Finite Group Databasebeta

← C2×#20261551 #27156502

Subgroups — C2×#20261551 #27156502

Order 12416.

1408 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15690858C211yesmaximal, characteristic, complemented
2C2×#15690857C211yesmaximal, characteristic, complemented
3#20261551C2116yesmaximal, direct product factor
4C2×#15690843C211yesmaximal, characteristic, complemented
5C2×#15690842C212yesmaximal, complemented
6C2×#15690839C212yesmaximal, complemented
7C2×#15690831C212yesmaximal, complemented
8C2×#15690830C211yesmaximal, characteristic, complemented
9C22×#13095501C211yesmaximal, characteristic, complemented
10C2×Q16×D97C212yesmaximal, complemented
11C2×#15690818C211yesmaximal, characteristic, complemented
12C2×#137×C97C211yesmaximal, characteristic, complemented
13#15690858—24no—
14#15690858C2218yescomplemented
15#15690857—24no—
16#15690857C2218yescomplemented
17C2×#13095512—21no—
18C2×#13095512C2212yes—
19C2×#13095511—21no—
20C2×#13095511—22no—
21C2×#13095511C2211yescharacteristic
22C2×#13095511C2211yescharacteristic
23C2×#13095510—22no—
24C2×#13095510C2212yes—
25C22×Dic194—21no—
26C22×Dic194C2211yescharacteristic
27C2×Q8×D97—21no—
28C2×Q8×D97C2212yes—
29C2×Q8×D97C2212yes—
30#15690843C2218yescomplemented
31#15690842C22116yescomplemented
32#15690839C22116yescomplemented
33#15690831C22116yescomplemented
34#15690830C2218yescomplemented
35C2×Dic388C2212yes—
36C2×#13095501C2211yescharacteristic
37C2×#13095501C2212yes—
38C2×#13095501C2212yes—
39C2×#13095501C2218yescomplemented
40Q16×D97C22116yescomplemented
41C2×#13095500C2212yes—
42C2×#13095499C2211yescharacteristic
43C2×#13095498C2211yescharacteristic
44C2×#13095496C2211yescharacteristic
45#15690818C2218yescomplemented
46C2×#13095494C2212yes—
47C22×#11715895C2211yescharacteristic
48C2×C8×D97C2212yes—
49C2×#41×C97C2211yescharacteristic
50C22×Q8×C97C2211yescharacteristic
51#137×C97C2218yescomplemented
52C2×Q16×C97C2212yes—
53C2×SD16×C97C2212yes—
54C2×M16×C97C2211yescharacteristic
55#13095512—28no—
56#13095512—24no—
57#13095512C2318yes—
58#13095511—22no—
59#13095511—216no—
60#13095511—28no—
61#13095511D414yes—
62#13095511C2314yes—
63#13095511C2314yes—
64#13095510—216no—
65#13095510—24no—
66#13095510C2318yes—
67C2×Dic194—21no—
68C2×Dic194—22no—
69C2×Dic194—22no—
70C2×Dic194—22no—
71C2×Dic194—28no—
72C2×Dic194D411yescharacteristic
73C2×Dic194C2311yescharacteristic
74C2×Dic194C2312yes—
75C2×Dic194C2314yes—
76Q8×D97—24no—
77Q8×D97—24no—
78Q8×D97—28no—
79Q8×D97C2318yes—
80Q8×D97C2318yes—
81C2×D388—22no—
82C2×D388D411yescharacteristic
83C2×D388C2311yescharacteristic
84C2×C22⋊D97—21no—
85C2×C22⋊D97—22no—
86C2×C22⋊D97D412yes—
87C22×Dic97—21no—
88C22×Dic97—22no—
89C2×C4×D97—22no—
90C2×C4×D97—22no—
91C2×C4×D97—21no—
92C2×C4×D97—22no—
93C2×C4×D97C2312yes—
94Dic388C2318yes—
95#13095501C2314yes—
96#13095501C2318yes—
97#13095501C2318yes—
98#13095500C2318yes—
99#13095499C2314yes—
100#13095498C2314yes—
101#13095496C2314yes—
102#13095494C2318yes—
103C2×#11715895C2311yescharacteristic
104C2×#11715895C2311yescharacteristic
105C2×#11715895C2314yes—
106C8×D97C2318yes—
107#41×C97—22no—
108#41×C97C2314yes—
109C2×Q8×C97—21no—
110C2×Q8×C97—22no—
111C2×Q8×C97C2311yescharacteristic
112C2×Q8×C97C2312yes—
