Finite Group Databasebeta

← #2373×C211 #41009993

Subgroups — #2373×C211 #41009993

Order 27008.

496 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#126×C211C211yesmaximal, characteristic
2#389×C211C211yesmaximal, characteristic, complemented
3#388×C211C211yesmaximal, characteristic
4#388×C211C211yesmaximal, characteristic
5#388×C211C211yesmaximal, characteristic
6#387×C211C211yesmaximal, characteristic, complemented
7#387×C211C211yesmaximal, characteristic, complemented
8#387×C211C211yesmaximal, characteristic, complemented
9#385×C211C211yesmaximal, characteristic
10C4×C4⋊C4×C211C211yesmaximal, characteristic, complemented
11C4×C4⋊C4×C211C211yesmaximal, characteristic, complemented
12C4×C4⋊C4×C211C211yesmaximal, characteristic, complemented
13C4×C22⋊C4×C211C211yesmaximal, characteristic
14C4×C22⋊C4×C211C211yesmaximal, characteristic
15#376×C211C211yesmaximal, characteristic, complemented
16C23×C4×C211C2211yescharacteristic
17#126×C211C418yescomplemented
18C2×C4⋊C4×C211C2211yescharacteristic
19C2×C4⋊C4×C211C2211yescharacteristic
20C2×C4⋊C4×C211C2211yescharacteristic
21C2×C4⋊C4×C211C2211yescharacteristic
22C2×C4⋊C4×C211C2211yescharacteristic
23C2×C4⋊C4×C211C2211yescharacteristic
24C2×C4⋊C4×C211C2211yescharacteristic
25C2×C4⋊C4×C211C2211yescharacteristic
26C2×C22⋊C4×C211C2211yescharacteristic
27C2×C22⋊C4×C211C2211yescharacteristic
28C2×C22⋊C4×C211C2211yescharacteristic
29C2×C22⋊C4×C211C2211yescharacteristic
30C2×C22⋊C4×C211C2211yescharacteristic
31C2×C22⋊C4×C211C2211yescharacteristic
32C2×C42×C211C2211yescharacteristic
33C2×C42×C211C2211yescharacteristic
34C2×C42×C211C2211yescharacteristic
35C2×C42×C211C2211yescharacteristic
36C2×C42×C211C2211yescharacteristic
37C2×C42×C211C2211yescharacteristic
38C2×C42×C211C2211yescharacteristic
39C2×C42×C211C2211yescharacteristic
40#95×C211C2211yescharacteristic
41#95×C211C2211yescharacteristic
42#95×C211C2211yescharacteristic
43#95×C211C2211yescharacteristic
44#95×C211C2211yescharacteristic
45#95×C211C2211yescharacteristic
46#95×C211C2211yescharacteristic
47#95×C211C2211yescharacteristic
48#95×C211C2211yescharacteristic
49#95×C211C2211yescharacteristic
50#95×C211C2211yescharacteristic
51#95×C211C2211yescharacteristic
52C24×C211C2311yescharacteristic
53C22×C4×C211—22no—
54C22×C4×C211—22no—
55C22×C4×C211—41no—
56C22×C4×C211C2311yescharacteristic
57C22×C4×C211C2311yescharacteristic
58C22×C4×C211C2311yescharacteristic
59C22×C4×C211C2311yescharacteristic
60C22×C4×C211C2311yescharacteristic
61C22×C4×C211C2311yescharacteristic
62C22×C4×C211C2311yescharacteristic
63C22×C4×C211C2311yescharacteristic
64C22×C4×C211C2311yescharacteristic
65C22×C4×C211C2311yescharacteristic
66C22×C4×C211C2311yescharacteristic
67C22×C4×C211C2311yescharacteristic
68C22×C4×C211C2311yescharacteristic
69C22×C4×C211C2311yescharacteristic
70C4⋊C4×C211—22no—
71C4⋊C4×C211—22no—
72C4⋊C4×C211—22no—
73C4⋊C4×C211—22no—
74C4⋊C4×C211—41no—
75C4⋊C4×C211C2×C414yescomplemented
76C4⋊C4×C211C2×C414yescomplemented
77C4⋊C4×C211C2×C414yescomplemented
78C22⋊C4×C211—22no—
79C22⋊C4×C211—22no—
80C22⋊C4×C211—41no—
81C22⋊C4×C211C2×C414yes—
82C22⋊C4×C211C2×C414yes—
83C22⋊C4×C211C2×C414yes—
84C42×C211—22no—
85C42×C211—22no—
86C42×C211—22no—
87C42×C211—22no—
88C42×C211—41no—
89C42×C211—41no—
90C42×C211—41no—
91C42×C211C2×C414yescomplemented
92C23×C211—21no—
93C23×C211—21no—
94C23×C211—21no—
95C23×C211—21no—
96C23×C211—21no—
97C23×C211—21no—
98C23×C211C22×C412yes—
99C23×C211C2411yescharacteristic
100C2×C4×C211—21no—
101C2×C4×C211—21no—
102C2×C4×C211—21no—
103C2×C4×C211—21no—
104C2×C4×C211—21no—
105C2×C4×C211—21no—
106C2×C4×C211—21no—
107C2×C4×C211—21no—
108C2×C4×C211—21no—
109C2×C4×C211—21no—
110C2×C4×C211—21no—
111C2×C4×C211—21no—
112C2×C4×C211—21no—
113C2×C4×C211—21no—
114C2×C4×C211—21no—
115C2×C4×C211—21no—
116C2×C4×C211—21no—
117C2×C4×C211—21no—
118C2×C4×C211—21no—
119C2×C4×C211—21no—
120C2×C4×C211—21no—
121C2×C4×C211—21no—
122C2×C4×C211—21no—
123C2×C4×C211—21no—
124C2×C4×C211—21no—
125C2×C4×C211—21no—
126C2×C4×C211—21no—
127C2×C4×C211—21no—
128C2×C4×C211—21no—
129C2×C4×C211—21no—
130C2×C4×C211—21no—
131C2×C4×C211—21no—
132C2×C4×C211—44no—
133C2×C4×C211C22×C412yes—
134C2×C4×C211C22×C412yes—
135C2×C4×C211C22×C412yes—
136C2×C4×C211C22×C412yes—
137C2×C4×C211C22×C412yes—
138C2×C4×C211C22×C412yes—
139C2×C4×C211#4112yes—
140C2×C4×C211#4112yes—
141C2×C4×C211#4112yes—
142C2×C4×C211#4112yes—
143C22×C211—22no—
144C22×C211—22no—
145C22×C211—22no—
146C22×C211—41no—
147C22×C211—41no—
148C22×C211—41no—
149C22×C211—41no—
150C22×C211C23×C411yescharacteristic
151C22×C211C2×#4111yescharacteristic
152C22×C211C2×#4111yescharacteristic
153C22×C211C2×#4111yescharacteristic
154C22×C211C2×#4111yescharacteristic
155C22×C211H(5,𝔽2)11yescharacteristic
156C22×C2112−1+411yescharacteristic
157C4×C211—22no—
158C4×C211—22no—
159C4×C211—22no—
160C4×C211—22no—
161C4×C211—22no—
162C4×C211—22no—
163C4×C211—22no—
