Finite Group Databasebeta

← C2×#20261725 #27158352

Subgroups — C2×#20261725 #27158352

Order 12416.

792 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13095519C211yesmaximal, characteristic, complemented
2C2×#15690889C211yesmaximal, characteristic, complemented
3#20261725C218yesmaximal, direct product factor
4C2×#15690873C212yesmaximal, complemented
5C2×#15690872C212yesmaximal, complemented
6C2×#41×D97C211yesmaximal, characteristic
7C2×#13095519—22no—
8C2×#13095519—21no—
9C2×#13095519C2212yes—
10C2×#13095519C2214yescomplemented
11C22×#11715897—21no—
12#15690889—22no—
13#15690889C2214yescomplemented
14C2×C4⋊#87604C2212yes—
15C2×C22⋊#87604—21no—
16C2×C4×#87604—21no—
17#15690873C2218yescomplemented
18#15690872C2218yescomplemented
19C2×#13095512C411yescharacteristic, complemented
20C2×#13095511—21no—
21C2×#13095511C411yescharacteristic, complemented
22C2×#13095510—21no—
23C2×#13095510C411yescharacteristic, complemented
24#41×D97—22no—
25#41×D97—22no—
26#41×D97C414yescomplemented
27#41×D97C2214yes—
28C2×Q8×D97C2211yescharacteristic
29C2×D4×D97—21no—
30C2×D4×D97C2211yescharacteristic
31C22×C4×D97—21no—
32C22×C4×D97C2211yescharacteristic
33C2×#41×C97C411yescharacteristic, complemented
34C22×#87604—22no—
35#13095519—22no—
36#13095519—24no—
37#13095519C2314yes—
38C2×#11715897—22no—
39C2×#11715897—24no—
40C4⋊#87604—22no—
41C4⋊#87604C2314yes—
42C22⋊#87604—24no—
43C4×#87604—24no—
44C23×D97—21no—
45C23×D97D411yescharacteristic
46#13095512—22no—
47#13095512C2×C414yescomplemented
48#13095511—22no—
49#13095511—24no—
50#13095511—24no—
51#13095511C2×C414yescomplemented
52#13095510—24no—
53#13095510—24no—
54#13095510—22no—
55#13095510C2×C414yescomplemented
56C2×Dic194—21no—
57C2×Dic194C2×C411yescharacteristic
58Q8×D97—21no—
59Q8×D97—21no—
60Q8×D97C2×C412yes—
61Q8×D97C2312yes—
62C2×D388—21no—
63C2×D388C2×C411yescharacteristic
64C2×C22⋊D97—21no—
65C2×C22⋊D97—21no—
66C2×C22⋊D97—21no—
67D4×D97—24no—
68D4×D97—24no—
69D4×D97—21no—
70D4×D97—21no—
71D4×D97C2×C412yes—
72D4×D97C2312yes—
73C22×Dic97—21no—
74C22×Dic97D411yescharacteristic
75C2×C4×D97—21no—
76C2×C4×D97—21no—
77C2×C4×D97—21no—
78C2×C4×D97—24no—
79C2×C4×D97—24no—
80C2×C4×D97C2×C412yes—
81C2×C4×D97C2×C411yescharacteristic
82C2×C4×D97D411yescharacteristic
83C2×C4×D97D412yes—
84C2×C4×D97D412yes—
85C2×C4×D97D411yescharacteristic
86C2×C4×D97C2311yescharacteristic
87C2×C4×D97C2312yes—
88#41×C97—22no—
89#41×C97C2×C414yescomplemented
90C2×Q8×C97C2×C411yescharacteristic
91C2×D4×C97—21no—
92C2×D4×C97C2×C411yescharacteristic
93C22×C4×C97—21no—
94C22×C4×C97C2×C411yescharacteristic
95C23×C97—21no—
96C23×C97C22⋊C411yescharacteristic
97C22×D97—24no—
98C22×D97—21no—
99C22×D97—22no—
100C22×D97—22no—
101C22×D97—44no—
102C22×D97C22⋊C412yes—
103C22×D97C22⋊C411yescharacteristic
104C22×D97C2×D412yes—
105C22×D97C2×D411yescharacteristic
106C2×#87604—24no—
107C2×#87604—44no—
108Q8×C97—21no—
109Q8×C97C22×C412yes—
110D4×C97—24no—
111D4×C97—21no—
112D4×C97C22×C412yes—
113C2×C4×C97—21no—
114C2×C4×C97—24no—
