Finite Group Databasebeta

← C2×#27161699 #38756089

Subgroups — C2×#27161699 #38756089

Order 25216.

1132 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161699C2116yesmaximal, direct product factor
2C2×#20263324C211yesmaximal, characteristic, complemented
3C2×#20263321C211yesmaximal, characteristic, complemented
4C2×#20263319C211yesmaximal, characteristic, complemented
5C2×#20263315C211yesmaximal, characteristic, complemented
6C2×#20263312C211yesmaximal, characteristic, complemented
7C2×#20263312C211yesmaximal, characteristic, complemented
8C2×#20263311C211yesmaximal, characteristic, complemented
9C2×#20263309C211yesmaximal, characteristic, complemented
10C2×#20263304C211yesmaximal, characteristic, complemented
11C2×#20263304C211yesmaximal, characteristic, complemented
12C2×#20263299C211yesmaximal, characteristic
13C2×#20263291C211yesmaximal, characteristic, complemented
14C2×C4×C22⋊D197C211yesmaximal, characteristic, complemented
15C2×#20263284C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C197C211yesmaximal, characteristic, complemented
17C22×Dic394C2211yescharacteristic
18C22×Dic394C2211yescharacteristic
19C22×D788C2211yescharacteristic
20C22×C22⋊D197C2211yescharacteristic
21C22×C22⋊D197C2211yescharacteristic
22C22×C4×D197C2211yescharacteristic
23C22×C4×D197C2211yescharacteristic
24#20263324C2218yescomplemented
25#20263321C2218yescomplemented
26#20263319C2218yescomplemented
27#20263315C2218yescomplemented
28#20263312C2218yescomplemented
29#20263312C2218yescomplemented
30#20263311C2218yescomplemented
31#20263309C2218yescomplemented
32#20263304C2218yescomplemented
33#20263304C2218yescomplemented
34#20263299C2218yes—
35#20263291C2218yescomplemented
36C4×C22⋊D197C2218yescomplemented
37#20263284C2218yescomplemented
38C2×C4⋊Dic197C2211yescharacteristic
39C2×#15691337C2211yescharacteristic
40C2×#15691337C2211yescharacteristic
41C2×#15691337C2211yescharacteristic
42C2×#15691337C2211yescharacteristic
43C2×#15691337C2211yescharacteristic
44C2×#15691337C2211yescharacteristic
45C2×C22⋊Dic197C2211yescharacteristic
46C2×C22⋊Dic197C2211yescharacteristic
47C2×#15691335C2211yescharacteristic
48C2×#15691335C2211yescharacteristic
49C2×#15691335C2211yescharacteristic
50C2×#15691335C2211yescharacteristic
51C2×#15691335C2211yescharacteristic
52C2×#15691335C2211yescharacteristic
53C2×#15691335C2211yescharacteristic
54C2×#15691335C2211yescharacteristic
55C2×C4×Dic197C2211yescharacteristic
56C2×C4×Dic197C2211yescharacteristic
57C2×C4×Dic197C2211yescharacteristic
58C2×C4×Dic197C2211yescharacteristic
59C22×Q8×C197C2211yescharacteristic
60C23×C4×C197C2211yescharacteristic
61C22⋊Q8×C197C2218yescomplemented
62C2×C4⋊C4×C197C2211yescharacteristic
63C2×C4⋊C4×C197C2211yescharacteristic
64C2×C4⋊C4×C197C2211yescharacteristic
65C2×C22⋊C4×C197C2211yescharacteristic
66C2×C22⋊C4×C197C2211yescharacteristic
67C23×D197C2311yescharacteristic
