Finite Group Databasebeta

← C2×#27161681 #38756069

Subgroups — C2×#27161681 #38756069

Order 25216.

1260 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161681C2116yesmaximal, direct product factor
2C2×#20263322C211yesmaximal, characteristic, complemented
3C2×#20263316C211yesmaximal, characteristic, complemented
4C2×#20263311C211yesmaximal, characteristic, complemented
5C2×#20263310C211yesmaximal, characteristic, complemented
6C2×#20263308C211yesmaximal, characteristic, complemented
7C2×#20263301C211yesmaximal, characteristic, complemented
8C2×#20263300C211yesmaximal, characteristic, complemented
9C2×#20263299C211yesmaximal, characteristic, complemented
10C2×#20263299C211yesmaximal, characteristic, complemented
11C2×#20263299C211yesmaximal, characteristic, complemented
12C2×#20263298C211yesmaximal, characteristic, complemented
13C2×C4×C22⋊D197C211yesmaximal, characteristic, complemented
14C2×#20263286C211yesmaximal, characteristic, complemented
15C2×#20263284C211yesmaximal, characteristic, complemented
16C2×#123×C197C211yesmaximal, characteristic, complemented
17C22×D788C2211yescharacteristic
18C22×D788C2211yescharacteristic
19C22×D788C2211yescharacteristic
20C22×C22⋊D197C2211yescharacteristic
21C22×C22⋊D197C2211yescharacteristic
22C22×C22⋊D197C2211yescharacteristic
23C22×C22⋊D197C2211yescharacteristic
24C22×C22⋊D197C2211yescharacteristic
25C22×C22⋊D197C2211yescharacteristic
26C23×Dic197C2211yescharacteristic
27C22×C4×D197C2211yescharacteristic
28C22×C4×D197C2211yescharacteristic
29C22×C4×D197C2211yescharacteristic
30#20263322C2218yescomplemented
31#20263316C2218yescomplemented
32#20263311C2218yescomplemented
33#20263310C2218yescomplemented
34#20263308C2218yescomplemented
35#20263301C2218yescomplemented
36#20263300C2218yescomplemented
37#20263299C2218yescomplemented
38#20263299C2218yescomplemented
39#20263299C2218yescomplemented
40#20263298C2218yescomplemented
41C4×C22⋊D197C2218yescomplemented
42#20263286C2218yescomplemented
43#20263284C2218yescomplemented
44C2×C4⋊Dic197C2211yescharacteristic
45C2×C4⋊Dic197C2211yescharacteristic
46C2×#15691337C2211yescharacteristic
47C2×#15691337C2211yescharacteristic
48C2×#15691337C2211yescharacteristic
49C2×#15691337C2211yescharacteristic
50C2×C22⋊Dic197C2211yescharacteristic
51C2×#15691335C2211yescharacteristic
52C2×#15691335C2211yescharacteristic
53C2×#15691335C2211yescharacteristic
54C2×#15691335C2211yescharacteristic
55C2×#15691335C2211yescharacteristic
56C2×#15691335C2211yescharacteristic
57C2×C4×Dic197C2211yescharacteristic
58C2×C4×Dic197C2211yescharacteristic
59C22×D4×C197C2211yescharacteristic
60C23×C4×C197C2211yescharacteristic
61#123×C197C2218yescomplemented
62C2×C4⋊C4×C197C2211yescharacteristic
63C2×C4⋊C4×C197C2211yescharacteristic
64C2×C22⋊C4×C197C2211yescharacteristic
65C2×C22⋊C4×C197C2211yescharacteristic
66C2×C22⋊C4×C197C2211yescharacteristic
67C23×D197C2311yescharacteristic
68C23×D197C2311yescharacteristic
69C23×D197C2311yescharacteristic
70C2×D788—42no—
71C2×D788—42no—
72C2×D788—42no—
73C2×D788C2314yes—
74C2×D788C2314yes—
75C2×D788C2314yes—
