Finite Group Databasebeta

← C2×#20261436 #27155260

Subgroups — C2×#20261436 #27155260

Order 12416.

1252 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#20261436C2116yesmaximal, direct product factor
2C2×#15690805C211yesmaximal, characteristic, complemented
3C2×#15690800C211yesmaximal, characteristic, complemented
4C2×#15690795C211yesmaximal, characteristic, complemented
5C2×#15690794C211yesmaximal, characteristic, complemented
6C2×#15690785C211yesmaximal, characteristic, complemented
7C2×#15690783C211yesmaximal, characteristic, complemented
8C2×#15690783C211yesmaximal, characteristic, complemented
9C2×#15690783C211yesmaximal, characteristic, complemented
10C2×C4×D388C211yesmaximal, characteristic, complemented
11C2×#15690772C211yesmaximal, characteristic, complemented
12C2×#15690770C211yesmaximal, characteristic, complemented
13C2×#15690770C211yesmaximal, characteristic, complemented
14C2×#15690767C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
16C2×#126×C97C211yesmaximal, characteristic, complemented
17C22×D388C2211yescharacteristic
18C22×D388C2211yescharacteristic
19C22×D388C2211yescharacteristic
20C22×C22⋊D97C2211yescharacteristic
21C22×C22⋊D97C2211yescharacteristic
22C22×C22⋊D97C2211yescharacteristic
23C23×Dic97C2211yescharacteristic
24C22×C4×D97C2211yescharacteristic
25C22×C4×D97C2211yescharacteristic
26C22×C4×D97C2211yescharacteristic
27C22×C4×D97C2211yescharacteristic
28C22×C4×D97C2211yescharacteristic
29#15690805C2218yescomplemented
30#15690800C2218yescomplemented
31#15690795C2218yescomplemented
32#15690794C2218yescomplemented
33#15690785C2218yescomplemented
34#15690783C2218yescomplemented
35#15690783C2218yescomplemented
36#15690783C2218yescomplemented
37C4×D388C2218yescomplemented
38#15690772C2218yescomplemented
39#15690770C2218yescomplemented
40#15690770C2218yescomplemented
41#15690767C2218yescomplemented
42C2×C4⋊Dic97C2211yescharacteristic
43C2×C4⋊Dic97C2211yescharacteristic
44C2×#13095490C2211yescharacteristic
45C2×#13095490C2211yescharacteristic
46C2×#13095490C2211yescharacteristic
47C2×#13095490C2211yescharacteristic
48C2×#13095490C2211yescharacteristic
49C4⋊C4×D97C2218yescomplemented
50C2×#13095488C2211yescharacteristic
51C2×#13095488C2211yescharacteristic
52C2×#13095488C2211yescharacteristic
53C2×#13095488C2211yescharacteristic
54C2×#13095488C2211yescharacteristic
55C2×#13095488C2211yescharacteristic
56C2×#13095488C2211yescharacteristic
57C2×C4×Dic97C2211yescharacteristic
58C2×C4×Dic97C2211yescharacteristic
59#126×C97C2218yescomplemented
60C2×C4⋊C4×C97C2211yescharacteristic
61C2×C4⋊C4×C97C2211yescharacteristic
62C2×C4⋊C4×C97C2211yescharacteristic
63C2×C22⋊C4×C97C2211yescharacteristic
64C2×C22⋊C4×C97C2211yescharacteristic
65C2×C22⋊C4×C97C2211yescharacteristic
66C2×C42×C97C2211yescharacteristic
67C23×D97C2311yescharacteristic
68C23×D97C2311yescharacteristic
69C23×D97C2311yescharacteristic
70C2×D388—24no—
71C2×D388—24no—
72C2×D388—42no—
73C2×D388C2314yes—
74C2×D388C2314yes—
75C2×D388C2314yes—
76C2×C22⋊D97—24no—
77C2×C22⋊D97—24no—
78C2×C22⋊D97—42no—
79C2×C22⋊D97C2314yes—
80C2×C22⋊D97C2314yes—
81C2×C22⋊D97C2314yes—
82C22×Dic97—42no—
83C22×Dic97C2311yescharacteristic
84C22×Dic97C2311yescharacteristic
85C22×Dic97C2311yescharacteristic
86C22×Dic97C2311yescharacteristic
87C22×Dic97C2314yes—
88C22×Dic97C2311yescharacteristic
89C2×C4×D97—24no—
90C2×C4×D97—24no—
91C2×C4×D97—42no—
92C2×C4×D97—42no—
93C2×C4×D97—42no—
94C2×C4×D97C2314yes—
