Finite Group Databasebeta

← C2×#20261439 #27155264

Subgroups — C2×#20261439 #27155264

Order 12416.

1200 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261439C2116yesmaximal, direct product factor
2C2×#15690808C211yesmaximal, characteristic, complemented
3C2×#15690800C211yesmaximal, characteristic, complemented
4C2×#15690800C211yesmaximal, characteristic, complemented
5C2×#15690788C211yesmaximal, characteristic, complemented
6C2×#15690788C211yesmaximal, characteristic, complemented
7C2×#15690787C211yesmaximal, characteristic, complemented
8C2×#15690787C211yesmaximal, characteristic, complemented
9C2×#15690786C211yesmaximal, characteristic, complemented
10C2×#15690775C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊D97C211yesmaximal, characteristic
12C2×#15690770C211yesmaximal, characteristic
13C2×#15690770C211yesmaximal, characteristic
14C22×#13095490C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C97C211yesmaximal, characteristic, complemented
17C22×Dic194C2211yescharacteristic
18C22×Dic194C2211yescharacteristic
19C22×C22⋊D97C2211yescharacteristic
20C23×Dic97C2211yescharacteristic
21C23×Dic97C2211yescharacteristic
22C22×C4×D97C2211yescharacteristic
23C22×C4×D97C2211yescharacteristic
24C22×C4×D97C2211yescharacteristic
25#15690808C2218yescomplemented
26#15690800C2218yescomplemented
27#15690800C2218yescomplemented
28#15690788C2218yescomplemented
29#15690788C2218yescomplemented
30#15690787C2218yescomplemented
31#15690787C2218yescomplemented
32#15690786C2218yescomplemented
33#15690775C2218yescomplemented
34C4×C22⋊D97C2218yes—
35#15690770C2218yes—
36#15690770C2218yes—
37C2×C4⋊Dic97C2211yescharacteristic
38C2×C4⋊Dic97C2211yescharacteristic
39C2×#13095490C2211yescharacteristic
40C2×#13095490C2211yescharacteristic
41C2×#13095490C2211yescharacteristic
42C2×#13095490C2211yescharacteristic
43C2×#13095490C2211yescharacteristic
44C2×#13095490C2211yescharacteristic
45C2×#13095490C2211yescharacteristic
46C2×#13095490C2211yescharacteristic
47C2×#13095490C2211yescharacteristic
48C2×#13095490C2211yescharacteristic
49C2×#13095490C2211yescharacteristic
50C2×#13095490C2218yescomplemented
51C4⋊C4×D97C2218yescomplemented
52C2×C22⋊Dic97C2211yescharacteristic
53C2×#13095488C2211yescharacteristic
54C2×#13095488C2211yescharacteristic
55C2×#13095488C2211yescharacteristic
56C2×C4×Dic97C2211yescharacteristic
57C2×C4×Dic97C2211yescharacteristic
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
68C2×Dic194—24no—
69C2×Dic194—42no—
70C2×Dic194C2314yes—
71C2×Dic194C2314yes—
72C2×C22⋊D97Q818yescomplemented
73C2×C22⋊D97C2314yes—
74C22×Dic97—24no—
75C22×Dic97—42no—
76C22×Dic97C2311yescharacteristic