76C2×C22⋊D197—24no—
77C2×C22⋊D197—24no—
78C2×C22⋊D197—42no—
79C2×C22⋊D197—42no—
80C2×C22⋊D197—42no—
81C2×C22⋊D197—42no—
82C2×C22⋊D197C2314yes—
83C2×C22⋊D197C2314yes—
84C2×C22⋊D197C2314yes—
85C2×C22⋊D197C2314yes—
86C2×C22⋊D197C2314yes—
87C2×C22⋊D197C2314yes—
88C22×Dic197—42no—
89C22×Dic197C2311yescharacteristic
90C22×Dic197C2311yescharacteristic
91C22×Dic197C2311yescharacteristic
92C22×Dic197C2311yescharacteristic
93C22×Dic197C2314yes—
94C22×Dic197C2311yescharacteristic
95C2×C4×D197—24no—
96C2×C4×D197—42no—
97C2×C4×D197—42no—
98C2×C4×D197C2314yes—
99C2×C4×D197C2314yes—
100C2×C4×D197C2314yes—
101C4⋊Dic197C2314yes—
102C4⋊Dic197C2314yes—
103#15691337C2314yes—
104#15691337C2314yes—
105#15691337C2314yes—
106#15691337C2314yes—
107C22⋊Dic197C2314yes—
108#15691335C2314yes—
109#15691335C2314yes—
110#15691335C2314yes—
111#15691335C2314yes—
112#15691335C2314yes—
113#15691335C2314yes—
114C4×Dic197C2314yes—
115C4×Dic197C2314yes—
116C24×C197C2311yescharacteristic
117C24×C197C2311yescharacteristic
118C2×D4×C197—42no—
119C2×D4×C197C2314yes—
120C22×C4×C197—42no—
121C22×C4×C197C2311yescharacteristic
122C22×C4×C197C2311yescharacteristic
123C22×C4×C197C2311yescharacteristic
124C22×C4×C197C2311yescharacteristic
125C22×C4×C197C2311yescharacteristic
126C22×C4×C197C2314yes—
127C4⋊C4×C197C2314yes—
128C4⋊C4×C197C2314yes—
129C22⋊C4×C197C2314yes—
130C22⋊C4×C197C2314yes—
131C22⋊C4×C197C2314yes—
132C22×D197—22no—
133C22×D197—22no—
134C22×D197—22no—
135C22×D197—22no—
136C22×D197—22no—
137C22×D197—22no—
138C22×D197—22no—
139C22×D197—22no—
140C22×D197—22no—
141C22×D197C2412yes—
142C22×D197C2412yes—
143C22×D197C2412yes—
144D788—44no—
145D788—44no—
146D788—44no—
147C22⋊D197—28no—
148C22⋊D197—28no—
149C22⋊D197—44no—
150C22⋊D197—44no—
151C22⋊D197—44no—
152C22⋊D197—44no—
153C2×Dic197—22no—
154C2×Dic197—22no—
155C2×Dic197—22no—
156C2×Dic197—22no—
157C2×Dic197—44no—
158C2×Dic197#4114yes—
159C2×Dic197C2412yes—
160C2×Dic197C2412yes—
161C2×Dic197C2412yes—
162C2×Dic197C2412yes—
163C2×Dic197C2412yes—
164C4×D197—28no—
165C4×D197—44no—
166C4×D197—44no—
167C23×C197—22no—
168C23×C197—22no—
169C23×C197—22no—
170C23×C197—22no—
171C23×C197—22no—
172C23×C197—22no—
173C23×C197C2412yes—
174C23×C197C2412yes—
175C23×C197C2411yescharacteristic
176D4×C197—44no—
177C2×C4×C197—22no—
178C2×C4×C197—22no—
179C2×C4×C197—22no—
180C2×C4×C197—22no—
181C2×C4×C197—22no—
182C2×C4×C197—44no—
183C2×C4×C197C2412yes—
184C2×C4×C197C2412yes—
185C2×C4×C197C2412yes—
186C2×C4×C197C2412yes—
187C2×C4×C197C2412yes—
188C22×C197—22no—
189C22×C197—22no—
190C22×C197—22no—
191C22×C197—22no—
192C22×C197—22no—
193C22×C197—22no—
194C22×C197—44no—
195C22×C197—44no—
196C22×C197C2×#4112yes—
197C22×C197H(5,𝔽2)12yes—
198C22×C197H(5,𝔽2)12yes—
199C22×C197C2511yescharacteristic
200C2×D197—22no—
201C2×D197—22no—
202C2×D197—22no—
203C2×D197—22no—
204C2×D197—22no—
205C2×D197—22no—
206C2×D197—22no—
207C2×D197—22no—
208C2×D197—22no—
209C2×D197—44no—
210C2×D197—44no—
211C2×D197—44no—
212C4×C197—22no—
213C4×C197—22no—
214C4×C197—22no—
215C4×C197—22no—
216C4×C197—22no—
217Dic197—22no—
