Finite Group Databasebeta

← #27158225

Subgroups — #27158225

Order 12416.

696 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#87604C212yesmaximal, complemented
2#20261685C212yesmaximal, complemented
3#20261684C212yesmaximal, complemented
4#20261683C212yesmaximal, complemented
5#20261683C212yesmaximal, complemented
6#20261682C212yesmaximal, complemented
7#20261677C211yesmaximal, characteristic, complemented
8C2×(C2×#87604)⋊C4C211yesmaximal, characteristic, complemented
9#121×D97C211yesmaximal, characteristic
10C23×#87604C2211yescharacteristic
11C23×#87604C2212yes—
12C23×#87604C2211yescharacteristic
13D4×#87604—28no—
14C2×C4⋊#87604C2212yes—
15C2×C4⋊#87604C2211yescharacteristic
16C2×C4⋊#87604C2211yescharacteristic
17C2×C22⋊#87604C2212yes—
18C2×C22⋊#87604C2212yes—
19C2×C22⋊#87604C2212yes—
20C2×C22⋊#87604C2212yes—
21C2×C4×#87604C2212yes—
22(C2×#87604)⋊C4C2212yes—
23(C2×#87604)⋊C4C2212yes—
24(C2×#87604)⋊C4C2212yes—
25(C2×#87604)⋊C4C2212yes—
26(C2×#87604)⋊C4C2212yes—
27C2×D4×D97C2212yes—
28C2×D4×D97C2211yescharacteristic
29C22×C4×D97C2211yescharacteristic
30#15690785C411yescharacteristic, complemented
31#15690784C411yescharacteristic, complemented
32#15690783C412yescomplemented
33#15690782C412yescomplemented
34#15690781C411yescharacteristic, complemented
35C4⋊C4×D97C2211yescharacteristic
36C22⋊C4×D97C2212yes—
37#121×C97C411yescharacteristic, complemented
38C22×#87604—22no—
39C22×#87604—24no—
40C22×#87604—41no—
41C22×#87604—41no—
42C22×#87604—42no—
43C22×#87604—42no—
44C22×#87604—41no—
45C22×#87604—41no—
46C22×#87604D414yescomplemented
47C22×#87604C2312yes—
48C22×#87604C2312yes—
49C22×#87604C2312yes—
50C22×#87604C2312yes—
51C4⋊#87604—24no—
52C4⋊#87604—41no—
53C4⋊#87604—41no—
54C22⋊#87604—24no—
55C22⋊#87604—24no—
56C22⋊#87604—42no—
57C22⋊#87604—42no—
58C4×#87604—24no—
59C23×D97C2312yes—
60C23×D97C2311yescharacteristic
61C2×D388C2×C412yescomplemented
62C2×D388C2×C411yescharacteristic, complemented
63C2×C22⋊D97C2×C412yescomplemented
64C2×C22⋊D97C2×C412yescomplemented
65C2×C22⋊D97C2×C412yescomplemented
66D4×D97—24no—
67D4×D97—24no—
68D4×D97—41no—
69D4×D97—41no—
70C22×Dic97C2×C411yescharacteristic, complemented
71C2×C4×D97—41no—
72C2×C4×D97—41no—
73C2×C4×D97C2×C411yescharacteristic, complemented
74C2×C4×D97C2×C411yescharacteristic, complemented
75C2×C4×D97C2311yescharacteristic
76C2×C4×D97C2312yes—
77C2×C4×D97C2311yescharacteristic
78C4⋊Dic97C2×C411yescharacteristic, complemented
79#13095490C2×C412yescomplemented
80C22⋊Dic97C2×C412yescomplemented
81#13095488C2×C412yescomplemented
82#13095488C2×C412yescomplemented
83C2×D4×C97C2×C412yescomplemented
84C2×D4×C97C2×C411yescharacteristic, complemented
85C22×C4×C97C2×C411yescharacteristic, complemented
86C4⋊C4×C97C2×C411yescharacteristic, complemented
87C22⋊C4×C97C2×C412yescomplemented
88C23×C97C22×C412yes—
89C23×C97C22×C411yescharacteristic
90C22×D97—21no—
91C22×D97—21no—
92C22×D97—21no—
93C22×D97—21no—
94C22×D97—22no—
95C22×D97—22no—
96C22×D97—22no—
97C22×D97—22no—
98C22×D97—22no—
99C22×D97—22no—
100C22×D97C22×C412yes—
101C22×D97C22×C411yescharacteristic
102C22×D97C2×D412yes—
103C22×D97#4112yes—
104C22×D97C2411yescharacteristic
105C2×#87604—22no—
106C2×#87604—22no—
107C2×#87604—22no—
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—44no—
117C2×#87604—48no—
118C2×#87604—44no—
119C2×#87604C2×D414yes—
120D4×C97—24no—
121D4×C97—41no—
122C2×C4×C97—41no—
123C2×C4×C97C22×C411yescharacteristic
124C2×C4×C97C22×C412yes—
125C2×C4×C97C22×C411yescharacteristic
126C22⋊D97—24no—
127C22⋊D97—24no—
128C22⋊D97—42no—
129C2×Dic97—41no—
130C2×Dic97C22×C411yescharacteristic
131C2×Dic97C22×C412yes—
132C2×Dic97C22×C411yescharacteristic
133D388—24no—
134D388—41no—
135C4×D97—22no—
136C4×D97—22no—
137C4×D97—21no—
138C4×D97—21no—
139C4×D97—21no—
140C4×D97—21no—
141C4×D97—41no—
142C4×D97—41no—
143C22×C97—21no—
144C22×C97—21no—
145C22×C97—22no—
146C22×C97—22no—
147C22×C97—22no—
148C22×C97C4×D412yes—
149C22×C97C23×C411yescharacteristic
150C2×D97—21no—
151C2×D97—21no—
152C2×D97—22no—
153C2×D97—22no—
154C2×D97—22no—
155C2×D97—22no—
156C2×D97—22no—
157C2×D97—22no—
158C2×D97—22no—
159C2×D97—42no—
160C2×D97—42no—
161C2×D97—42no—
162C2×D97—42no—
163C2×D97C4×D412yes—
164C2×D97C22×D411yescharacteristic
165C2×D97C2×#4111yescharacteristic
