Finite Group Databasebeta

← C2×#20261481 #27155861

Subgroups — C2×#20261481 #27155861

Order 12416.

1152 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261481C2116yesmaximal, direct product factor
2C2×#15690800C211yesmaximal, characteristic, complemented
3C2×#15690799C211yesmaximal, characteristic, complemented
4C2×#15690796C211yesmaximal, characteristic, complemented
5C2×#15690795C211yesmaximal, characteristic, complemented
6C2×#15690795C211yesmaximal, characteristic, complemented
7C2×#15690793C211yesmaximal, characteristic, complemented
8C2×#15690792C211yesmaximal, characteristic, complemented
9C2×#15690790C211yesmaximal, characteristic, complemented
10C2×#15690788C211yesmaximal, characteristic, complemented
11C2×#15690786C211yesmaximal, characteristic, complemented
12C2×#15690785C211yesmaximal, characteristic, complemented
13C2×#15690784C211yesmaximal, characteristic, complemented
14C2×#15690783C211yesmaximal, characteristic, complemented
15C2×#15690783C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C97C211yesmaximal, characteristic, complemented
17C22×Dic194C2211yescharacteristic
18C22×D388C2211yescharacteristic
19C22×D388C2211yescharacteristic
20C22×D388C2211yescharacteristic
21C22×C22⋊D97C2211yescharacteristic
22C22×C22⋊D97C2211yescharacteristic
23C22×C22⋊D97C2211yescharacteristic
24C22×C4×D97C2211yescharacteristic
25C22×C4×D97C2211yescharacteristic
26C22×C4×D97C2211yescharacteristic
27#15690800C2218yescomplemented
28#15690799C2218yescomplemented
29#15690796C2218yescomplemented
30#15690795C2218yescomplemented
31#15690795C2218yescomplemented
32#15690793C2218yescomplemented
33#15690792C2218yescomplemented
34#15690790C2218yescomplemented
35#15690788C2218yescomplemented
36#15690786C2218yescomplemented
37#15690785C2218yescomplemented
38#15690784C2218yescomplemented
39#15690783C2218yescomplemented
40#15690783C2218yescomplemented
41C2×C4⋊Dic97C2211yescharacteristic
42C2×C4⋊Dic97C2211yescharacteristic
43C2×#13095490C2211yescharacteristic
44C2×#13095490C2211yescharacteristic
45C2×#13095490C2211yescharacteristic
46C2×#13095490C2211yescharacteristic
47C2×#13095490C2211yescharacteristic
48C2×C22⋊Dic97C2211yescharacteristic
49C2×#13095488C2211yescharacteristic
50C2×#13095488C2211yescharacteristic
51C2×#13095488C2211yescharacteristic
52C2×#13095488C2211yescharacteristic
53C2×#13095488C2211yescharacteristic
54C2×#13095488C2211yescharacteristic
55C2×#13095488C2211yescharacteristic
56C2×#13095488C2211yescharacteristic
57C2×#13095488C2211yescharacteristic
58C2×C4×Dic97C2211yescharacteristic
59C22×Q8×C97C2211yescharacteristic
60C23×C4×C97C2211yescharacteristic
61C22⋊Q8×C97C2218yescomplemented
62C2×C4⋊C4×C97C2211yescharacteristic
63C2×C4⋊C4×C97C2211yescharacteristic
64C2×C4⋊C4×C97C2211yescharacteristic
65C2×C22⋊C4×C97C2211yescharacteristic
66C2×C22⋊C4×C97C2211yescharacteristic
67C23×D97C2311yescharacteristic
68C23×D97C2311yescharacteristic