115C2×C4×C97C22⋊C412yes—
116C2×C4×C97C22⋊C411yescharacteristic
117C2×C4×C97C22×C411yescharacteristic
118C2×C4×C97C22×C412yes—
119C22⋊D97—24no—
120C22⋊D97—24no—
121C22⋊D97—22no—
122C22⋊D97—22no—
123C2×Dic97—21no—
124C2×Dic97—24no—
125C2×Dic97C22⋊C411yescharacteristic
126C2×Dic97C22⋊C412yes—
127C2×Dic97C2×D411yescharacteristic
128C2×Dic97C2×D412yes—
129D388—24no—
130D388—21no—
131D388C22×C412yes—
132C4×D97—22no—
133C4×D97—22no—
134C4×D97—24no—
135C4×D97—24no—
136C4×D97C22⋊C412yes—
137C4×D97C22⋊C412yes—
138C4×D97C22⋊C412yes—
139C4×D97C22×C411yescharacteristic
140C4×D97C22×C412yes—
141C4×D97C2×D412yes—
142C4×D97C2×D411yescharacteristic
143C4×D97C2×D411yescharacteristic
144C4×D97C2×D412yes—
145C4×D97C2411yescharacteristic
146Dic194—21no—
147Dic194—22no—
148Dic194—22no—
149Dic194C22×C412yes—
150#11715897—24no—
151C22×C97—22no—
152C22×C97—22no—
153C22×C97—42no—
154C22×C97C2×C22⋊C412yes—
155C22×C97C2×C22⋊C411yescharacteristic
156C2×D97—22no—
157C2×D97—24no—
158C2×D97—24no—
159C2×D97—22no—
160C2×D97—42no—
161C2×D97—44no—
162C2×D97—44no—
163C2×D97C2×C22⋊C412yes—
164C2×D97C2×C22⋊C411yescharacteristic
165C2×D97#13514yes—
166C2×D97C22×D411yescharacteristic
167#87604—44no—
168C4×C97—22no—
169C4×C97C2×C22⋊C411yescharacteristic
170C4×C97C2×C22⋊C412yes—
171C4×C97C23×C411yescharacteristic
172Dic97—22no—
173Dic97C2×C22⋊C411yescharacteristic
174Dic97C2×C22⋊C412yes—
175Dic97C22×D411yescharacteristic
176C2×C97—22no—
177C2×C97—42no—
178C2×C97#41612yescomplemented
179C2×C97C22×C22⋊C411yescharacteristic
180D97—22no—
181D97—42no—
182D97#41612yescomplemented
183D97C2×#13512yes—
184C2×#416—971nomaximal, Sylow
185C97C2×#41611yescharacteristic, complemented, Sylow
186C22×#41—971no—
187C23×C8—971no—
188C2×#117—971no—
189#416—978no—
190C2×#103—972no—
191C2×#102—972no—
192C2×#41—974no—
193C2×#41—974no—
194C2×#41—971no—
195C2×#41—971no—
196C2×#41—971no—
197C2×#41—1941no—
198C2×#41—1941no—
199C2×#41—1942no—
200C2×#41—1942no—
201C2×#41#8760411yescharacteristic, complemented
202C22×Q8—971no—
203C22×D4—971no—
204C22×D4—1941no—
205C23×C4—971no—
206C23×C4—1941no—
207C22×C8—972no—
208C22×C8—974no—
209C22×C8—1941no—
210C22×C8—1942no—
211C22×C8—1941no—
212#117—974no—
213#117—1942no—
214C2×C4⋊C4—972no—
215C2×C22⋊C4—1941no—
216C2×C42—1941no—
217#103—978no—
218#102—978no—
219C24—971no—
220C24—1941no—
221#41—22no—
222#41—974no—
223#41—974no—
224#41—974no—
225#41—1944no—
226#41—1944no—
227#41—1944no—
228#41—1944no—
229#41—1942no—
230#41—1942no—
231#41—1942no—
232#41C2×#8760414yescomplemented
233C2×Q8—972no—
234C2×Q8—972no—
235C2×Q8—971no—
236C2×Q8—1941no—
237C2×Q8—1941no—
238C2×Q8—1941no—
239C2×Q8C2×#8760411yescharacteristic
240C2×D4—21no—
241C2×D4—972no—
242C2×D4—972no—
243C2×D4—971no—
244C2×D4—1941no—
245C2×D4—1941no—
246C2×D4—1941no—
247C2×D4—1941no—
248C2×D4—1944no—
249C2×D4—1944no—
250C2×D4—1941no—
251C2×D4—1941no—
252C2×D4C2×#8760411yescharacteristic
253C22×C4—21no—
254C22×C4—971no—
255C22×C4—972no—
256C22×C4—972no—
257C22×C4—971no—
258C22×C4—971no—
259C22×C4—971no—