166C2×D97H(5,𝔽2)11yescharacteristic
167C2×D97H(5,𝔽2)11yescharacteristic
168C2×D97H(5,𝔽2)11yescharacteristic
169C2×D972−1+411yescharacteristic
170#87604—24no—
171#87604—42no—
172#87604—42no—
173#87604—42no—
174C4×C97—22no—
175C4×C97—21no—
176C4×C97—21no—
177Dic97—22no—
178Dic97—21no—
179Dic97—21no—
180C2×C97—22no—
181C2×C97—42no—
182C2×C97C2×C4×D411yescharacteristic
183C2×C97#51811yescharacteristic
184C2×C97#52011yescharacteristic
185D97—22no—
186D97—42no—
187D97#53411yescharacteristic
188D97#53711yescharacteristic
189D97#53811yescharacteristic
190D97#53911yescharacteristic
191#2325—971nomaximal, Sylow
192C97#232511yescharacteristic, complemented, Sylow
193C2×#121—971no—
194C2×C4×D4—972no—
195#388—972no—
196#387—972no—
197#386—972no—
198#386—972no—
199#385—972no—
200#380—971no—
201C2×#95—971no—
202C22×D4—971no—
203C22×D4—972no—
204C23×C4—971no—
205C23×C4—971no—
206C23×C4—972no—
207C23×C4—971no—
208#121—971no—
209#121—972no—
210#121—971no—
211#121—971no—
212#121—972no—
213#121#8760411yescharacteristic, complemented
214C4×D4—1948no—
215C2×C4⋊C4—971no—
216C2×C4⋊C4—971no—
217C2×C4⋊C4—971no—
218C2×C4⋊C4—972no—
219C2×C22⋊C4—972no—
220C2×C22⋊C4—972no—
221C2×C22⋊C4—972no—
222C2×C22⋊C4—972no—
223C2×C22⋊C4—972no—
224C2×C42—972no—
225#95—972no—
226#95—972no—
227#95—972no—
228#95—972no—
229#95—972no—
230C24—971no—
231C24—972no—
232C2×D4—972no—
233C2×D4—972no—
234C2×D4—972no—
235C2×D4—971no—
236C2×D4—972no—
237C2×D4—3881no—
238C2×D4—3881no—
239C2×D4—1944no—
240C2×D4—1944no—
241C2×D4C2×#8760412yescomplemented
242C2×D4C2×#8760411yescharacteristic, complemented
243C22×C4—974no—
244C22×C4—971no—
245C22×C4—972no—
246C22×C4—971no—
247C22×C4—971no—
248C22×C4—971no—
249C22×C4—972no—
250C22×C4—971no—
251C22×C4—972no—
252C22×C4—972no—
253C22×C4—972no—
254C22×C4—3881no—
255C22×C4—3881no—
256C22×C4—3881no—
257C22×C4—3881no—
258C22×C4—3882no—
259C22×C4—3882no—
260C22×C4—3881no—
261C22×C4—3881no—
262C22×C4—1944no—
263C22×C4—1942no—
264C22×C4C2×#8760411yescharacteristic, complemented
265C4⋊C4—972no—
266C4⋊C4—971no—
267C4⋊C4—3881no—
268C4⋊C4—3881no—
269C4⋊C4—1944no—
270C4⋊C4C2×#8760411yescharacteristic, complemented
271C22⋊C4—972no—
272C22⋊C4—972no—
273C22⋊C4—972no—
274C22⋊C4—3882no—
275C22⋊C4—3882no—
276C22⋊C4—1944no—
277C22⋊C4—1944no—
278C22⋊C4C2×#8760412yescomplemented
279C42—1944no—
280C23—971no—
281C23—972no—
282C23—972no—
283C23—971no—
284C23—972no—
285C23—1942no—
286C23—1942no—
287C23—1942no—
288C23—1942no—
289C23—1942no—
290C23—1942no—
291C23—1941no—
292C23—1941no—
293C23—1941no—
294C23—1941no—
295C23C22×#8760412yes—
296C23C22×#8760411yescharacteristic
297D4—24no—
298D4—41no—
299D4—3881no—
300D4—3882no—
301D4—1944no—
302D4—1944no—
303D4—1944no—
304C2×C4—41no—
305C2×C4—974no—
306C2×C4—971no—
307C2×C4—972no—
308C2×C4—971no—
309C2×C4—3881no—
310C2×C4—3881no—
311C2×C4—3884no—
312C2×C4—3888no—
313C2×C4—3881no—
314C2×C4—3884no—
315C2×C4—1942no—
316C2×C4—1941no—
317C2×C4—1941no—
318C2×C4—1941no—
319C2×C4—1941no—
320C2×C4—1942no—
321C2×C4—1942no—
322C2×C4—1942no—
323C2×C4—1942no—
324C2×C4—1942no—
325C2×C4—1942no—
326C2×C4—1942no—
327C2×C4—1942no—
328C2×C4—1942no—
329C2×C4—1942no—
330C2×C4—1942no—
331C2×C4—1942no—
332C2×C4C22×#8760411yescharacteristic
333C2×C4C22×#8760412yes—
334C2×C4C22×#8760411yescharacteristic
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—971no—
341C22—971no—
342C22—971no—
343C22—971no—
344C22—971no—
345C22—971no—
346C22—972no—
347C22—3882no—
348C22—3882no—
349C22—3882no—
350C22—3882no—
351C22—1942no—
352C22—1942no—
353C22—1942no—
354C22—1942no—
355C22—1942no—
356C22—1942no—
357C22—1942no—
358C22—1941no—
359C22—1941no—
360C22D4×#8760412yes—
361C22C23×#8760411yescharacteristic
362C4—22no—
363C4—21no—
364C4—21no—
365C4—3882no—
366C4—3882no—
367C4—3882no—
368C4—1944no—
369C4—1942no—
370C4—1941no—
371C4—1941no—
372C2—22no—
373C2—42no—
374C2—971no—
375C2—971no—
376C2—971no—
377C2—971no—
378C2—3882no—
379C2—1942no—
380C2C2×D4×#8760411yescharacteristic
381C2#2026180211yescharacteristic
382C2#2026180411yescharacteristic
subgroupG/H
center, FrattiniC22C23×#87604
commutator, socleC22×C97C23×C4
Fitting#121×C97C4
prime radical#121×D97C2
cosocleC22×D97C24
hypercenter#121#87604
hypocenterC97#2325
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #121

