Finite Group Databasebeta

← C2×#27161727 #38756166

Subgroups — C2×#27161727 #38756166

Order 25216.

1632 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
1C22×#15691358C211yesmaximal, characteristic, complemented
2C22×#15691357C211yesmaximal, characteristic, complemented
3#27161727C2116yesmaximal, direct product factor
4C2×#20263325C211yesmaximal, characteristic, complemented
5C2×#20263311C211yesmaximal, characteristic, complemented
6C2×#20263310C211yesmaximal, characteristic, complemented
7C2×#20263308C211yesmaximal, characteristic, complemented
8C2×#20263304C211yesmaximal, characteristic, complemented
9C2×#20263303C211yesmaximal, characteristic, complemented
10C2×#20263299C211yesmaximal, characteristic, complemented
11C2×#20263298C211yesmaximal, characteristic, complemented
12C2×#20263286C211yesmaximal, characteristic, complemented
13C2×#20263286C211yesmaximal, characteristic, complemented
14C22×#15691337C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D197C211yesmaximal, characteristic, complemented
16C2×#123×C197C211yesmaximal, characteristic, complemented
17C2×#15691358—28no—
18C2×#15691358C2218yescomplemented
19C2×#15691357—28no—
20C2×#15691357C2218yescomplemented
21C22×Dic394C2211yescharacteristic
22C22×Dic394C2211yescharacteristic
23C22×D788C2211yescharacteristic
24C22×C22⋊D197C2211yescharacteristic
25C22×C22⋊D197C2211yescharacteristic
26C22×C22⋊D197C2211yescharacteristic
27C22×C22⋊D197C2211yescharacteristic
28C23×Dic197C2211yescharacteristic
29C23×Dic197C2211yescharacteristic
30C23×Dic197C2211yescharacteristic
31C22×C4×D197C2211yescharacteristic
32C22×C4×D197C2211yescharacteristic
33C22×C4×D197C2211yescharacteristic
34C22×C4×D197C2211yescharacteristic
35#20263325C2218yescomplemented
36#20263311C2218yescomplemented
37#20263310C2218yescomplemented
38#20263308C2218yescomplemented
39#20263304C2218yescomplemented
40#20263303C2218yescomplemented
41#20263299C2218yescomplemented
42#20263298C2218yescomplemented
43#20263286C2218yescomplemented
44#20263286C2218yescomplemented
45C2×C4⋊Dic197C2211yescharacteristic
46C2×C4⋊Dic197C2211yescharacteristic
47C2×#15691337C2211yescharacteristic
48C2×#15691337C2211yescharacteristic
49C2×#15691337C2211yescharacteristic
50C2×#15691337C2211yescharacteristic
51C2×#15691337C2211yescharacteristic
52C2×#15691337C2211yescharacteristic
53C2×#15691337C2218yescomplemented
54C4⋊C4×D197C2218yescomplemented
55C2×C22⋊Dic197C2211yescharacteristic
56C2×#15691335C2211yescharacteristic
57C2×#15691335C2211yescharacteristic
58C2×#15691335C2211yescharacteristic
59C2×#15691335C2211yescharacteristic
60C2×C4×Dic197C2211yescharacteristic
61C22×D4×C197C2211yescharacteristic
62C23×C4×C197C2211yescharacteristic
63#123×C197C2218yescomplemented
64C2×C4⋊C4×C197C2211yescharacteristic
65C2×C4⋊C4×C197C2211yescharacteristic
66C2×C22⋊C4×C197C2211yescharacteristic
67C2×C22⋊C4×C197C2211yescharacteristic
68C2×C22⋊C4×C197C2211yescharacteristic
69C23×D197C2311yescharacteristic
70C23×D197C2311yescharacteristic
71#15691358—232no—
72#15691357—232no—
73C2×Dic394—24no—
74C2×Dic394—24no—
75C2×Dic394C2314yes—
76C2×Dic394C2314yes—
77C2×D788—24no—
78C2×D788C2314yes—
79C2×C22⋊D197—24no—