95C2×C4×D97C2314yes—
96C2×C4×D97C2314yes—
97C2×C4×D97C2314yes—
98C2×C4×D97C2314yes—
99C4⋊Dic97C2314yes—
100C4⋊Dic97C2314yes—
101#13095490C2314yes—
102#13095490C2314yes—
103#13095490C2314yes—
104#13095490C2314yes—
105#13095490C2314yes—
106#13095488C2314yes—
107#13095488C2314yes—
108#13095488C2314yes—
109#13095488C2314yes—
110#13095488C2314yes—
111#13095488C2314yes—
112#13095488C2314yes—
113C4×Dic97C2314yes—
114C4×Dic97C2314yes—
115C24×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117C22×C4×C97C2311yescharacteristic
118C22×C4×C97C2311yescharacteristic
119C22×C4×C97C2311yescharacteristic
120C22×C4×C97C2311yescharacteristic
121C22×C4×C97C2311yescharacteristic
122C4⋊C4×C97C2314yes—
123C4⋊C4×C97C2314yes—
124C4⋊C4×C97C2314yes—
125C22⋊C4×C97C2314yes—
126C22⋊C4×C97C2314yes—
127C22⋊C4×C97C2314yes—
128C42×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#4114yes—
143C22×D97C2412yes—
144C22×D97C2412yes—
145C22×D97C2412yes—
146C2×C4×C97—22no—
147C2×C4×C97—22no—
148C2×C4×C97—22no—
149C2×C4×C97—22no—
150C2×C4×C97—22no—
151C2×C4×C97—22no—
152C2×C4×C97C2412yes—
153C2×C4×C97C2412yes—
154C2×C4×C97C2412yes—
155C2×C4×C97C2412yes—
156C2×C4×C97C2412yes—
157C2×C4×C97C2412yes—
158C22⋊D97—28no—
159C22⋊D97—28no—
160C22⋊D97—44no—
161C2×Dic97—22no—
162C2×Dic97—22no—
163C2×Dic97—22no—
164C2×Dic97—22no—
165C2×Dic97—44no—
166C2×Dic97#4114yes—
167C2×Dic97C2412yes—
168C2×Dic97C2412yes—
169C2×Dic97C2412yes—
170C2×Dic97C2412yes—
171C2×Dic97C2412yes—
172D388—28no—
173D388—28no—
174D388—44no—
175C4×D97—28no—
176C4×D97—28no—
177C4×D97—44no—
178C4×D97—44no—
179C4×D97—44no—
180C22×C97—22no—
181C22×C97—22no—
182C22×C97—22no—
183C22×C97—44no—
184C22×C97C2×#4112yes—
185C22×C97C2×#4112yes—
186C22×C97H(5,𝔽2)12yes—
187C22×C97C2511yescharacteristic
188C2×D97—28no—
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×D97C2×#4114yes—
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—
210Dic97C2×#4114yes—
211C2×C97—42no—
212C2×C97#55314yescomplemented
213C2×C97C22×#4111yescharacteristic
214C2×C97C22×#4111yescharacteristic
215C2×C97C2×H(5,𝔽2)11yescharacteristic
216D97—24no—
217D97—42no—
218D97—42no—
219C2×#553—971nomaximal, Sylow
220C97C2×#55311yescharacteristic, complemented, Sylow
221#553—9716no—
222C2×#126—971no—
223C2×#126D9711yescharacteristic, complemented
224C2×#125—971no—
225C2×#123—971no—
226C2×#123—971no—
227C2×#121—971no—
228C2×#121—971no—
229C2×#121—971no—
230C2×#121—971no—
231C2×C4×D4—971no—
232C2×C4×D4—971no—
233C2×C4×D4—971no—
234C2×C4×D4—971no—
235C2×#117—971no—
236C22×C4⋊C4—971no—
237C22×D4—971no—
238C22×D4—971no—
239C22×D4—971no—
240C22×D4—971no—
241C22×D4—971no—
242C22×D4—971no—
243C23×C4—971no—
244C23×C4—971no—
245C23×C4—971no—
246C23×C4—971no—
247C23×C4—971no—
248C23×C4—971no—
249#126—978no—
250#126C2×D9718yescomplemented
251#125—978no—
252#123—978no—
253#123—978no—
254#121—978no—
255#121—978no—
256#121—978no—
257#121—978no—
258C4×D4—978no—
259C4×D4—978no—
260C4×D4—978no—
261C4×D4—978no—
262#117—978no—
263C2×C4⋊C4—971no—
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—978no—
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⋊C4C2×D9711yescharacteristic
282C2×C22⋊C4C2×D9711yescharacteristic
283C2×C22⋊C4C2×D9711yescharacteristic
284C2×C42—971no—
285C2×C42—971no—
286C2×C42C2×D9711yescharacteristic
287C24—971no—
288C24—971no—
289C24—971no—
290C24C22×D9711yescharacteristic
291C2×D4—974no—
292C2×D4—974no—
293C2×D4—974no—
294C2×D4—974no—
295C2×D4—974no—
296C2×D4—974no—