Complements

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

NH ≅ G/Ntype
1. C2×D4×#87604C2semidirect
2. #20261685C2semidirect
3. #20261684C2semidirect
4. #20261683C2semidirect
5. #20261683C2semidirect
6. #20261682C2semidirect
7. #20261677C2semidirect
8. C2×(C2×#87604)⋊C4C2semidirect
30. #15690785C4semidirect
31. #15690784C4semidirect
32. #15690783C4semidirect
33. #15690782C4semidirect
34. #15690781C4semidirect
37. #121×C97C4semidirect
46. C22×#87604D4semidirect
61. C2×D388C2×C4semidirect
62. C2×D388C2×C4semidirect
63. C2×C22⋊D97C2×C4semidirect
64. C2×C22⋊D97C2×C4semidirect
65. C2×C22⋊D97C2×C4semidirect
70. C22×Dic97C2×C4semidirect
73. C2×C4×D97C2×C4semidirect
74. C2×C4×D97C2×C4semidirect
78. C4⋊Dic97C2×C4semidirect
79. #13095490C2×C4semidirect
80. C22⋊Dic97C2×C4semidirect
81. #13095488C2×C4semidirect
82. #13095488C2×C4semidirect
83. C2×D4×C97C2×C4semidirect
84. C2×D4×C97C2×C4semidirect
85. C22×C4×C97C2×C4semidirect
86. C4⋊C4×C97C2×C4semidirect
87. C22⋊C4×C97C2×C4semidirect
192. C97#2325semidirect
213. #121#87604semidirect
241. C2×D4C2×#87604semidirect
242. C2×D4C2×#87604semidirect
264. C22×C4C2×#87604semidirect
270. C4⋊C4C2×#87604semidirect
278. C22⋊C4C2×#87604semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#876042
#202616852
#202616842
#202616832
#202616822
#202616772
C2×(C2×#87604)⋊C42
#121×D972
#232597