69C23×D97C2311yescharacteristic
70C2×Dic194—42no—
71C2×Dic194C2314yes—
72C2×D388—42no—
73C2×D388—42no—
74C2×D388—42no—
75C2×D388C2314yes—
76C2×D388C2314yes—
77C2×D388C2314yes—
78C2×C22⋊D97—42no—
79C2×C22⋊D97—42no—
80C2×C22⋊D97—42no—
81C2×C22⋊D97C2314yes—
82C2×C22⋊D97C2314yes—
83C2×C22⋊D97C2314yes—
84C22×Dic97C2311yescharacteristic
85C22×Dic97C2311yescharacteristic
86C22×Dic97C2311yescharacteristic
87C22×Dic97C2311yescharacteristic
88C22×Dic97C2311yescharacteristic
89C2×C4×D97—42no—
90C2×C4×D97—42no—
91C2×C4×D97—42no—
92C2×C4×D97C2314yes—
93C2×C4×D97C2314yes—
94C2×C4×D97C2314yes—
95C4⋊Dic97C2314yes—
96C4⋊Dic97C2314yes—
97#13095490C2314yes—
98#13095490C2314yes—
99#13095490C2314yes—
100#13095490C2314yes—
101#13095490C2314yes—
102C22⋊Dic97C2314yes—
103#13095488C2314yes—
104#13095488C2314yes—
105#13095488C2314yes—
106#13095488C2314yes—
107#13095488C2314yes—
108#13095488C2314yes—
109#13095488C2314yes—
110#13095488C2314yes—
111#13095488C2314yes—
112C4×Dic97C2314yes—
113C24×C97C2311yescharacteristic
114C2×Q8×C97—42no—
115C2×Q8×C97C2314yes—
116C22×C4×C97—42no—
117C22×C4×C97C2311yescharacteristic
118C22×C4×C97C2311yescharacteristic
119C22×C4×C97C2311yescharacteristic
120C22×C4×C97C2311yescharacteristic
121C22×C4×C97C2311yescharacteristic
122C22×C4×C97C2311yescharacteristic
123C22×C4×C97C2314yes—
124C4⋊C4×C97C2314yes—
125C4⋊C4×C97C2314yes—
126C4⋊C4×C97C2314yes—
127C22⋊C4×C97C2314yes—
128C22⋊C4×C97C2314yes—
129C23×C97—22no—
130C23×C97—22no—
131C23×C97—22no—
132C23×C97C2412yes—
133C23×C97C2411yescharacteristic
134C22×D97—22no—
135C22×D97—22no—
136C22×D97—22no—
137C22×D97—22no—
138C22×D97—22no—
139C22×D97—22no—
140C22×D97—22no—
141C22×D97—22no—
142C22×D97—22no—
143C22×D97C2412yes—
144C22×D97C2412yes—
145C22×D97C2412yes—
146Q8×C97—44no—
147C2×C4×C97—22no—
148C2×C4×C97—22no—
149C2×C4×C97—22no—
150C2×C4×C97—22no—
151C2×C4×C97—22no—
152C2×C4×C97—22no—
153C2×C4×C97—44no—
154C2×C4×C97C2412yes—
155C2×C4×C97C2412yes—
156C2×C4×C97C2412yes—
157C2×C4×C97C2412yes—
158C2×C4×C97C2412yes—
159C2×C4×C97C2412yes—
160C22⋊D97—44no—
161C22⋊D97—44no—
162C22⋊D97—44no—
163C2×Dic97—22no—
164C2×Dic97—22no—
165C2×Dic97—22no—
166C2×Dic97—22no—
167C2×Dic97—22no—
168C2×Dic97C2412yes—
169C2×Dic97C2412yes—
170C2×Dic97C2412yes—
171C2×Dic97C2412yes—
172C2×Dic97C2412yes—
173D388—44no—
174D388—44no—
175D388—44no—
176C4×D97—44no—
177C4×D97—44no—
178C4×D97—44no—
179Dic194—44no—
180C22×C97—22no—
181C22×C97—22no—
182C22×C97—22no—
183C22×C97—44no—
184C22×C97H(5,𝔽2)12yes—
185C22×C97H(5,𝔽2)12yes—
186C22×C972−1+412yes—
187C22×C97C2511yescharacteristic
188C2×D97—22no—
189C2×D97—22no—
190C2×D97—22no—
191C2×D97—22no—
192C2×D97—22no—
193C2×D97—22no—
194C2×D97—22no—
195C2×D97—22no—
196C2×D97—22no—
197C2×D97—44no—
198C2×D97—44no—
199C2×D97—44no—
200C4×C97—22no—
201C4×C97—22no—
202C4×C97—22no—
203C4×C97—22no—
204C4×C97—22no—
205C4×C97—22no—
206Dic97—22no—
207Dic97—22no—
