Finite Group Databasebeta

← C2×#20261428 #27155252

Subgroups — C2×#20261428 #27155252

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#20261428C2116yesmaximal, direct product factor
2C2×#15690805C211yesmaximal, characteristic, complemented
3C2×#15690803C211yesmaximal, characteristic, complemented
4C2×#15690800C211yesmaximal, characteristic, complemented
5C2×#15690792C211yesmaximal, characteristic, complemented
6C2×#15690791C211yesmaximal, characteristic, complemented
7C2×#15690782C211yesmaximal, characteristic, complemented
8C2×#15690782C211yesmaximal, characteristic, complemented
9C2×#15690782C211yesmaximal, characteristic, complemented
10C2×D4×Dic97C211yesmaximal, characteristic, complemented
11C2×D4×Dic97C211yesmaximal, characteristic, complemented
12C2×C4×C22⋊D97C211yesmaximal, characteristic, complemented
13C2×#15690770C211yesmaximal, characteristic, complemented
14C2×#15690766C211yesmaximal, characteristic, complemented
15C22×#13095490C211yesmaximal, characteristic, complemented
16C2×#121×C97C211yesmaximal, characteristic, complemented
17C22×C22⋊D97C2211yescharacteristic
18C22×C22⋊D97C2211yescharacteristic
19C22×C22⋊D97C2211yescharacteristic
20C23×Dic97C2211yescharacteristic
21C23×Dic97C2211yescharacteristic
22C23×Dic97C2211yescharacteristic
23C23×Dic97C2211yescharacteristic
24C22×C4×D97C2211yescharacteristic
25#15690805C2218yescomplemented
26#15690803C2218yescomplemented
27#15690800C2218yescomplemented
28#15690792C2218yescomplemented
29#15690791C2218yescomplemented
30#15690782C2218yescomplemented
31#15690782C2218yescomplemented
32#15690782C2218yescomplemented
33D4×Dic97C2218yescomplemented
34D4×Dic97C2218yescomplemented
35C4×C22⋊D97C2218yescomplemented
36#15690770C2218yescomplemented
37#15690766C2218yescomplemented
38C2×C4⋊Dic97C2211yescharacteristic
39C2×C4⋊Dic97C2211yescharacteristic
40C2×#13095490C2211yescharacteristic
41C2×#13095490C2211yescharacteristic
42C2×#13095490C2211yescharacteristic
43C2×#13095490C2211yescharacteristic
44C2×#13095490C2211yescharacteristic
45C2×#13095490C2211yescharacteristic
46C2×#13095490C2211yescharacteristic
47C2×#13095490C2218yescomplemented
48C2×C22⋊Dic97C2211yescharacteristic
49C2×C22⋊Dic97C2211yescharacteristic
50C2×C22⋊Dic97C2211yescharacteristic
51C2×C22⋊Dic97C2211yescharacteristic
52C2×C22⋊Dic97C2211yescharacteristic
53C2×#13095488C2211yescharacteristic
54C2×#13095488C2211yescharacteristic
55C2×#13095488C2211yescharacteristic
56C2×C4×Dic97C2211yescharacteristic
57C2×C4×Dic97C2211yescharacteristic
58C2×C4×Dic97C2211yescharacteristic
59C22×D4×C97C2211yescharacteristic
60C22×D4×C97C2211yescharacteristic
61C22×D4×C97C2211yescharacteristic
62C23×C4×C97C2211yescharacteristic
63#121×C97C2218yescomplemented
64C2×C4⋊C4×C97C2211yescharacteristic
65C2×C22⋊C4×C97C2211yescharacteristic
66C2×C22⋊C4×C97C2211yescharacteristic
67C23×D97C2311yescharacteristic
68C2×C22⋊D97—24no—
69C2×C22⋊D97—24no—
70C2×C22⋊D97—42no—
71C2×C22⋊D97C2314yes—
72C2×C22⋊D97C2314yes—
73C2×C22⋊D97C2314yes—
74C22×Dic97—24no—
75C22×Dic97—24no—
76C22×Dic97—42no—
77C22×Dic97—42no—
78C22×Dic97C2311yescharacteristic
79C22×Dic97C2311yescharacteristic
80C22×Dic97C2311yescharacteristic
81C22×Dic97C2311yescharacteristic
82C22×Dic97C2311yescharacteristic
83C22×Dic97C2311yescharacteristic
84C22×Dic97C2314yes—
85C22×Dic97C2314yes—
86C22×Dic97C2314yes—
87C22×Dic97C2314yes—
88C22×Dic97C2311yescharacteristic
89C2×C4×D97—42no—
90C2×C4×D97C2314yes—
91C4⋊Dic97C2314yes—
92C4⋊Dic97C2314yes—
93#13095490C2314yes—
94#13095490C2314yes—