164C4×C211—22no—
165C4×C211—22no—
166C4×C211—22no—
167C4×C211—41no—
168C4×C211—41no—
169C4×C211—41no—
170C4×C211—41no—
171C2×C211—42no—
172C2×C211C4×#4111yescharacteristic
173C2×C211C4×#4111yescharacteristic
174C2×C211#52011yescharacteristic
175C2×C211#55111yescharacteristic
176C2×C211#55211yescharacteristic
177C2×C211#55311yescharacteristic
178C2×C211#55611yescharacteristic
179C211#237311yescharacteristic, direct product factor, Sylow
180#2373C21111yesmaximal, characteristic, direct product factor, Sylow
181C2×#126C2×C21111yescharacteristic
182#389C2×C21111yescharacteristic, complemented
183#388C2×C21111yescharacteristic
184#388C2×C21111yescharacteristic
185#388C2×C21111yescharacteristic
186#387C2×C21111yescharacteristic, complemented
187#387C2×C21111yescharacteristic, complemented
188#387C2×C21111yescharacteristic, complemented
189#385C2×C21111yescharacteristic
190C4×C4⋊C4C2×C21111yescharacteristic, complemented
191C4×C4⋊C4C2×C21111yescharacteristic, complemented
192C4×C4⋊C4C2×C21111yescharacteristic, complemented
193C4×C22⋊C4C2×C21111yescharacteristic
194C4×C22⋊C4C2×C21111yescharacteristic
195#376C2×C21111yescharacteristic, complemented
196C23×C4C22×C21111yescharacteristic
197#126C4×C21118yescomplemented
198C2×C4⋊C4C22×C21111yescharacteristic
199C2×C4⋊C4C22×C21111yescharacteristic
200C2×C4⋊C4C22×C21111yescharacteristic
201C2×C4⋊C4C22×C21111yescharacteristic
202C2×C4⋊C4C22×C21111yescharacteristic
203C2×C4⋊C4C22×C21111yescharacteristic
204C2×C4⋊C4C22×C21111yescharacteristic
205C2×C4⋊C4C22×C21111yescharacteristic
206C2×C22⋊C4C22×C21111yescharacteristic
207C2×C22⋊C4C22×C21111yescharacteristic
208C2×C22⋊C4C22×C21111yescharacteristic
209C2×C22⋊C4C22×C21111yescharacteristic
210C2×C22⋊C4C22×C21111yescharacteristic
211C2×C22⋊C4C22×C21111yescharacteristic
212C2×C42C22×C21111yescharacteristic
213C2×C42C22×C21111yescharacteristic
214C2×C42C22×C21111yescharacteristic
215C2×C42C22×C21111yescharacteristic
216C2×C42C22×C21111yescharacteristic
217C2×C42C22×C21111yescharacteristic
218C2×C42C22×C21111yescharacteristic
219C2×C42C22×C21111yescharacteristic
220#95C22×C21111yescharacteristic
221#95C22×C21111yescharacteristic
222#95C22×C21111yescharacteristic
223#95C22×C21111yescharacteristic
224#95C22×C21111yescharacteristic
225#95C22×C21111yescharacteristic
226#95C22×C21111yescharacteristic
227#95C22×C21111yescharacteristic
228#95C22×C21111yescharacteristic
229#95C22×C21111yescharacteristic
230#95C22×C21111yescharacteristic
231#95C22×C21111yescharacteristic
232C24C23×C21111yescharacteristic
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—41no—
236C22×C4C23×C21111yescharacteristic
237C22×C4C23×C21111yescharacteristic
238C22×C4C23×C21111yescharacteristic
239C22×C4C23×C21111yescharacteristic
240C22×C4C23×C21111yescharacteristic
241C22×C4C23×C21111yescharacteristic
242C22×C4C23×C21111yescharacteristic
243C22×C4C23×C21111yescharacteristic
244C22×C4C23×C21111yescharacteristic
245C22×C4C23×C21111yescharacteristic
246C22×C4C23×C21111yescharacteristic
247C22×C4C23×C21111yescharacteristic
248C22×C4C23×C21111yescharacteristic
249C22×C4C23×C21111yescharacteristic
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—41no—
255C4⋊C4C2×C4×C21114yescomplemented
256C4⋊C4C2×C4×C21114yescomplemented
257C4⋊C4C2×C4×C21114yescomplemented
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—41no—
261C22⋊C4C2×C4×C21114yes—
262C22⋊C4C2×C4×C21114yes—
263C22⋊C4C2×C4×C21114yes—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—41no—
269C42—41no—
270C42—41no—
271C42C2×C4×C21114yescomplemented
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C22×C4×C21112yes—
279C23C24×C21111yescharacteristic
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—44no—
313C2×C4C22×C4×C21112yes—
314C2×C4C22×C4×C21112yes—
315C2×C4C22×C4×C21112yes—
316C2×C4C22×C4×C21112yes—
317C2×C4C22×C4×C21112yes—
318C2×C4C22×C4×C21112yes—
319C2×C4#41×C21112yes—
320C2×C4#41×C21112yes—
321C2×C4#41×C21112yes—
322C2×C4#41×C21112yes—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C23×C4×C21111yescharacteristic
331C22C2×#41×C21111yescharacteristic
332C22C2×#41×C21111yescharacteristic
333C22C2×#41×C21111yescharacteristic
334C22C2×#41×C21111yescharacteristic
335C22H(5,𝔽2)×C21111yescharacteristic
336C222−1+4×C21111yescharacteristic
337C4—22no—
338C4—22no—
339C4—22no—
340C4—22no—
341C4—22no—
342C4—22no—
343C4—22no—
344C4—22no—
345C4—22no—
346C4—22no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—41no—
351C2—42no—
352C2C4×#41×C21111yescharacteristic
353C2C4×#41×C21111yescharacteristic
354C2#520×C21111yescharacteristic
355C2#551×C21111yescharacteristic
356C2#552×C21111yescharacteristic
357C2#553×C21111yescharacteristic
358C2#556×C21111yescharacteristic
subgroupG/H
center, socleC23×C211C24
commutatorC22C23×C4×C211
Frattini, cosocleC23C24×C211
prime radicalC24×C211C23
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C211 ◁ G