260C22×C4—972no—
261C22×C4—971no—
262C22×C4—972no—
263C22×C4—1941no—
264C22×C4—1944no—
265C22×C4—1944no—
266C22×C4—1941no—
267C22×C4—1941no—
268C22×C4—1942no—
269C22×C4—1941no—
270C22×C4C2×#8760411yescharacteristic
271C2×C8—974no—
272C2×C8—1942no—
273C2×C8—1944no—
274C2×C8—1942no—
275C2×C8—1944no—
276C4⋊C4—974no—
277C4⋊C4—1942no—
278C22⋊C4—1944no—
279C42—1944no—
280C23—21no—
281C23—971no—
282C23—972no—
283C23—972no—
284C23—971no—
285C23—3884no—
286C23—1942no—
287C23—1942no—
288C23—1944no—
289C23—1941no—
290C23C22⋊#8760411yescharacteristic
291Q8—21no—
292Q8—972no—
293Q8—1942no—
294Q8—1942no—
295Q8—1941no—
296Q8C22×#8760412yes—
297D4—24no—
298D4—21no—
299D4—972no—
300D4—1944no—
301D4—1942no—
302D4—1942no—
303D4—1944no—
304D4—1944no—
305D4—1941no—
306D4C22×#8760412yes—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—971no—
310C2×C4—972no—
311C2×C4—971no—
312C2×C4—972no—
313C2×C4—972no—
314C2×C4—971no—
315C2×C4—971no—
316C2×C4—972no—
317C2×C4—972no—
318C2×C4—972no—
319C2×C4—972no—
320C2×C4—971no—
321C2×C4—971no—
322C2×C4—972no—
323C2×C4—3884no—
324C2×C4—1944no—
325C2×C4—1942no—
326C2×C4—1942no—
327C2×C4—1944no—
328C2×C4—1944no—
329C2×C4—1941no—
330C2×C4—1944no—
331C2×C4C22⋊#8760412yes—
332C2×C4C22⋊#8760411yescharacteristic
333C2×C4C22×#8760411yescharacteristic
334C2×C4C22×#8760412yes—
335C8—1944no—
336C22—22no—
337C22—22no—
338C22—42no—
339C22—974no—
340C22—971no—
341C22—971no—
342C22—972no—
343C22—3884no—
344C22—3884no—
345C22—3882no—
346C22—1944no—
347C22—1944no—
348C22—1942no—
349C22—1942no—
350C22C2×C22⋊#8760412yes—
351C22C2×C22⋊#8760411yescharacteristic
352C4—22no—
353C4—971no—
354C4—972no—
355C4—971no—
356C4—3884no—
357C4—1942no—
358C4C2×C22⋊#8760412yes—
359C4C2×C22⋊#8760411yescharacteristic
360C4C23×#8760411yescharacteristic
361C2—22no—
362C2—42no—
363C2—972no—
364C2—972no—
365C2—3882no—
366C2—1942no—
367C2#2026172512yesdirect product factor
368C2C22×C22⋊#8760411yescharacteristic
subgroupG/H
centerC2×C4C22⋊#87604
commutatorC4×C97C23×C4
FrattiniC4C23×#87604
FittingC2×#41×C97C4
socleC22×C97C2×C22⋊C4
prime radicalC2×#41×D97C2
cosocleC4×D97C24
hypercenterC2×#41#87604
hypocenterC97C2×#416
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
1. C22×#13095519C2semidirect
2. C2×#15690889C2semidirect
3. #20261725C2direct (N × H)
4. C2×#15690873C2semidirect
5. C2×#15690872C2semidirect
10. C2×#13095519C22semidirect
13. #15690889C22semidirect
17. #15690873C22semidirect
18. #15690872C22semidirect
19. C2×#13095512C4semidirect
21. C2×#13095511C4semidirect
23. C2×#13095510C4semidirect
26. #41×D97C4semidirect
33. C2×#41×C97C4semidirect
47. #13095512C2×C4semidirect
51. #13095511C2×C4semidirect
55. #13095510C2×C4semidirect
89. #41×C97C2×C4semidirect
178. C2×C97#416semidirect
182. D97#416semidirect
185. C97C2×#416semidirect
201. C2×#41#87604semidirect
232. #41C2×#87604semidirect
367. C2#20261725direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955192
C2×#156908892
#202617252
C2×#156908732
C2×#156908722
C2×#41×D972
C2×#41697