77C22×Dic97C2311yescharacteristic
78C22×Dic97C2311yescharacteristic
79C22×Dic97C2311yescharacteristic
80C22×Dic97C2311yescharacteristic
81C22×Dic97C2314yes—
82C22×Dic97C2314yes—
83C22×Dic97C2311yescharacteristic
84C22×Dic97C2311yescharacteristic
85C2×C4×D97—24no—
86C2×C4×D97—42no—
87C2×C4×D97—42no—
88C2×C4×D97C2314yes—
89C2×C4×D97C2314yes—
90C2×C4×D97C2314yes—
91C4⋊Dic97C2314yes—
92C4⋊Dic97C2314yes—
93#13095490C2314yes—
94#13095490C2314yes—
95#13095490C2314yes—
96#13095490C2314yes—
97#13095490C2314yes—
98#13095490C2314yes—
99#13095490C2314yes—
100#13095490C2314yes—
101#13095490C2314yes—
102#13095490C2314yes—
103#13095490C2314yes—
104C22⋊Dic97C2314yes—
105#13095488C2314yes—
106#13095488C2314yes—
107#13095488C2314yes—
108C4×Dic97C2314yes—
109C4×Dic97C2314yes—
110C4×Dic97C2314yes—
111C24×C97C2311yescharacteristic
112C2×Q8×C97—42no—
113C2×Q8×C97C2314yes—
114C22×C4×C97—42no—
115C22×C4×C97C2311yescharacteristic
116C22×C4×C97C2311yescharacteristic
117C22×C4×C97C2311yescharacteristic
118C22×C4×C97C2311yescharacteristic
119C22×C4×C97C2311yescharacteristic
120C22×C4×C97C2311yescharacteristic
121C22×C4×C97C2314yes—
122C4⋊C4×C97C2314yes—
123C4⋊C4×C97C2314yes—
124C4⋊C4×C97C2314yes—
125C22⋊C4×C97C2314yes—
126C22⋊C4×C97C2314yes—
127C23×C97—22no—
128C23×C97—22no—
129C23×C97C2×Q814yes—
130C23×C97C2412yes—
131C23×C97C2411yescharacteristic
132C22×D97—22no—
133C22×D97—22no—
134C22×D97C2×Q814yes—
135C22×D97C2412yes—
136Q8×C97—44no—
137C2×C4×C97—22no—
138C2×C4×C97—22no—
139C2×C4×C97—22no—
140C2×C4×C97—22no—
141C2×C4×C97—22no—
142C2×C4×C97—22no—
143C2×C4×C97—44no—
144C2×C4×C97C2412yes—
145C2×C4×C97C2412yes—
146C2×C4×C97C2412yes—
147C2×C4×C97C2412yes—
148C2×C4×C97C2412yes—
149C2×C4×C97C2412yes—
150C22⋊D97C2×Q8116yescomplemented
151C2×Dic97—22no—
152C2×Dic97—22no—
153C2×Dic97—22no—
154C2×Dic97—22no—
155C2×Dic97—22no—
156C2×Dic97—28no—
157C2×Dic97—44no—
158C2×Dic97C2×Q814yes—
159C2×Dic97#4114yes—
160C2×Dic97C2412yes—
161C2×Dic97C2412yes—
162C2×Dic97C2412yes—
163C2×Dic97C2412yes—
164C2×Dic97C2412yes—
165C2×Dic97C2412yes—
166C2×Dic97C2412yes—
167C4×D97—28no—
168C4×D97—44no—
169C4×D97—44no—
170Dic194—28no—
171Dic194—44no—
172C22×C97—28no—
173C22×C97—22no—
174C22×C97—22no—
175C22×C97C22×Q814yes—
176C22×C97C22×Q812yes—
177C22×C97C2×#4112yes—
178C22×C97H(5,𝔽2)12yes—
179C22×C97C2511yescharacteristic
180C2×D97—28no—
181C2×D97—22no—
182C2×D97—22no—
183C2×D97C22×Q814yes—
184C4×C97—22no—
185C4×C97—22no—
186C4×C97—22no—
187C4×C97—22no—
188C4×C97—22no—
189C4×C97—22no—
190Dic97—22no—
191Dic97—22no—
192Dic97—22no—
193Dic97—22no—
194Dic97—22no—
195Dic97C22×Q814yes—
196Dic97C2×#4114yes—
197C2×C97—24no—
198C2×C97#55414yescomplemented
199C2×C97C23×Q811yescharacteristic
