
163C22×C97#11712yescomplemented
164C22×C97C23×C411yescharacteristic
165C2×D97—21no—
166C2×D97—21no—
167C2×D97—24no—
168C2×D97—24no—
169C2×D97—21no—
170C2×D97—21no—
171C2×D97—21no—
172C2×D97—42no—
173C2×D97—42no—
174C2×D97#11712yescomplemented
175C2×D97C22×D411yescharacteristic
176C2×D97C2×#4112yes—
177C2×D97C2×#4112yes—
178C2×D97C2×#4111yescharacteristic
179#87604—24no—
180#87604—24no—
181#87604—42no—
182#87604—42no—
183C4×C97—21no—
184C4×C97—21no—
185C4×C97—21no—
186C4×C97—21no—
187C4×C97C4×D412yes—
188Dic97—21no—
189Dic97—21no—
190Dic97—21no—
191Dic97—21no—
192Dic97C4×D412yes—
193C2×C97—22no—
194C2×C97—41no—
195C2×C97C2×#11711yescharacteristic
196C2×C97C2×C4×D411yescharacteristic
197C2×C97C4×#4111yescharacteristic
198D97—22no—
199D97—41no—
200D97#52512yes—
201D97#52912yes—
202#2290—971nomaximal, Sylow
203C97#229011yescharacteristic, complemented, Sylow
204C2×C4×D4—971no—
205C2×#117—971no—
206C22×C42—971no—
207#388—972no—
208#387—972no—
209#386—972no—
210#385—972no—
211C4×C4⋊C4—971no—
212C4×C22⋊C4—971no—
213C4×C22⋊C4—971no—
214#376—971no—
215C22×D4—971no—
216C23×C4—971no—
217C23×C4—971no—
218C23×C4—972no—
219C4×D4—971no—
220C4×D4—971no—
221C4×D4—971no—
222C4×D4—971no—
223C4×D4—971no—
224C4×D4—971no—
225C4×D4—971no—
226C4×D4#8760411yescharacteristic, complemented
227#117—1944no—
228C2×C4⋊C4—971no—
229C2×C4⋊C4—972no—
230C2×C4⋊C4—972no—
231C2×C22⋊C4—971no—
232C2×C22⋊C4—971no—
233C2×C22⋊C4—972no—
234C2×C22⋊C4—972no—
235C2×C22⋊C4—972no—
236C2×C42—971no—
237C2×C42—971no—
238C2×C42—971no—
239C2×C42—971no—
240C2×C42—971no—
241C2×C42—972no—
242C2×C42—1944no—
243#95—972no—
244#95—972no—
245#95—972no—
246#95—972no—
247#95—972no—
248C24—971no—
249C24—971no—
250C2×D4—971no—
251C2×D4—971no—
252C2×D4—971no—
253C2×D4—1944no—
254C2×D4C2×#8760411yescharacteristic
255C22×C4—971no—
256C22×C4—972no—
257C22×C4—972no—
258C22×C4—971no—
259C22×C4—971no—
260C22×C4—971no—
261C22×C4—971no—
262C22×C4—971no—
263C22×C4—971no—
264C22×C4—971no—
265C22×C4—971no—
266C22×C4—971no—
267C22×C4—971no—
268C22×C4—972no—
269C22×C4—972no—
270C22×C4—1942no—
271C22×C4—1942no—
272C22×C4—1942no—
273C22×C4—1942no—
274C22×C4—1944no—
275C22×C4—1944no—
276C22×C4—1944no—
277C22×C4C2×#8760411yescharacteristic
278C22×C4C2×#8760411yescharacteristic, complemented
279C4⋊C4—971no—
280C4⋊C4—971no—
281C4⋊C4—971no—
282C4⋊C4—1944no—
283C4⋊C4—1944no—
284C4⋊C4C2×#8760411yescharacteristic, complemented
285C22⋊C4—971no—
286C22⋊C4—971no—
287C22⋊C4—971no—
288C22⋊C4—971no—
289C22⋊C4—971no—
290C22⋊C4—971no—
291C22⋊C4—1944no—
292C22⋊C4—1944no—
293C22⋊C4—1944no—
294C22⋊C4C2×#8760411yescharacteristic
295C22⋊C4C2×#8760411yescharacteristic, complemented
296C42—974no—
297C42—971no—
298C42—971no—
299C42—971no—
300C42—1942no—
301C42—1942no—
302C42—1944no—
303C42—1942no—
304C42C2×#8760411yescharacteristic, complemented
305C23—971no—
306C23—974no—
307C23—971no—
308C23—971no—
309C23—1941no—
310C23—1942no—
311C23—1941no—
312C23—1942no—
313C23—1941no—
314C23—1941no—
315C23—1942no—
316C23C22×#8760411yescharacteristic
317C23C22×#8760411yescharacteristic
318D4—22no—
319D4—1942no—
320D4—1942no—
321D4—1942no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—971no—