Complements

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

NH ≅ G/Ntype
2. #389×C211C2semidirect
6. #387×C211C2semidirect
7. #387×C211C2semidirect
8. #387×C211C2semidirect
10. C4×C4⋊C4×C211C2semidirect
11. C4×C4⋊C4×C211C2semidirect
12. C4×C4⋊C4×C211C2semidirect
15. #376×C211C2semidirect
17. #126×C211C4semidirect
75. C4⋊C4×C211C2×C4semidirect
76. C4⋊C4×C211C2×C4semidirect
77. C4⋊C4×C211C2×C4semidirect
91. C42×C211C2×C4semidirect
179. C211#2373direct (N × H)
180. #2373C211direct (N × H)
182. #389C2×C211semidirect
186. #387C2×C211semidirect
187. #387C2×C211semidirect
188. #387C2×C211semidirect
190. C4×C4⋊C4C2×C211semidirect
191. C4×C4⋊C4C2×C211semidirect
192. C4×C4⋊C4C2×C211semidirect
195. #376C2×C211semidirect
197. #126C4×C211semidirect
255. C4⋊C4C2×C4×C211semidirect
256. C4⋊C4C2×C4×C211semidirect
257. C4⋊C4C2×C4×C211semidirect
271. C42C2×C4×C211semidirect

Maximal subgroups

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

subgroupindex
C2×#126×C2112
#389×C2112
#388×C2112
#387×C2112
#385×C2112
C4×C4⋊C4×C2112
C4×C22⋊C4×C2112
#376×C2112
#2373211