218Dic197—22no—
219Dic197—22no—
220Dic197—22no—
221Dic197C2×#4114yes—
222C2×C197—42no—
223C2×C197—42no—
224C2×C197#54014yescomplemented
225C2×C197C22×#4111yescharacteristic
226C2×C197C2×H(5,𝔽2)11yescharacteristic
227C2×C197C2×H(5,𝔽2)11yescharacteristic
228D197—42no—
229D197—42no—
230D197—42no—
231C197C2×#54011yescharacteristic, complemented, Sylow
232C2×#540—1971nomaximal, Sylow
233#540—19716no—
234C2×#127—1971no—
235C2×#125—1971no—
236C2×#123—1971no—
237C2×#123—1971no—
238C2×#123—1971no—
239C2×#123D19711yescharacteristic, complemented
240C2×#121—1971no—
241C2×#121—1971no—
242C2×#121—1971no—
243C2×#121—1971no—
244C2×#121—1971no—
245C2×#121—1971no—
246C2×C4×D4—1971no—
247C2×C4×D4—1971no—
248C2×#117—1971no—
249C22×D4—1971no—
250C22×D4—1971no—
251C22×D4—1971no—
252C22×D4—1971no—
253C22×D4—1971no—
254C22×D4—1971no—
255C22×D4—1971no—
256C22×D4—1971no—
257C22×D4—1971no—
258C22×D4C2×D19711yescharacteristic
259C23×C4—1971no—
260C23×C4—1971no—
261C23×C4—1971no—
262C23×C4—1971no—
263C23×C4C2×D19711yescharacteristic
264#127—1978no—
265#125—1978no—
266#123—1978no—
267#123—1978no—
268#123—1978no—
269#123C2×D19718yescomplemented
270#121—1978no—
271#121—1978no—
272#121—1978no—
273#121—1978no—
274#121—1978no—
275#121—1978no—
276C4×D4—1978no—
277C4×D4—1978no—
278#117—1978no—
279C2×C4⋊C4—1971no—
280C2×C4⋊C4—1971no—
281C2×C4⋊C4—1971no—
282C2×C4⋊C4—1971no—
283C2×C4⋊C4—1971no—
284C2×C4⋊C4—1971no—
285C2×C4⋊C4C2×D19711yescharacteristic
286C2×C4⋊C4C2×D19711yescharacteristic
287C2×C22⋊C4—1971no—
288C2×C22⋊C4—1971no—
289C2×C22⋊C4—1971no—
290C2×C22⋊C4—1971no—
291C2×C22⋊C4—1971no—
292C2×C22⋊C4—1971no—
293C2×C22⋊C4—1971no—
294C2×C22⋊C4C2×D19711yescharacteristic
295C2×C22⋊C4C2×D19711yescharacteristic
296C2×C22⋊C4C2×D19711yescharacteristic
297C2×C42—1971no—
298C2×C42—1971no—
299C24—1971no—
300C24—1971no—
301C24—1971no—
302C24C22×D19711yescharacteristic
303C24C22×D19711yescharacteristic
304C2×D4—42no—
305C2×D4—7882no—
306C2×D4—7882no—
307C2×D4—7882no—
308C2×D4—7882no—
309C2×D4—7882no—
310C2×D4—7882no—
311C2×D4—7882no—
312C2×D4—3944no—
313C2×D4—3944no—
314C2×D4—1974no—
315C2×D4—1974no—
316C2×D4—1974no—
317C2×D4—1974no—
318C2×D4—1974no—
319C2×D4—1974no—
320C2×D4—1974no—
321C2×D4—1974no—
322C2×D4—1974no—
323C2×D4C22×D19714yes—
324C22×C4—42no—
325C22×C4—7882no—
326C22×C4—7882no—
327C22×C4—7882no—
328C22×C4—3944no—
329C22×C4—1974no—
330C22×C4—1974no—
331C22×C4—1974no—
332C22×C4—1971no—
333C22×C4—1971no—
334C22×C4—1971no—
335C22×C4—1971no—
336C22×C4—1974no—
337C22×C4—1971no—
338C22×C4C22×D19711yescharacteristic
339C22×C4C22×D19711yescharacteristic
340C22×C4C22×D19711yescharacteristic
341C22×C4C22×D19711yescharacteristic
342C22×C4C22×D19711yescharacteristic
343C22×C4C22×D19714yes—
344C4⋊C4—1974no—
345C4⋊C4—1974no—
346C4⋊C4—1974no—
347C4⋊C4—1974no—
348C4⋊C4—1974no—
349C4⋊C4—1974no—
350C4⋊C4C22×D19714yes—
351C4⋊C4C22×D19714yes—
352C22⋊C4—1974no—
353C22⋊C4—1974no—
354C22⋊C4—1974no—
355C22⋊C4—1974no—
356C22⋊C4—1974no—
357C22⋊C4—1974no—
358C22⋊C4—1974no—
359C22⋊C4C22×D19714yes—
360C22⋊C4C22×D19714yes—
361C22⋊C4C22×D19714yes—
362C42—1974no—