95#13095490C2314yes—
96#13095490C2314yes—
97#13095490C2314yes—
98#13095490C2314yes—
99#13095490C2314yes—
100C22⋊Dic97C2314yes—
101C22⋊Dic97C2314yes—
102C22⋊Dic97C2314yes—
103C22⋊Dic97C2314yes—
104C22⋊Dic97C2314yes—
105#13095488C2314yes—
106#13095488C2314yes—
107#13095488C2314yes—
108C4×Dic97C2314yes—
109C4×Dic97C2314yes—
110C4×Dic97C2314yes—
111C24×C97C2311yescharacteristic
112C24×C97C2311yescharacteristic
113C24×C97C2311yescharacteristic
114C2×D4×C97—24no—
115C2×D4×C97—24no—
116C2×D4×C97—42no—
117C2×D4×C97C2314yes—
118C2×D4×C97C2314yes—
119C2×D4×C97C2314yes—
120C22×C4×C97—42no—
121C22×C4×C97C2311yescharacteristic
122C22×C4×C97C2311yescharacteristic
123C22×C4×C97C2311yescharacteristic
124C22×C4×C97C2311yescharacteristic
125C22×C4×C97C2314yes—
126C4⋊C4×C97C2314yes—
127C22⋊C4×C97C2314yes—
128C22⋊C4×C97C2314yes—
129C23×C97—22no—
130C23×C97—22no—
131C23×C97—22no—
132C23×C97—22no—
133C23×C97—22no—
134C23×C97—22no—
135C23×C97—22no—
136C23×C97—22no—
137C23×C97#4114yes—
138C23×C97C2412yes—
139C23×C97C2412yes—
140C23×C97C2412yes—
141C23×C97C2411yescharacteristic
142C22×D97—22no—
143C22×D97—22no—
144C22×D97—22no—
145C22×D97C2412yes—
146D4×C97—28no—
147D4×C97—28no—
148D4×C97—44no—
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—
158C22⋊D97—28no—
159C22⋊D97—28no—
160C22⋊D97—44no—
161C2×Dic97—22no—
162C2×Dic97—22no—
163C2×Dic97—22no—
164C2×Dic97—22no—
165C2×Dic97—22no—
166C2×Dic97—22no—
167C2×Dic97—28no—
168C2×Dic97—28no—
169C2×Dic97—44no—
170C2×Dic97—44no—
171C2×Dic97#4114yes—
172C2×Dic97C2412yes—
173C2×Dic97C2412yes—
174C2×Dic97C2412yes—
175C2×Dic97C2412yes—
176C2×Dic97C2412yes—
177C2×Dic97C2412yes—
178C2×Dic97C2412yes—
179C4×D97—44no—
180C22×C97—28no—
181C22×C97—22no—
182C22×C97—22no—
183C22×C97—22no—
184C22×C97—22no—
185C22×C97—22no—
186C22×C97—22no—
187C22×C97—22no—
188C22×C97—22no—
189C22×C97—44no—
190C22×C97—44no—
191C22×C97C2×#4112yes—
192C22×C97C2×#4114yes—
193C22×C97C2×#4112yes—
194C22×C97H(5,𝔽2)12yes—
195C22×C97C2511yescharacteristic
196C2×D97—22no—
197C2×D97—22no—
198C2×D97—22no—
199C2×D97—44no—
200C4×C97—22no—
201C4×C97—22no—
202C4×C97—22no—
203C4×C97—22no—
204Dic97—22no—
205Dic97—22no—
206Dic97—22no—
207Dic97—22no—
208Dic97—22no—
209Dic97—22no—
210Dic97C2×#4114yes—
211C2×C97—24no—
212C2×C97—42no—
213C2×C97—42no—
214C2×C97#55314yescomplemented
215C2×C97C22×#4111yescharacteristic
216C2×C97C22×#4111yescharacteristic
217C2×C97C2×H(5,𝔽2)11yescharacteristic
218D97—42no—
219C2×#553—971nomaximal, Sylow
220C97C2×#55311yescharacteristic, complemented, Sylow
221#553—9716no—
222C2×#126—971no—
223C2×#126—971no—
224C2×#125—971no—
225C2×#123—971no—
226C2×#123—971no—
227C2×#121—971no—
228C2×#121—971no—
229C2×#121—971no—
230C2×#121D9711yescharacteristic, complemented
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×D4C2×D9711yescharacteristic
241C22×D4C2×D9711yescharacteristic
242C22×D4C2×D9711yescharacteristic
243C23×C4—971no—
244C23×C4—971no—
245C23×C4—971no—
246C23×C4—971no—
247C23×C4—971no—
248C23×C4C2×D9711yescharacteristic
249#126—978no—
250#126—978no—
251#125—978no—
252#123—978no—
253#123—978no—
254#121—978no—
255#121—978no—
256#121—978no—
257#121C2×D9718yescomplemented
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—978no—
268C2×C4⋊C4—971no—
269C2×C4⋊C4—971no—
270C2×C4⋊C4—971no—
271C2×C4⋊C4—971no—
272C2×C4⋊C4—971no—
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⋊C4C2×D9711yescharacteristic
283C2×C22⋊C4C2×D9711yescharacteristic
284C2×C42—971no—
285C2×C42—971no—
286C2×C42—971no—
287C24—971no—
288C24C22×D9711yescharacteristic