208Dic97—22no—
209Dic97—22no—
210Dic97—22no—
211C2×C97—42no—
212C2×C97#56214yescomplemented
213C2×C97C2×H(5,𝔽2)11yescharacteristic
214C2×C97C2×H(5,𝔽2)11yescharacteristic
215C2×C97C2×2−1+411yescharacteristic
216D97—42no—
217D97—42no—
218D97—42no—
219C2×#562—971nomaximal, Sylow
220C97C2×#56211yescharacteristic, complemented, Sylow
221#562—9716no—
222C2×#125—971no—
223C2×#124—971no—
224C2×#124—971no—
225C2×#123—971no—
226C2×#123—971no—
227C2×#123—971no—
228C2×#123—971no—
229C2×C22⋊Q8—971no—
230C2×C22⋊Q8—971no—
231C2×C22⋊Q8—971no—
232C2×C22⋊Q8D9711yescharacteristic, complemented
233C2×#121—971no—
234C2×#121—971no—
235C2×#121—971no—
236C2×#121—971no—
237C22×Q8—971no—
238C22×Q8C2×D9711yescharacteristic
239C22×D4—971no—
240C22×D4—971no—
241C22×D4—971no—
242C22×D4—971no—
243C22×D4—971no—
244C22×D4—971no—
245C23×C4—971no—
246C23×C4—971no—
247C23×C4—971no—
248C23×C4C2×D9711yescharacteristic
249#125—978no—
250#124—978no—
251#124—978no—
252#123—978no—
253#123—978no—
254#123—978no—
255#123—978no—
256C22⋊Q8—978no—
257C22⋊Q8—978no—
258C22⋊Q8—978no—
259C22⋊Q8C2×D9718yescomplemented
260#121—978no—
261#121—978no—
262#121—978no—
263#121—978no—
264C2×C4⋊C4—971no—
265C2×C4⋊C4—971no—
266C2×C4⋊C4—971no—
267C2×C4⋊C4—971no—
268C2×C4⋊C4—971no—
269C2×C4⋊C4—971no—
270C2×C4⋊C4—971no—
271C2×C4⋊C4C2×D9711yescharacteristic
272C2×C4⋊C4C2×D9711yescharacteristic
273C2×C4⋊C4C2×D9711yescharacteristic
274C2×C22⋊C4—971no—
275C2×C22⋊C4—971no—
276C2×C22⋊C4—971no—
277C2×C22⋊C4—971no—
278C2×C22⋊C4—971no—
279C2×C22⋊C4—971no—
280C2×C22⋊C4—971no—
281C2×C22⋊C4—971no—
282C2×C22⋊C4—971no—
283C2×C22⋊C4—971no—
284C2×C22⋊C4C2×D9711yescharacteristic
285C2×C22⋊C4C2×D9711yescharacteristic
286C2×C42—971no—
287C24—971no—
288C24—971no—
289C24—971no—
290C24C22×D9711yescharacteristic
291C2×Q8—42no—
292C2×Q8—974no—
293C2×Q8—3882no—
294C2×Q8C22×D9714yes—
295C2×D4—974no—
296C2×D4—974no—
297C2×D4—974no—
298C2×D4—974no—
299C2×D4—974no—
300C2×D4—974no—
301C2×D4—3882no—
302C2×D4—3882no—
303C2×D4—3882no—
304C2×D4—3882no—
305C2×D4—3882no—
306C2×D4—3882no—
307C22×C4—42no—
308C22×C4—971no—
309C22×C4—971no—
310C22×C4—971no—
311C22×C4—971no—
312C22×C4—971no—
313C22×C4—974no—
314C22×C4—974no—
315C22×C4—974no—
316C22×C4—3882no—
317C22×C4—3882no—
318C22×C4—3882no—
319C22×C4C22×D9711yescharacteristic
320C22×C4C22×D9711yescharacteristic
321C22×C4C22×D9711yescharacteristic
322C22×C4C22×D9711yescharacteristic
323C22×C4C22×D9711yescharacteristic
324C22×C4C22×D9711yescharacteristic
325C22×C4C22×D9714yes—
326C4⋊C4—974no—
327C4⋊C4—974no—
328C4⋊C4—974no—
329C4⋊C4—974no—
330C4⋊C4—974no—
331C4⋊C4—974no—
332C4⋊C4—974no—
333C4⋊C4C22×D9714yes—
334C4⋊C4C22×D9714yes—
335C4⋊C4C22×D9714yes—
336C22⋊C4—974no—
337C22⋊C4—974no—
338C22⋊C4—974no—
339C22⋊C4—974no—
340C22⋊C4—974no—
341C22⋊C4—974no—
342C22⋊C4—974no—
343C22⋊C4—974no—
344C22⋊C4—974no—
345C22⋊C4—974no—
346C22⋊C4C22×D9714yes—
347C22⋊C4C22×D9714yes—