297C2×D4—3882no—
298C2×D4—3882no—
299C2×D4—1944no—
300C2×D4—1944no—
301C2×D4—1944no—
302C2×D4—1944no—
303C22×C4—971no—
304C22×C4—974no—
305C22×C4—971no—
306C22×C4—971no—
307C22×C4—971no—
308C22×C4—971no—
309C22×C4—974no—
310C22×C4—974no—
311C22×C4—974no—
312C22×C4—974no—
313C22×C4—974no—
314C22×C4—3882no—
315C22×C4—3882no—
316C22×C4—3882no—
317C22×C4—3882no—
318C22×C4—1944no—
319C22×C4—1944no—
320C22×C4C22×D9711yescharacteristic
321C22×C4C22×D9711yescharacteristic
322C22×C4C22×D9711yescharacteristic
323C22×C4C22×D9711yescharacteristic
324C22×C4C22×D9711yescharacteristic
325C22×C4C22×D9711yescharacteristic
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⋊C4C22×D9714yes—
344C22⋊C4C22×D9714yes—
345C22⋊C4C22×D9714yes—
346C42—974no—
347C42—974no—
348C42C22×D9714yes—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—974no—
353C23—972no—
354C23—972no—
355C23—972no—
356C23—1942no—
357C23—1942no—
358C23—1942no—
359C23—1942no—
360C23—1942no—
361C23—1942no—
362C23—1942no—
363C23—1942no—
364C23C23×D9712yes—
365C23C23×D9711yescharacteristic
366D4—3884no—
367D4—3884no—
368D4—1948no—
369D4—1948no—
370D4—1948no—
371D4—1948no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—974no—
379C2×C4—972no—
380C2×C4—972no—
381C2×C4—972no—
382C2×C4—972no—
383C2×C4—972no—
384C2×C4—3884no—
385C2×C4—3884no—
386C2×C4—3884no—
387C2×C4—3884no—
388C2×C4—1948no—
389C2×C4—1948no—
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—974no—
405C22—3884no—
406C22—3884no—
407C22—1942no—
408C22—1942no—
409C22—1942no—
410C22—1942no—
411C22—1942no—
412C22—1942no—
413C22—1948no—
414C22—1942no—
415C22—1942no—
416C22#41×D9712yes—
417C22#41×D9712yes—
418C22#1569085612yes—
419C22C24×D9711yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—974no—
427C4—1942no—
428C4—1942no—
429C4—1942no—
430C4—1942no—
431C2—42no—
432C2—3882no—
433C2—3882no—
434C2—1944no—
435C2#2026143614yesdirect product factor
436C2C2×#41×D9711yescharacteristic
437C2C2×#41×D9711yescharacteristic
438C2C2×#1569085611yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#126×C97C2
socleC23×C97C24
hypercenterC2×#126D97
hypocenterC97C2×#553
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. #20261436C2direct (N × H)
2. C2×#15690805C2semidirect
3. C2×#15690800C2semidirect
4. C2×#15690795C2semidirect
5. C2×#15690794C2semidirect
6. C2×#15690785C2semidirect
7. C2×#15690783C2semidirect
8. C2×#15690783C2semidirect
9. C2×#15690783C2semidirect
10. C2×C4×D388C2semidirect
11. C2×#15690772C2semidirect
12. C2×#15690770C2semidirect
13. C2×#15690770C2semidirect
14. C2×#15690767C2semidirect
15. C2×C4⋊C4×D97C2semidirect
16. C2×#126×C97C2semidirect
29. #15690805C22semidirect
30. #15690800C22semidirect
31. #15690795C22semidirect
32. #15690794C22semidirect
33. #15690785C22semidirect
34. #15690783C22semidirect
35. #15690783C22semidirect
36. #15690783C22semidirect
37. C4×D388C22semidirect
38. #15690772C22semidirect
39. #15690770C22semidirect
40. #15690770C22semidirect
41. #15690767C22semidirect
49. C4⋊C4×D97C22semidirect
59. #126×C97C22semidirect
212. C2×C97#553semidirect
220. C97C2×#553semidirect
223. C2×#126D97semidirect
250. #126C2×D97semidirect
435. C2#20261436direct (N × H)

Maximal subgroups

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

subgroupindex
#202614362
C2×#156908052
C2×#156908002
C2×#156907952
C2×#156907942
C2×#156907852
C2×#156907832
C2×C4×D3882
C2×#156907722
C2×#156907702
C2×#156907672
C2×C4⋊C4×D972
C2×#126×C972
C2×#55397