289C24C22×D9711yescharacteristic
290C24C22×D9711yescharacteristic
291C2×D4—24no—
292C2×D4—24no—
293C2×D4—42no—
294C2×D4—974no—
295C2×D4—974no—
296C2×D4—974no—
297C2×D4—3882no—
298C2×D4—1944no—
299C2×D4—1944no—
300C2×D4C22×D9714yes—
301C2×D4C22×D9714yes—
302C2×D4C22×D9714yes—
303C22×C4—42no—
304C22×C4—971no—
305C22×C4—974no—
306C22×C4—974no—
307C22×C4—974no—
308C22×C4—974no—
309C22×C4—971no—
310C22×C4—971no—
311C22×C4—971no—
312C22×C4—971no—
313C22×C4—971no—
314C22×C4—971no—
315C22×C4—974no—
316C22×C4—3882no—
317C22×C4—3882no—
318C22×C4—3882no—
319C22×C4—1944no—
320C22×C4—1944no—
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⋊C4—974no—
334C4⋊C4—974no—
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⋊C4C22×D9714yes—
345C22⋊C4C22×D9714yes—
346C42—974no—
347C42—974no—
348C42—974no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—972no—
358C23—1942no—
359C23—1942no—
360C23—1942no—
361C23#1309551014yes—
362C23C23×D9712yes—
363C23C23×D9712yes—
364C23C23×D9712yes—
365C23C23×D9711yescharacteristic
366D4—28no—
367D4—28no—
368D4—44no—
369D4—3884no—
370D4—1948no—
371D4—1948no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—44no—
377C2×C4—974no—
378C2×C4—972no—
379C2×C4—972no—
380C2×C4—972no—
381C2×C4—972no—
382C2×C4—972no—
383C2×C4—972no—
384C2×C4—972no—
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×C4—1942no—
395C2×C4—1942no—
396C2×C4C23×D9712yes—
397C2×C4C23×D9712yes—
398C2×C4C23×D9712yes—
399C2×C4C23×D9712yes—
400C22—28no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—44no—
410C22—44no—
411C22—3884no—
412C22—1942no—
413C22—1942no—
414C22—1942no—
415C22#41×D9712yes—
416C22C2×#1309551014yes—
417C22C2×#1309551012yes—
418C22#1569085512yes—
419C22C24×D9711yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—974no—
425C4—1942no—
426C4—1942no—
427C4—1942no—
428C4—1942no—
429C4—1942no—
430C4—1942no—
431C2—24no—
432C2—42no—
433C2—42no—
434C2—3882no—
435C2#2026142814yesdirect product factor
436C2C2×#41×D9711yescharacteristic
437C2C22×#1309551011yescharacteristic
438C2C2×#1569085511yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#121×C97C2
socleC23×C97C24
hypercenterC2×#121D97
hypocenterC97C2×#553
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20261428C2direct (N × H)
2. C2×#15690805C2semidirect
3. C2×#15690803C2semidirect
4. C2×#15690800C2semidirect
5. C2×#15690792C2semidirect
6. C2×#15690791C2semidirect
7. C2×#15690782C2semidirect
8. C2×#15690782C2semidirect
9. C2×#15690782C2semidirect
10. C2×D4×Dic97C2semidirect
11. C2×D4×Dic97C2semidirect
12. C2×C4×C22⋊D97C2semidirect
13. C2×#15690770C2semidirect
14. C2×#15690766C2semidirect
15. C22×#13095490C2semidirect
16. C2×#121×C97C2semidirect
25. #15690805C22semidirect
26. #15690803C22semidirect
27. #15690800C22semidirect
28. #15690792C22semidirect
29. #15690791C22semidirect
30. #15690782C22semidirect
31. #15690782C22semidirect
32. #15690782C22semidirect
33. D4×Dic97C22semidirect
34. D4×Dic97C22semidirect
35. C4×C22⋊D97C22semidirect
36. #15690770C22semidirect
37. #15690766C22semidirect
47. C2×#13095490C22semidirect
63. #121×C97C22semidirect
214. C2×C97#553semidirect
220. C97C2×#553semidirect
230. C2×#121D97semidirect
257. #121C2×D97semidirect
435. C2#20261428direct (N × H)

Maximal subgroups

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

subgroupindex
#202614282
C2×#156908052
C2×#156908032
C2×#156908002
C2×#156907922
C2×#156907912
C2×#156907822
C2×D4×Dic972
C2×C4×C22⋊D972
C2×#156907702
C2×#156907662
C22×#130954902
C2×#121×C972
C2×#55397