348C42—974no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—972no—
353C23—972no—
354C23—972no—
355C23—1942no—
356C23—1942no—
357C23—1942no—
358C23—1942no—
359C23—1942no—
360C23—1942no—
361C23—1942no—
362C23—1942no—
363C23—1942no—
364C23C23×D9712yes—
365C23C23×D9711yescharacteristic
366Q8—44no—
367Q8—3884no—
368D4—3884no—
369D4—3884no—
370D4—3884no—
371D4—3884no—
372D4—3884no—
373D4—3884no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—44no—
381C2×C4—972no—
382C2×C4—972no—
383C2×C4—972no—
384C2×C4—972no—
385C2×C4—972no—
386C2×C4—3884no—
387C2×C4—3884no—
388C2×C4—3884no—
389C2×C4—1942no—
390C2×C4—1942no—
391C2×C4—1942no—
392C2×C4—1942no—
393C2×C4—1942no—
394C2×C4C23×D9712yes—
395C2×C4C23×D9712yes—
396C2×C4C23×D9712yes—
397C2×C4C23×D9712yes—
398C2×C4C23×D9712yes—
399C2×C4C23×D9712yes—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—44no—
404C22—3884no—
405C22—3884no—
406C22—3884no—
407C22—1942no—
408C22—1942no—
409C22—1942no—
410C22—1942no—
411C22—1942no—
412C22—1942no—
413C22—1942no—
414C22—1942no—
415C22—1942no—
416C22#1569085512yes—
417C22#1569085612yes—
418C22#1569085812yes—
419C22C24×D9711yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—1942no—
427C4—1942no—
428C4—1942no—
429C4—1942no—
430C4—1942no—
431C2—42no—
432C2—3882no—
433C2—3882no—
434C2—3882no—
435C2#2026148114yesdirect product factor
436C2C2×#1569085511yescharacteristic
437C2C2×#1569085611yescharacteristic
438C2C2×#1569085811yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×C22⋊Q8×C97C2
socleC23×C97C24
hypercenterC2×C22⋊Q8D97
hypocenterC97C2×#562
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20261481C2direct (N × H)
2. C2×#15690800C2semidirect
3. C2×#15690799C2semidirect
4. C2×#15690796C2semidirect
5. C2×#15690795C2semidirect
6. C2×#15690795C2semidirect
7. C2×#15690793C2semidirect
8. C2×#15690792C2semidirect
9. C2×#15690790C2semidirect
10. C2×#15690788C2semidirect
11. C2×#15690786C2semidirect
12. C2×#15690785C2semidirect
13. C2×#15690784C2semidirect
14. C2×#15690783C2semidirect
15. C2×#15690783C2semidirect
16. C2×C22⋊Q8×C97C2semidirect
27. #15690800C22semidirect
28. #15690799C22semidirect
29. #15690796C22semidirect
30. #15690795C22semidirect
31. #15690795C22semidirect
32. #15690793C22semidirect
33. #15690792C22semidirect
34. #15690790C22semidirect
35. #15690788C22semidirect
36. #15690786C22semidirect
37. #15690785C22semidirect
38. #15690784C22semidirect
39. #15690783C22semidirect
40. #15690783C22semidirect
61. C22⋊Q8×C97C22semidirect
212. C2×C97#562semidirect
220. C97C2×#562semidirect
232. C2×C22⋊Q8D97semidirect
259. C22⋊Q8C2×D97semidirect
435. C2#20261481direct (N × H)

Maximal subgroups

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

subgroupindex
#202614812
C2×#156908002
C2×#156907992
C2×#156907962
C2×#156907952
C2×#156907932
C2×#156907922
C2×#156907902
C2×#156907882
C2×#156907862
C2×#156907852
C2×#156907842
C2×#156907832
C2×C22⋊Q8×C972
C2×#56297