325C2×C4—972no—
326C2×C4—972no—
327C2×C4—974no—
328C2×C4—974no—
329C2×C4—971no—
330C2×C4—971no—
331C2×C4—971no—
332C2×C4—971no—
333C2×C4—1942no—
334C2×C4—1942no—
335C2×C4—1941no—
336C2×C4—1941no—
337C2×C4—1942no—
338C2×C4—1941no—
339C2×C4—1941no—
340C2×C4—1942no—
341C2×C4—1942no—
342C2×C4—1942no—
343C2×C4—1942no—
344C2×C4—1942no—
345C2×C4—1942no—
346C2×C4—1942no—
347C2×C4—1942no—
348C2×C4—1942no—
349C2×C4—1942no—
350C2×C4—1942no—
351C2×C4—1942no—
352C2×C4—1942no—
353C2×C4—19416no—
354C2×C4—1942no—
355C2×C4—1942no—
356C2×C4C22×#8760411yescharacteristic
357C2×C4C22×#8760411yescharacteristic
358C2×C4C22×#8760411yescharacteristic
359C2×C4C22×#8760411yescharacteristic
360C2×C4C22×#8760411yescharacteristic
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—972no—
367C22—971no—
368C22—971no—
369C22—972no—
370C22—972no—
371C22—3882no—
372C22—3882no—
373C22—1944no—
374C22—1944no—
375C22—1941no—
376C22—1941no—
377C22—1941no—
378C22—1941no—
379C22—1941no—
380C22#1569088912yescomplemented
381C22C23×#8760411yescharacteristic
382C4—21no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—972no—
387C4—3882no—
388C4—3882no—
389C4—1944no—
390C4—1944no—
391C4—1941no—
392C4—1941no—
393C4—1941no—
394C4—1941no—
395C4D4×#8760412yes—
396C2—22no—
397C2—41no—
398C2—972no—
399C2—972no—
400C2—3881no—
401C2—1942no—
402C2C2×#1569088911yescharacteristic
403C2C2×D4×#8760411yescharacteristic
404C2#41×#8760411yescharacteristic
subgroupG/H
centerC2×C4C22×#87604
commutator, socleC22×C97C23×C4
FrattiniC22C23×#87604
FittingC4×D4×C97C4
prime radicalC2×D4×D97C22
cosocleC22×D97C24
hypercenterC4×D4#87604
hypocenterC97#2290
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#15690889C2semidirect
2. C22×C4×#87604C2semidirect
3. #20261685C2semidirect
4. #20261684C2semidirect
6. #20261682C2semidirect
7. C4×C4⋊#87604C2semidirect
8. C4×C22⋊#87604C2semidirect
9. C4×C22⋊#87604C2semidirect
10. #20261671C2semidirect
33. C4×D388C4semidirect
34. D4×Dic97C4semidirect
35. #15690772C4semidirect
36. C4×C22⋊D97C4semidirect
37. C4×C22⋊D97C4semidirect
38. #15690770C4semidirect
39. #15690770C4semidirect
44. C4×D4×C97C4semidirect
61. C4×#87604D4semidirect
69. C22×Dic97C2×C4semidirect
76. C2×C4×D97C2×C4semidirect
77. C2×C4×D97C2×C4semidirect
83. C4⋊Dic97C2×C4semidirect
84. #13095490C2×C4semidirect
85. #13095490C2×C4semidirect
87. C22⋊Dic97C2×C4semidirect
90. #13095488C2×C4semidirect
91. #13095488C2×C4semidirect
92. C4×Dic97C2×C4semidirect
93. C4×Dic97C2×C4semidirect
94. C4×Dic97C2×C4semidirect
97. C22×C4×C97C2×C4semidirect
98. C4⋊C4×C97C2×C4semidirect
100. C22⋊C4×C97C2×C4semidirect
101. C42×C97C2×C4semidirect
163. C22×C97#117semidirect
174. C2×D97#117semidirect
203. C97#2290semidirect
226. C4×D4#87604semidirect
278. C22×C4C2×#87604semidirect
284. C4⋊C4C2×#87604semidirect
295. C22⋊C4C2×#87604semidirect
304. C42C2×#87604semidirect
380. C22#15690889semidirect

Maximal subgroups

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

subgroupindex
C2×#156908892
C22×C4×#876042
#202616852
#202616842
#202616832
#202616822
C4×C4⋊#876042
C4×C22⋊#876042
#202616712
C4×D4×D972
#229097