80C2×C22⋊D197—24no—
81C2×C22⋊D197—24no—
82C2×C22⋊D197D418yescomplemented
83C2×C22⋊D197C2314yes—
84C2×C22⋊D197C2314yes—
85C2×C22⋊D197C2314yes—
86C2×C22⋊D197C2314yes—
87C22×Dic197—24no—
88C22×Dic197—24no—
89C22×Dic197—24no—
90C22×Dic197C2311yescharacteristic
91C22×Dic197C2311yescharacteristic
92C22×Dic197C2311yescharacteristic
93C22×Dic197C2311yescharacteristic
94C22×Dic197C2314yes—
95C22×Dic197C2314yes—
96C22×Dic197C2314yes—
97C22×Dic197C2311yescharacteristic
98C22×Dic197C2311yescharacteristic
99C2×C4×D197—24no—
100C2×C4×D197—24no—
101C2×C4×D197—24no—
102C2×C4×D197—24no—
103C2×C4×D197C2314yes—
104C2×C4×D197C2314yes—
105C2×C4×D197C2314yes—
106C2×C4×D197C2314yes—
107C4⋊Dic197C2314yes—
108C4⋊Dic197C2314yes—
109#15691337C2314yes—
110#15691337C2314yes—
111#15691337C2314yes—
112#15691337C2314yes—
113#15691337C2314yes—
114#15691337C2314yes—
115C22⋊Dic197C2314yes—
116#15691335C2314yes—
117#15691335C2314yes—
118#15691335C2314yes—
119#15691335C2314yes—
120C4×Dic197C2314yes—
121C24×C197C2311yescharacteristic
122C24×C197C2311yescharacteristic
123C2×D4×C197—24no—
124C2×D4×C197C2314yes—
125C22×C4×C197—24no—
126C22×C4×C197C2311yescharacteristic
127C22×C4×C197C2311yescharacteristic
128C22×C4×C197C2311yescharacteristic
129C22×C4×C197C2311yescharacteristic
130C22×C4×C197C2311yescharacteristic
131C22×C4×C197C2314yes—
132C4⋊C4×C197C2314yes—
133C4⋊C4×C197C2314yes—
134C22⋊C4×C197C2314yes—
135C22⋊C4×C197C2314yes—
136C22⋊C4×C197C2314yes—
137C22×D197—22no—
138C22×D197—22no—
139C22×D197—22no—
140C22×D197—22no—
141C22×D197—22no—
142C22×D197C2×D414yes—
143C22×D197C2412yes—
144C22×D197C2412yes—
145Dic394—28no—
146Dic394—28no—
147D788—28no—
148C22⋊D197—28no—
149C22⋊D197—28no—
150C22⋊D197—28no—
151C22⋊D197C2×D4116yescomplemented
152C2×Dic197—22no—
153C2×Dic197—22no—
154C2×Dic197—22no—
155C2×Dic197—28no—
156C2×Dic197—22no—
157C2×Dic197—28no—
158C2×Dic197—28no—
159C2×Dic197C2×D414yes—
160C2×Dic197#4114yescomplemented
161C2×Dic197C2412yes—
162C2×Dic197C2412yes—
163C2×Dic197C2412yes—
164C2×Dic197C2412yes—
165C2×Dic197C2412yes—
166C2×Dic197C2412yes—
167C4×D197—28no—
168C4×D197—28no—
169C4×D197—28no—
170C4×D197—28no—
171C23×C197—22no—
172C23×C197—22no—
173C23×C197—22no—
174C23×C197—22no—
175C23×C197—22no—
176C23×C197C2×D414yes—
177C23×C197C2412yes—
178C23×C197C2412yes—
179C23×C197C2411yescharacteristic
180D4×C197—28no—
181C2×C4×C197—22no—
182C2×C4×C197—22no—
183C2×C4×C197—28no—
184C2×C4×C197—22no—
185C2×C4×C197—22no—
186C2×C4×C197—22no—
187C2×C4×C197C2412yes—
188C2×C4×C197C2412yes—
189C2×C4×C197C2412yes—
190C2×C4×C197C2412yes—
191C2×C4×C197C2412yes—
192C22×C197—28no—
193C22×C197—22no—
194C22×C197—22no—
195C22×C197—22no—
196C22×C197—22no—
197C22×C197—22no—
198C22×C197—44no—
199C22×C197C22×D414yes—
200C22×C197C22×D412yes—
201C22×C197C2×#4112yes—
202C22×C1972−1+412yes—
203C22×C197C2511yescharacteristic
204C2×D197—28no—
205C2×D197—22no—
206C2×D197—22no—
207C2×D197—22no—
208C2×D197—22no—
209C2×D197—22no—
210C2×D197—44no—
211C2×D197C22×D414yes—
212C4×C197—22no—
213C4×C197—22no—
214C4×C197—22no—
215C4×C197—22no—
216C4×C197—22no—
217Dic197—22no—
218Dic197—22no—
219Dic197—22no—
220Dic197—22no—
221Dic197C22×D414yes—