113C2×Q8×C97C2314yes—
114C2×D4×C97—21no—
115C2×D4×C97C2311yescharacteristic
116C22×C4×C97—21no—
117C22×C4×C97—21no—
118C22×C4×C97C2311yescharacteristic
119Q16×C97C2318yes—
120SD16×C97C2318yes—
121M16×C97C2314yes—
122C2×C8×C97C2312yes—
123C23×C97—21no—
124C23×C97C2×D411yescharacteristic
125C22×D97—21no—
126C22×D97C2×D412yes—
127Q8×C97—22no—
128Q8×C97—24no—
129Q8×C97—21no—
130Q8×C97C2414yes—
131Q8×C97C2412yes—
132D4×C97—24no—
133D4×C97—21no—
134D4×C97C2412yes—
135C2×C4×C97—21no—
136C2×C4×C97—22no—
137C2×C4×C97—24no—
138C2×C4×C97—24no—
139C2×C4×C97C2×D411yescharacteristic
140C2×C4×C97C2×D412yes—
141C2×C4×C97C2411yescharacteristic
142C2×C4×C97C2412yes—
143C22⋊D97—24no—
144C22⋊D97—28no—
145C22⋊D97C2×D418yes—
146C2×Dic97—21no—
147C2×Dic97—22no—
148C2×Dic97—24no—
149C2×Dic97—28no—
150C2×Dic97C2×D412yes—
151D388—28no—
152D388—21no—
153D388C2×D414yes—
154D388C2412yes—
155C4×D97—28no—
156C4×D97—28no—
157C4×D97—24no—
158C4×D97—28no—
159C4×D97C2×D414yes—
160C4×D97C2414yes—
161Dic194—21no—
162Dic194—22no—
163Dic194—24no—
164Dic194—28no—
165Dic194—28no—
166Dic194C2×D414yes—
167Dic194C2412yes—
168Dic194C2414yes—
169C8×C97C2414yes—
170#11715895C2412yes—
171#11715895C2412yes—
172C22×C97—22no—
173C22×C97—22no—
174C22×C97—42no—
175C22×C97C22×D412yes—
176C22×C97C22×D411yescharacteristic
177C2×D97—24no—
178C2×D97—22no—
179C2×D97—42no—
180C2×D97C22×D414yes—
181C4×C97—22no—
182C4×C97—24no—
183C4×C97C22×D411yescharacteristic
184C4×C97C22×D412yes—
185C4×C97C2511yescharacteristic
186Dic97—22no—
187Dic97—24no—
188Dic97C22×D414yes—
189C2×C97—22no—
190C2×C97—42no—
191C2×C97#57812yescomplemented
192C2×C97C23×D411yescharacteristic
193D97—24no—
194D97—42no—
195C2×#578—971nomaximal, Sylow
196C97C2×#57811yescharacteristic, complemented, Sylow
197C2×2−1+4—971no—
198C2×2−1+4—971no—
199#578—9716no—
200C2×#137—972no—
201C2×#137—972no—
202C2×#137—971no—
203C2×#137D9711yescharacteristic, complemented
204C2×#135—971no—
205C2×#135—972no—
206C22×Q16—971no—
207C22×Q16—972no—
208C2×#131—971no—
2092−1+4—978no—
2102−1+4—978no—
2112−1+4—1944no—
2122−1+4—1944no—
213C2×#41—971no—
214C2×#41—972no—
215C2×#41—972no—
216C2×#41—971no—
217C2×#41—1942no—
218C2×#41—1941no—
219C2×#41—1942no—
220C2×#41—1941no—
221C2×#41C2×D9711yescharacteristic
222C22×Q8—972no—
223C22×Q8—972no—
224C22×Q8—971no—
225C22×Q8—1941no—
226C22×Q8—1941no—
227C22×Q8C2×D9711yescharacteristic
228#137—9716no—
229#137—9716no—
230#137—978no—
231#137C2×D9718yescomplemented
232#135—978no—
233#135—9716no—
234C2×Q16—971no—
235C2×Q16—972no—
236C2×Q16—972no—
237C2×Q16—978no—
238C2×Q16—9716no—
239C2×Q16—972no—
240C2×Q16C2×D9712yes—
241C2×SD16—971no—
242C2×SD16—971no—
243C2×SD16—972no—
244C2×SD16C2×D9712yes—
245C2×D8—971no—
246#131—978no—
247C2×M16—972no—
248C2×M16C2×D9711yescharacteristic
249C22×C8—971no—
250C22×C8—972no—
251#41—22no—
252#41—974no—
253#41—974no—
254#41—978no—
255#41—978no—
256#41—974no—
257#41—19416no—
258#41—1948no—
259#41—1944no—
260#41—1944no—
261#41—19416no—
262#41—1948no—
263#41—1942no—
264#41C22×D9714yes—
265C2×Q8—21no—
266C2×Q8—22no—
267C2×Q8—971no—
268C2×Q8—978no—
269C2×Q8—978no—
270C2×Q8—971no—
271C2×Q8—972no—
272C2×Q8—974no—
273C2×Q8—1942no—
274C2×Q8—1942no—
275C2×Q8—1948no—
276C2×Q8—1944no—