200C2×C97C22×#4111yescharacteristic
201C2×C97C2×H(5,𝔽2)11yescharacteristic
202D97—24no—
203C2×#554—971nomaximal, Sylow
204C97C2×#55411yescharacteristic, complemented, Sylow
205#554—9716no—
206C2×C4⋊Q8—971no—
207C2×#125—971no—
208C2×#125—971no—
209C2×C22⋊Q8—971no—
210C2×C22⋊Q8—971no—
211C2×C22⋊Q8—971no—
212C2×C22⋊Q8—971no—
213C2×C22⋊Q8—971no—
214C2×C22⋊Q8D9711yescharacteristic, complemented
215C2×C4×Q8—971no—
216C2×C4×D4—971no—
217C2×C4×D4—971no—
218C2×C4×D4—971no—
219C22×C4⋊C4—971no—
220C22×C4⋊C4—971no—
221C22×Q8—971no—
222C22×Q8—971no—
223C22×Q8C2×D9711yescharacteristic
224C22×D4—971no—
225C23×C4—971no—
226C23×C4—971no—
227C23×C4—971no—
228C23×C4—971no—
229C23×C4—971no—
230C23×C4C2×D9711yescharacteristic
231C4⋊Q8—978no—
232#125—978no—
233#125—978no—
234C22⋊Q8—978no—
235C22⋊Q8—978no—
236C22⋊Q8—978no—
237C22⋊Q8—978no—
238C22⋊Q8—978no—
239C22⋊Q8C2×D9718yescomplemented
240C4×Q8—978no—
241C4×D4—978no—
242C4×D4—978no—
243C4×D4—978no—
244C2×C4⋊C4—971no—
245C2×C4⋊C4—971no—
246C2×C4⋊C4—971no—
247C2×C4⋊C4—971no—
248C2×C4⋊C4—971no—
249C2×C4⋊C4—971no—
250C2×C4⋊C4—978no—
251C2×C4⋊C4—971no—
252C2×C4⋊C4—971no—
253C2×C4⋊C4—971no—
254C2×C4⋊C4—971no—
255C2×C4⋊C4—971no—
256C2×C4⋊C4—971no—
257C2×C4⋊C4—971no—
258C2×C4⋊C4—978no—
259C2×C4⋊C4C2×D9711yescharacteristic
260C2×C4⋊C4C2×D9711yescharacteristic
261C2×C4⋊C4C2×D9711yescharacteristic
262C2×C22⋊C4—971no—
263C2×C22⋊C4—971no—
264C2×C22⋊C4—971no—
265C2×C22⋊C4—971no—
266C2×C22⋊C4C2×D9711yescharacteristic
267C2×C22⋊C4C2×D9711yescharacteristic
268C2×C42—971no—
269C2×C42—971no—
270C2×C42—971no—
271C24—971no—
272C24C22×D9711yescharacteristic
273C2×Q8—42no—
274C2×Q8—974no—
275C2×Q8—974no—
276C2×Q8—3882no—
277C2×Q8—1944no—
278C2×Q8C22×D9714yes—
279C2×D4—978no—
280C2×D4—974no—
281C22×C4—42no—
282C22×C4—971no—
283C22×C4—971no—
284C22×C4—974no—
285C22×C4—974no—
286C22×C4—971no—
287C22×C4—971no—
288C22×C4—971no—
289C22×C4—971no—
290C22×C4—971no—
291C22×C4—974no—
292C22×C4—974no—
293C22×C4—974no—
294C22×C4—3882no—
295C22×C4—3882no—
296C22×C4—3882no—
297C22×C4—1944no—
298C22×C4—1944no—
299C22×C4C22×D9711yescharacteristic
300C22×C4C22×D9711yescharacteristic
301C22×C4C22×D9711yescharacteristic
302C22×C4C22×D9711yescharacteristic
303C22×C4C22×D9711yescharacteristic
304C22×C4C22×D9711yescharacteristic
305C22×C4C22×D9714yes—
306C4⋊C4—974no—
307C4⋊C4—974no—
308C4⋊C4—974no—
309C4⋊C4—974no—
310C4⋊C4—974no—
311C4⋊C4—974no—
312C4⋊C4—974no—
313C4⋊C4—974no—
314C4⋊C4—974no—
315C4⋊C4—974no—
316C4⋊C4—974no—
317C4⋊C4—974no—
318C4⋊C4—974no—
319C4⋊C4C22×D9714yes—
320C4⋊C4C22×D9714yes—
321C4⋊C4C22×D9714yes—
322C22⋊C4—974no—
323C22⋊C4—974no—
324C22⋊C4—974no—
325C22⋊C4—974no—