222Dic197C2×#4114yescomplemented
223C2×C197—24no—
224C2×C197—42no—
225C2×C197#54714yescomplemented
226C2×C197C23×D411yescharacteristic
227C2×C197C22×#4111yescharacteristic
228C2×C197C2×2−1+411yescharacteristic
229D197—24no—
230D197—42no—
231C197C2×#54711yescharacteristic, complemented, Sylow
232C2×#547—1971nomaximal, Sylow
233C22×#41—1971no—
234C22×#41—1971no—
235#547—19716no—
236C2×C4⋊Q8—1971no—
237C2×#123—1971no—
238C2×#123—1971no—
239C2×#123—1971no—
240C2×#123D19711yescharacteristic, complemented
241C2×C22⋊Q8—1971no—
242C2×C22⋊Q8—1971no—
243C2×#121—1971no—
244C2×#121—1971no—
245C2×C4×D4—1971no—
246C2×C4×D4—1971no—
247C22×C4⋊C4—1971no—
248C22×C4⋊C4—1971no—
249C2×#41—3948no—
250C2×#41—3948no—
251C2×#41—1978no—
252C2×#41—1978no—
253C22×Q8—1971no—
254C22×Q8—1971no—
255C22×D4—1971no—
256C22×D4—1971no—
257C22×D4—1971no—
258C22×D4—1971no—
259C22×D4—1971no—
260C22×D4C2×D19711yescharacteristic
261C23×C4—1971no—
262C23×C4—1971no—
263C23×C4—1971no—
264C23×C4—1971no—
265C23×C4—1971no—
266C23×C4—1971no—
267C23×C4—1971no—
268C23×C4C2×D19711yescharacteristic
269C4⋊Q8—1978no—
270#123—1978no—
271#123—1978no—
272#123—1978no—
273#123C2×D19718yescomplemented
274C22⋊Q8—1978no—
275C22⋊Q8—1978no—
276#121—1978no—
277#121—1978no—
278C4×D4—1978no—
279C4×D4—1978no—
280C2×C4⋊C4—1971no—
281C2×C4⋊C4—1971no—
282C2×C4⋊C4—1971no—
283C2×C4⋊C4—1971no—
284C2×C4⋊C4—1971no—
285C2×C4⋊C4—1971no—
286C2×C4⋊C4—1978no—
287C2×C4⋊C4—1971no—
288C2×C4⋊C4—1971no—
289C2×C4⋊C4—1978no—
290C2×C4⋊C4C2×D19711yescharacteristic
291C2×C4⋊C4C2×D19711yescharacteristic
292C2×C22⋊C4—1971no—
293C2×C22⋊C4—1971no—
294C2×C22⋊C4—1971no—
295C2×C22⋊C4—1971no—
296C2×C22⋊C4—1971no—
297C2×C22⋊C4C2×D19711yescharacteristic
298C2×C22⋊C4C2×D19711yescharacteristic
299C2×C22⋊C4C2×D19711yescharacteristic
300C2×C42—1971no—
301C24—1971no—
302C24—1971no—
303C24C22×D19711yescharacteristic
304C24C22×D19711yescharacteristic
305#41—39432no—
306#41—39432no—
307C2×Q8—3944no—
308C2×Q8—3944no—
309C2×Q8—1974no—
310C2×Q8—1974no—
311C2×D4—24no—
312C2×D4—3944no—
313C2×D4—3944no—
314C2×D4—3944no—
315C2×D4—3944no—
316C2×D4—1974no—
317C2×D4—1974no—
318C2×D4—1974no—
319C2×D4—1978no—
320C2×D4—1974no—
321C2×D4—1974no—
322C2×D4C22×D19714yes—
323C22×C4—24no—
324C22×C4—3944no—
325C22×C4—3944no—
326C22×C4—3944no—
327C22×C4—3944no—
328C22×C4—3944no—
329C22×C4—3944no—
330C22×C4—3944no—
331C22×C4—1974no—
332C22×C4—1974no—
333C22×C4—1974no—
334C22×C4—1971no—
335C22×C4—1971no—
336C22×C4—1971no—
337C22×C4—1971no—
338C22×C4—1974no—
339C22×C4—1974no—
340C22×C4—1974no—
341C22×C4—1974no—
342C22×C4—1971no—
343C22×C4—1971no—
344C22×C4C22×D19714yes—
345C22×C4C22×D19711yescharacteristic
346C22×C4C22×D19711yescharacteristic
347C22×C4C22×D19711yescharacteristic
348C22×C4C22×D19711yescharacteristic
349C22×C4C22×D19711yescharacteristic
350C4⋊C4—1974no—
351C4⋊C4—1974no—
352C4⋊C4—1974no—
353C4⋊C4—1974no—
354C4⋊C4—1974no—
355C4⋊C4—1974no—
356C4⋊C4—1974no—
357C4⋊C4—1974no—
358C4⋊C4C22×D19714yes—
359C4⋊C4C22×D19714yes—
360C22⋊C4—1974no—
361C22⋊C4—1974no—
362C22⋊C4—1974no—
363C22⋊C4—1974no—
364C22⋊C4—1974no—
365C22⋊C4C22×D19714yes—
366C22⋊C4C22×D19714yes—
367C22⋊C4C22×D19714yes—
368C42—1974no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—3942no—