68C23×D197C2311yescharacteristic
69C2×Dic394—24no—
70C2×Dic394—42no—
71C2×Dic394C2314yes—
72C2×Dic394C2314yes—
73C2×D788—42no—
74C2×D788C2314yes—
75C2×C22⋊D197—24no—
76C2×C22⋊D197—42no—
77C2×C22⋊D197C2314yes—
78C2×C22⋊D197C2314yes—
79C22×Dic197C2311yescharacteristic
80C22×Dic197C2311yescharacteristic
81C22×Dic197C2311yescharacteristic
82C22×Dic197C2311yescharacteristic
83C22×Dic197C2311yescharacteristic
84C22×Dic197C2311yescharacteristic
85C2×C4×D197—24no—
86C2×C4×D197—42no—
87C2×C4×D197C2314yes—
88C2×C4×D197C2314yes—
89C4⋊Dic197C2314yes—
90#15691337C2314yes—
91#15691337C2314yes—
92#15691337C2314yes—
93#15691337C2314yes—
94#15691337C2314yes—
95#15691337C2314yes—
96C22⋊Dic197C2314yes—
97C22⋊Dic197C2314yes—
98#15691335C2314yes—
99#15691335C2314yes—
100#15691335C2314yes—
101#15691335C2314yes—
102#15691335C2314yes—
103#15691335C2314yes—
104#15691335C2314yes—
105#15691335C2314yes—
106C4×Dic197C2314yes—
107C4×Dic197C2314yes—
108C4×Dic197C2314yes—
109C4×Dic197C2314yes—
110C24×C197C2311yescharacteristic
111C2×Q8×C197—42no—
112C2×Q8×C197C2314yes—
113C22×C4×C197—42no—
114C22×C4×C197C2311yescharacteristic
115C22×C4×C197C2311yescharacteristic
116C22×C4×C197C2311yescharacteristic
117C22×C4×C197C2311yescharacteristic
118C22×C4×C197C2311yescharacteristic
119C22×C4×C197C2311yescharacteristic
120C22×C4×C197C2314yes—
121C4⋊C4×C197C2314yes—
122C4⋊C4×C197C2314yes—
123C4⋊C4×C197C2314yes—
124C22⋊C4×C197C2314yes—
125C22⋊C4×C197C2314yes—
126C22×D197—22no—
127C22×D197—22no—
128C22×D197—22no—
129C22×D197—22no—
130C22×D197—22no—
131C22×D197—22no—
132C22×D197C2412yes—
133C22×D197C2412yes—
134Dic394—28no—
135Dic394—44no—
136D788—44no—
137C22⋊D197—28no—
138C22⋊D197—44no—
139C2×Dic197—22no—
140C2×Dic197—22no—
141C2×Dic197—22no—
142C2×Dic197—22no—
143C2×Dic197—22no—
144C2×Dic197#4114yes—
145C2×Dic197C2412yes—
146C2×Dic197C2412yes—
147C2×Dic197C2412yes—
148C2×Dic197C2412yes—
149C2×Dic197C2412yes—
150C2×Dic197C2412yes—
151C4×D197—28no—
152C4×D197—44no—
153C23×C197—22no—
154C23×C197—22no—
155C23×C197—22no—
156C23×C197C2412yes—
157C23×C197C2411yescharacteristic
158Q8×C197—44no—
159C2×C4×C197—22no—
160C2×C4×C197—22no—
161C2×C4×C197—22no—
162C2×C4×C197—22no—
163C2×C4×C197—22no—
164C2×C4×C197—22no—
165C2×C4×C197—44no—
166C2×C4×C197C2412yes—
167C2×C4×C197C2412yes—
168C2×C4×C197C2412yes—
169C2×C4×C197C2412yes—
170C2×C4×C197C2412yes—
171C2×C4×C197C2412yes—
172C22×C197—22no—
173C22×C197—22no—
174C22×C197—22no—
175C22×C197—44no—
176C22×C197C2×#4112yes—
177C22×C197H(5,𝔽2)12yes—
178C22×C1972−1+412yes—
179C22×C197C2511yescharacteristic
180C2×D197—22no—
181C2×D197—22no—
182C2×D197—22no—
183C2×D197—22no—
184C2×D197—22no—
185C2×D197—22no—
186C2×D197—44no—
187C2×D197—44no—
188C4×C197—22no—
189C4×C197—22no—
190C4×C197—22no—
191C4×C197—22no—
192C4×C197—22no—
193C4×C197—22no—
194Dic197—22no—
195Dic197—22no—
196Dic197—22no—
197Dic197—22no—
198Dic197—22no—