277C2×Q8—1944no—
278C2×Q8—1948no—
279C2×Q8—1941no—
280C2×Q8—1942no—
281C2×Q8C22×D9711yescharacteristic
282C2×Q8C22×D9712yes—
283C2×Q8C22×D9714yes—
284C2×D4—21no—
285C2×D4—972no—
286C2×D4—971no—
287C2×D4—971no—
288C2×D4—1942no—
289C2×D4—1942no—
290C2×D4—1941no—
291C2×D4C22×D9711yescharacteristic
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—972no—
295C22×C4—1942no—
296C22×C4—1942no—
297C22×C4—1942no—
298C22×C4—1942no—
299C22×C4—1941no—
300C22×C4—1941no—
301C22×C4C22×D9711yescharacteristic
302Q16—974no—
303Q16—978no—
304Q16—978no—
305Q16—978no—
306Q16C22×D9718yes—
307SD16—974no—
308SD16—974no—
309SD16—978no—
310SD16C22×D9718yes—
311D8—974no—
312M16—978no—
313M16C22×D9714yes—
314C2×C8—971no—
315C2×C8—971no—
316C2×C8—974no—
317C2×C8—978no—
318C2×C8C22×D9712yes—
319C23—21no—
320C23—972no—
321C23—1941no—
322C23D4×D9711yescharacteristic
323Q8—24no—
324Q8—22no—
325Q8—21no—
326Q8—974no—
327Q8—972no—
328Q8—974no—
329Q8—1948no—
330Q8—1948no—
331Q8—1944no—
332Q8—1941no—
333Q8—1942no—
334Q8C23×D9712yes—
335Q8C23×D9714yes—
336D4—24no—
337D4—21no—
338D4—978no—
339D4—974no—
340D4—972no—
341D4—1948no—
342D4—1948no—
343D4—1944no—
344D4—1941no—
345D4C23×D9712yes—
346C2×C4—24no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—972no—
351C2×C4—974no—
352C2×C4—974no—
353C2×C4—1948no—
354C2×C4—1948no—
355C2×C4—1948no—
356C2×C4—1948no—
357C2×C4—1944no—
358C2×C4—1941no—
359C2×C4—1942no—
360C2×C4—1944no—
361C2×C4D4×D9712yes—
362C2×C4D4×D9711yescharacteristic
363C2×C4C23×D9712yes—
364C2×C4C23×D9711yescharacteristic
365C8—972no—
366C8—972no—
367C8C23×D9714yes—
368C22—22no—
369C22—22no—
370C22—42no—
371C22—974no—
372C22—3882no—
373C22—1942no—
374C22—1944no—
375C22C2×D4×D9712yes—
376C22C2×D4×D9711yescharacteristic
377C4—22no—
378C4—24no—
379C4—974no—
380C4—1942no—
381C4—1944no—
382C4C2×D4×D9712yes—
383C4C2×D4×D9711yescharacteristic
384C4C24×D9711yescharacteristic
385C2—22no—
386C2—42no—
387C2—3882no—
388C2—1944no—
389C2#2026155112yesdirect product factor
390C2C22×D4×D9711yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC4×C97C25
FrattiniC4C24×D97
FittingC2×#137×C97C2
socleC22×C97C22×D4
hypercenterC2×#137D97
hypocenterC97C2×#578
LCSG ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#15690858C2semidirect
2. C2×#15690857C2semidirect
3. #20261551C2direct (N × H)
4. C2×#15690843C2semidirect
5. C2×#15690842C2semidirect
6. C2×#15690839C2semidirect
7. C2×#15690831C2semidirect
8. C2×#15690830C2semidirect
9. C22×#13095501C2semidirect
10. C2×Q16×D97C2semidirect
11. C2×#15690818C2semidirect
12. C2×#137×C97C2semidirect
14. #15690858C22semidirect
16. #15690857C22semidirect
30. #15690843C22semidirect
31. #15690842C22semidirect
32. #15690839C22semidirect
33. #15690831C22semidirect
34. #15690830C22semidirect
39. C2×#13095501C22semidirect
40. Q16×D97C22semidirect
45. #15690818C22semidirect
51. #137×C97C22semidirect
191. C2×C97#578semidirect
196. C97C2×#578semidirect
203. C2×#137D97semidirect
231. #137C2×D97semidirect
389. C2#20261551direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#156908582
C2×#156908572
#202615512
C2×#156908432
C2×#156908422
C2×#156908392
C2×#156908312
C2×#156908302
C22×#130955012
C2×Q16×D972
C2×#156908182
C2×#137×C972
C2×#57897