375C23—3942no—
376C23—3942no—
377C23—3942no—
378C23—3942no—
379C23—1972no—
380C23—1974no—
381C23—1972no—
382C23D4×D19714yes—
383C23C23×D19712yes—
384C23C23×D19712yes—
385C23C23×D19711yescharacteristic
386Q8—3948no—
387Q8—3948no—
388D4—28no—
389D4—3948no—
390D4—3948no—
391D4—3948no—
392D4—3948no—
393D4—19716no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—28no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—3948no—
401C2×C4—3948no—
402C2×C4—3948no—
403C2×C4—3948no—
404C2×C4—3942no—
405C2×C4—3942no—
406C2×C4—3942no—
407C2×C4—3948no—
408C2×C4—3942no—
409C2×C4—3948no—
410C2×C4—3948no—
411C2×C4—1972no—
412C2×C4—1972no—
413C2×C4—1972no—
414C2×C4—1972no—
415C2×C4—1974no—
416C2×C4—1974no—
417C2×C4—1972no—
418C2×C4—1972no—
419C2×C4C23×D19712yes—
420C2×C4C23×D19712yes—
421C2×C4C23×D19712yes—
422C2×C4C23×D19712yes—
423C2×C4C23×D19712yes—
424C22—22no—
425C22—22no—
426C22—28no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—44no—
431C22—7884no—
432C22—3942no—
433C22—3942no—
434C22—3942no—
435C22—3942no—
436C22—3942no—
437C22—3948no—
438C22—1974no—
439C22C2×D4×D19714yes—
440C22C2×D4×D19712yes—
441C22#41×D19712yes—
442C22#2026337312yes—
443C22C24×D19711yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—3942no—
450C4—3942no—
451C4—3942no—
452C4—3942no—
453C4—1974no—
454C4—1974no—
455C2—24no—
456C2—42no—
457C2—7882no—
458C2—3944no—
459C2#2716172714yesdirect product factor
460C2C22×D4×D19711yescharacteristic
461C2C2×#41×D19711yescharacteristic
462C2C2×#2026337311yescharacteristic
subgroupG/H
centerC23C23×D197
commutator, cosocleC22×C197C25
FrattiniC22C24×D197
FittingC2×#123×C197C2
socleC23×C197C24
hypercenterC2×#123D197
hypocenterC197C2×#547
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×#15691358C2semidirect
2. C22×#15691357C2semidirect
3. #27161727C2direct (N × H)
4. C2×#20263325C2semidirect
5. C2×#20263311C2semidirect
6. C2×#20263310C2semidirect
7. C2×#20263308C2semidirect
8. C2×#20263304C2semidirect
9. C2×#20263303C2semidirect
10. C2×#20263299C2semidirect
11. C2×#20263298C2semidirect
12. C2×#20263286C2semidirect
13. C2×#20263286C2semidirect
14. C22×#15691337C2semidirect
15. C2×C4⋊C4×D197C2semidirect
16. C2×#123×C197C2semidirect
18. C2×#15691358C22semidirect
20. C2×#15691357C22semidirect
35. #20263325C22semidirect
36. #20263311C22semidirect
37. #20263310C22semidirect
38. #20263308C22semidirect
39. #20263304C22semidirect
40. #20263303C22semidirect
41. #20263299C22semidirect
42. #20263298C22semidirect
43. #20263286C22semidirect
44. #20263286C22semidirect
53. C2×#15691337C22semidirect
54. C4⋊C4×D197C22semidirect
63. #123×C197C22semidirect
82. C2×C22⋊D197D4semidirect
151. C22⋊D197C2×D4semidirect
160. C2×Dic197#41semidirect
222. Dic197C2×#41semidirect
225. C2×C197#547semidirect
231. C197C2×#547semidirect
240. C2×#123D197semidirect
273. #123C2×D197semidirect
459. C2#27161727direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#156913582
C22×#156913572
#271617272
C2×#202633252
C2×#202633112
C2×#202633102
C2×#202633082
C2×#202633042
C2×#202633032
C2×#202632992
C2×#202632982
C2×#202632862
C22×#156913372
C2×C4⋊C4×D1972
C2×#123×C1972
C2×#547197