199Dic197C2×#4114yes—
200C2×C197—42no—
201C2×C197#54214yescomplemented
202C2×C197C22×#4111yescharacteristic
203C2×C197C2×H(5,𝔽2)11yescharacteristic
204C2×C197C2×2−1+411yescharacteristic
205D197—42no—
206D197—42no—
207C197C2×#54211yescharacteristic, complemented, Sylow
208C2×#542—1971nomaximal, Sylow
209#542—19716no—
210C2×C4⋊Q8—1971no—
211C2×#126—1971no—
212C2×#126—1971no—
213C2×#124—1971no—
214C2×#124—1971no—
215C2×#124—1971no—
216C2×#123—1971no—
217C2×#123—1971no—
218C2×C22⋊Q8—1971no—
219C2×C22⋊Q8—1971no—
220C2×C22⋊Q8D19711yescharacteristic, complemented
221C2×#121—1971no—
222C2×C4×Q8—1971no—
223C2×C4×D4—1971no—
224C2×#117—1971no—
225C22×Q8—1971no—
226C22×Q8—1971no—
227C22×Q8C2×D19711yescharacteristic
228C22×D4—1971no—
229C22×D4—1971no—
230C22×D4—1971no—
231C23×C4—1971no—
232C23×C4—1971no—
233C23×C4C2×D19711yescharacteristic
234C4⋊Q8—1978no—
235#126—1978no—
236#126—1978no—
237#124—1978no—
238#124—1978no—
239#124—1978no—
240#123—1978no—
241#123—1978no—
242C22⋊Q8—1978no—
243C22⋊Q8—1978no—
244C22⋊Q8C2×D19718yescomplemented
245#121—1978no—
246C4×Q8—1978no—
247C4×D4—1978no—
248#117—1978no—
249C2×C4⋊C4—1971no—
250C2×C4⋊C4—1971no—
251C2×C4⋊C4—1971no—
252C2×C4⋊C4—1971no—
253C2×C4⋊C4—1971no—
254C2×C4⋊C4—1971no—
255C2×C4⋊C4—1971no—
256C2×C4⋊C4C2×D19711yescharacteristic
257C2×C4⋊C4C2×D19711yescharacteristic
258C2×C4⋊C4C2×D19711yescharacteristic
259C2×C22⋊C4—1971no—
260C2×C22⋊C4—1971no—
261C2×C22⋊C4—1971no—
262C2×C22⋊C4—1971no—
263C2×C22⋊C4—1971no—
264C2×C22⋊C4—1971no—
265C2×C22⋊C4—1971no—
266C2×C22⋊C4—1971no—
267C2×C22⋊C4—1971no—
268C2×C22⋊C4—1971no—
269C2×C22⋊C4C2×D19711yescharacteristic
270C2×C22⋊C4C2×D19711yescharacteristic
271C2×C42—1971no—
272C2×C42—1971no—
273C2×C42—1971no—
274C2×C42—1971no—
275C24—1971no—
276C24—1971no—
277C24C22×D19711yescharacteristic
278C2×Q8—42no—
279C2×Q8—7882no—
280C2×Q8—3944no—
281C2×Q8—1974no—
282C2×Q8—1974no—
283C2×Q8C22×D19714yes—
284C2×D4—7882no—
285C2×D4—7882no—
286C2×D4—3944no—
287C2×D4—1974no—
288C2×D4—1974no—
289C2×D4—1974no—
290C22×C4—42no—
291C22×C4—7882no—
292C22×C4—3944no—
293C22×C4—1974no—
294C22×C4—1971no—
295C22×C4—1971no—
296C22×C4—1971no—
297C22×C4—1971no—
298C22×C4—1971no—
299C22×C4—1974no—
300C22×C4—1971no—
301C22×C4C22×D19711yescharacteristic
302C22×C4C22×D19711yescharacteristic
303C22×C4C22×D19711yescharacteristic
304C22×C4C22×D19711yescharacteristic
305C22×C4C22×D19711yescharacteristic
306C22×C4C22×D19711yescharacteristic
307C22×C4C22×D19714yes—
308C4⋊C4—1974no—
309C4⋊C4—1974no—
310C4⋊C4—1974no—
311C4⋊C4—1974no—
312C4⋊C4—1974no—
313C4⋊C4—1974no—
314C4⋊C4—1974no—
315C4⋊C4C22×D19714yes—
316C4⋊C4C22×D19714yes—
317C4⋊C4C22×D19714yes—
318C22⋊C4—1974no—
319C22⋊C4—1974no—
320C22⋊C4—1974no—
321C22⋊C4—1974no—
322C22⋊C4—1974no—
323C22⋊C4—1974no—
324C22⋊C4—1974no—
325C22⋊C4—1974no—
326C22⋊C4—1974no—
327C22⋊C4—1974no—
328C22⋊C4C22×D19714yes—
329C22⋊C4C22×D19714yes—
330C42—1974no—
331C42—1974no—