363C42—1974no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—3942no—
371C23—3942no—
372C23—3942no—
373C23—3942no—
374C23—3942no—
375C23—3942no—
376C23—3942no—
377C23—3942no—
378C23—3942no—
379C23—1972no—
380C23—1972no—
381C23—1972no—
382C23C23×D19712yes—
383C23C23×D19712yes—
384C23C23×D19711yescharacteristic
385D4—44no—
386D4—7884no—
387D4—7884no—
388D4—7884no—
389D4—7884no—
390D4—7884no—
391D4—7884no—
392D4—7884no—
393D4—3948no—
394D4—3948no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—44no—
401C2×C4—7884no—
402C2×C4—7884no—
403C2×C4—7884no—
404C2×C4—3948no—
405C2×C4—3942no—
406C2×C4—3942no—
407C2×C4—3942no—
408C2×C4—3942no—
409C2×C4—1972no—
410C2×C4—1972no—
411C2×C4—1972no—
412C2×C4—1972no—
413C2×C4—1974no—
414C2×C4—1972no—
415C2×C4C23×D19712yes—
416C2×C4C23×D19712yes—
417C2×C4C23×D19712yes—
418C2×C4C23×D19712yes—
419C2×C4C23×D19712yes—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—44no—
427C22—44no—
428C22—7884no—
429C22—7884no—
430C22—7884no—
431C22—3942no—
432C22—3942no—
433C22—3942no—
434C22—3942no—
435C22—3942no—
436C22—3942no—
437C22—3942no—
438C22—3942no—
439C22—3942no—
440C22#41×D19712yes—
441C22#2026337112yes—
442C22#2026337212yes—
443C22C24×D19711yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—3942no—
450C4—3942no—
451C4—3942no—
452C4—3942no—
453C4—1974no—
454C2—42no—
455C2—42no—
456C2—7882no—
457C2—7882no—
458C2—7882no—
459C2#2716168114yesdirect product factor
460C2C2×#41×D19711yescharacteristic
461C2C2×#2026337111yescharacteristic
462C2C2×#2026337211yescharacteristic
subgroupG/H
centerC23C23×D197
commutator, cosocleC22×C197C25
FrattiniC22C24×D197
FittingC2×#123×C197C2
socleC23×C197C24
hypercenterC2×#123D197
hypocenterC197C2×#540
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #27161681C2direct (N × H)
2. C2×#20263322C2semidirect
3. C2×#20263316C2semidirect
4. C2×#20263311C2semidirect
5. C2×#20263310C2semidirect
6. C2×#20263308C2semidirect
7. C2×#20263301C2semidirect
8. C2×#20263300C2semidirect
9. C2×#20263299C2semidirect
10. C2×#20263299C2semidirect
11. C2×#20263299C2semidirect
12. C2×#20263298C2semidirect
13. C2×C4×C22⋊D197C2semidirect
14. C2×#20263286C2semidirect
15. C2×#20263284C2semidirect
16. C2×#123×C197C2semidirect
30. #20263322C22semidirect
31. #20263316C22semidirect
32. #20263311C22semidirect
33. #20263310C22semidirect
34. #20263308C22semidirect
35. #20263301C22semidirect
36. #20263300C22semidirect
37. #20263299C22semidirect
38. #20263299C22semidirect
39. #20263299C22semidirect
40. #20263298C22semidirect
41. C4×C22⋊D197C22semidirect
42. #20263286C22semidirect
43. #20263284C22semidirect
61. #123×C197C22semidirect
224. C2×C197#540semidirect
231. C197C2×#540semidirect
239. C2×#123D197semidirect
269. #123C2×D197semidirect
459. C2#27161681direct (N × H)

Maximal subgroups

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

subgroupindex
#271616812
C2×#202633222
C2×#202633162
C2×#202633112
C2×#202633102
C2×#202633082
C2×#202633012
C2×#202633002
C2×#202632992
C2×#202632982
C2×C4×C22⋊D1972
C2×#202632862
C2×#202632842
C2×#123×C1972
C2×#540197