326C22⋊C4C22×D9714yes—
327C22⋊C4C22×D9714yes—
328C42—974no—
329C42—974no—
330C42—974no—
331C23—22no—
332C23—22no—
333C23—974no—
334C23—972no—
335C23—1942no—
336C23—1942no—
337C23Q8×D9714yes—
338C23C23×D9712yes—
339C23C23×D9711yescharacteristic
340Q8—44no—
341Q8—3884no—
342Q8—1948no—
343D4—9716no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—974no—
352C2×C4—974no—
353C2×C4—972no—
354C2×C4—972no—
355C2×C4—972no—
356C2×C4—972no—
357C2×C4—972no—
358C2×C4—972no—
359C2×C4—972no—
360C2×C4—3884no—
361C2×C4—3884no—
362C2×C4—3884no—
363C2×C4—1948no—
364C2×C4—1948no—
365C2×C4—1942no—
366C2×C4—1942no—
367C2×C4—1942no—
368C2×C4—1942no—
369C2×C4—1942no—
370C2×C4C23×D9712yes—
371C2×C4C23×D9712yes—
372C2×C4C23×D9712yes—
373C2×C4C23×D9712yes—
374C2×C4C23×D9712yes—
375C2×C4C23×D9712yes—
376C22—28no—
377C22—22no—
378C22—22no—
379C22—974no—
380C22—1948no—
381C22—1942no—
382C22—1942no—
383C22C2×Q8×D9714yes—
384C22C2×Q8×D9712yes—
385C22#41×D9712yes—
386C22#1569085512yes—
387C22C24×D9711yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—974no—
395C4—974no—
396C4—1942no—
397C4—1942no—
398C4—1942no—
399C4—1942no—
400C4—1942no—
401C2—24no—
402C2—1944no—
403C2#2026143914yesdirect product factor
404C2C22×Q8×D9711yescharacteristic
405C2C2×#41×D9711yescharacteristic
406C2C2×#1569085511yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×C22⋊Q8×C97C2
socleC23×C97C24
prime radicalC22×C22⋊D97C22
hypercenterC2×C22⋊Q8D97
hypocenterC97C2×#554
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #20261439C2direct (N × H)
2. C2×#15690808C2semidirect
3. C2×#15690800C2semidirect
4. C2×#15690800C2semidirect
5. C2×#15690788C2semidirect
6. C2×#15690788C2semidirect
7. C2×#15690787C2semidirect
8. C2×#15690787C2semidirect
9. C2×#15690786C2semidirect
10. C2×#15690775C2semidirect
14. C22×#13095490C2semidirect
15. C2×C4⋊C4×D97C2semidirect
16. C2×C22⋊Q8×C97C2semidirect
25. #15690808C22semidirect
26. #15690800C22semidirect
27. #15690800C22semidirect
28. #15690788C22semidirect
29. #15690788C22semidirect
30. #15690787C22semidirect
31. #15690787C22semidirect
32. #15690786C22semidirect
33. #15690775C22semidirect
50. C2×#13095490C22semidirect
51. C4⋊C4×D97C22semidirect
61. C22⋊Q8×C97C22semidirect
72. C2×C22⋊D97Q8semidirect
150. C22⋊D97C2×Q8semidirect
198. C2×C97#554semidirect
204. C97C2×#554semidirect
214. C2×C22⋊Q8D97semidirect
239. C22⋊Q8C2×D97semidirect
403. C2#20261439direct (N × H)

Maximal subgroups

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

subgroupindex
#202614392
C2×#156908082
C2×#156908002
C2×#156907882
C2×#156907872
C2×#156907862
C2×#156907752
C2×C4×C22⋊D972
C2×#156907702
C22×#130954902
C2×C4⋊C4×D972
C2×C22⋊Q8×C972
C2×#55497