332C42—1974no—
333C42—1974no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—3942no—
338C23—3942no—
339C23—3942no—
340C23—3942no—
341C23—3942no—
342C23—3942no—
343C23—1972no—
344C23—1972no—
345C23C23×D19712yes—
346C23C23×D19711yescharacteristic
347Q8—44no—
348Q8—7884no—
349Q8—3948no—
350D4—7884no—
351D4—7884no—
352D4—3948no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—44no—
360C2×C4—7884no—
361C2×C4—3948no—
362C2×C4—3942no—
363C2×C4—3942no—
364C2×C4—3942no—
365C2×C4—3942no—
366C2×C4—3942no—
367C2×C4—1972no—
368C2×C4—1972no—
369C2×C4—1972no—
370C2×C4—1972no—
371C2×C4—1972no—
372C2×C4—1974no—
373C2×C4—1972no—
374C2×C4C23×D19712yes—
375C2×C4C23×D19712yes—
376C2×C4C23×D19712yes—
377C2×C4C23×D19712yes—
378C2×C4C23×D19712yes—
379C2×C4C23×D19712yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—44no—
384C22—7884no—
385C22—7884no—
386C22—3942no—
387C22—3942no—
388C22—3942no—
389C22—3942no—
390C22—3942no—
391C22—3942no—
392C22#41×D19712yes—
393C22#2026337112yes—
394C22#2026337412yes—
395C22C24×D19711yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—22no—
402C4—3942no—
403C4—3942no—
404C4—3942no—
405C4—3942no—
406C4—3942no—
407C4—1974no—
408C2—42no—
409C2—7882no—
410C2—7882no—
411C2#2716169914yesdirect product factor
412C2C2×#41×D19711yescharacteristic
413C2C2×#2026337111yescharacteristic
414C2C2×#2026337411yescharacteristic
subgroupG/H
centerC23C23×D197
commutator, cosocleC22×C197C25
FrattiniC22C24×D197
FittingC2×C22⋊Q8×C197C2
socleC23×C197C24
prime radicalC2×#20263299C2
hypercenterC2×C22⋊Q8D197
hypocenterC197C2×#542
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #27161699C2direct (N × H)
2. C2×#20263324C2semidirect
3. C2×#20263321C2semidirect
4. C2×#20263319C2semidirect
5. C2×#20263315C2semidirect
6. C2×#20263312C2semidirect
7. C2×#20263312C2semidirect
8. C2×#20263311C2semidirect
9. C2×#20263309C2semidirect
10. C2×#20263304C2semidirect
11. C2×#20263304C2semidirect
13. C2×#20263291C2semidirect
14. C2×C4×C22⋊D197C2semidirect
15. C2×#20263284C2semidirect
16. C2×C22⋊Q8×C197C2semidirect
24. #20263324C22semidirect
25. #20263321C22semidirect
26. #20263319C22semidirect
27. #20263315C22semidirect
28. #20263312C22semidirect
29. #20263312C22semidirect
30. #20263311C22semidirect
31. #20263309C22semidirect
32. #20263304C22semidirect
33. #20263304C22semidirect
35. #20263291C22semidirect
36. C4×C22⋊D197C22semidirect
37. #20263284C22semidirect
61. C22⋊Q8×C197C22semidirect
201. C2×C197#542semidirect
207. C197C2×#542semidirect
220. C2×C22⋊Q8D197semidirect
244. C22⋊Q8C2×D197semidirect
411. C2#27161699direct (N × H)

Maximal subgroups

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

subgroupindex
#271616992
C2×#202633242
C2×#202633212
C2×#202633192
C2×#202633152
C2×#202633122
C2×#202633112
C2×#202633092
C2×#202633042
C2×#202632992
C2×#202632912
C2×C4×C22⋊D1972
C2×#202632842
C2×C22⋊Q8×C1972
C2×#542197