Finite Group Databasebeta

← C2×#20261383 #27155190

Subgroups — C2×#20261383 #27155190

Order 12416.

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×#13095511C211yesmaximal, characteristic, complemented
2C22×#13095510C211yesmaximal, characteristic, complemented
3#20261383C2116yesmaximal, direct product factor
4C2×#15690809C211yesmaximal, characteristic, complemented
5C2×#15690795C211yesmaximal, characteristic, complemented
6C2×#15690794C211yesmaximal, characteristic, complemented
7C2×#15690792C211yesmaximal, characteristic, complemented
8C2×#15690788C211yesmaximal, characteristic, complemented
9C2×#15690787C211yesmaximal, characteristic, complemented
10C2×#15690783C211yesmaximal, characteristic, complemented
11C2×#15690782C211yesmaximal, characteristic, complemented
12C2×#15690770C211yesmaximal, characteristic, complemented
13C2×#15690770C211yesmaximal, characteristic, complemented
14C22×#13095490C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D97C211yesmaximal, characteristic, complemented
16C2×#123×C97C211yesmaximal, characteristic, complemented
17C2×#13095511—28no—
18C2×#13095511C2218yescomplemented
19C2×#13095510—28no—
20C2×#13095510C2218yescomplemented
21C22×Dic194C2211yescharacteristic
22C22×Dic194C2211yescharacteristic
23C22×D388C2211yescharacteristic
24C22×C22⋊D97C2211yescharacteristic
25C22×C22⋊D97C2211yescharacteristic
26C22×C22⋊D97C2211yescharacteristic
27C22×C22⋊D97C2211yescharacteristic
28C23×Dic97C2211yescharacteristic
29C23×Dic97C2211yescharacteristic
30C23×Dic97C2211yescharacteristic
31C22×C4×D97C2211yescharacteristic
32C22×C4×D97C2211yescharacteristic
33C22×C4×D97C2211yescharacteristic
34C22×C4×D97C2211yescharacteristic
35#15690809C2218yescomplemented
36#15690795C2218yescomplemented
37#15690794C2218yescomplemented
38#15690792C2218yescomplemented
39#15690788C2218yescomplemented
40#15690787C2218yescomplemented
41#15690783C2218yescomplemented
42#15690782C2218yescomplemented
43#15690770C2218yescomplemented
44#15690770C2218yescomplemented
45C2×C4⋊Dic97C2211yescharacteristic
46C2×C4⋊Dic97C2211yescharacteristic
47C2×#13095490C2211yescharacteristic
48C2×#13095490C2211yescharacteristic
49C2×#13095490C2211yescharacteristic
50C2×#13095490C2211yescharacteristic
51C2×#13095490C2211yescharacteristic
52C2×#13095490C2211yescharacteristic
53C2×#13095490C2218yescomplemented
54C4⋊C4×D97C2218yescomplemented
55C2×C22⋊Dic97C2211yescharacteristic
56C2×#13095488C2211yescharacteristic
57C2×#13095488C2211yescharacteristic
58C2×#13095488C2211yescharacteristic
59C2×#13095488C2211yescharacteristic
60C2×C4×Dic97C2211yescharacteristic
61C22×D4×C97C2211yescharacteristic
62C23×C4×C97C2211yescharacteristic
63#123×C97C2218yescomplemented
64C2×C4⋊C4×C97C2211yescharacteristic
65C2×C4⋊C4×C97C2211yescharacteristic
66C2×C22⋊C4×C97C2211yescharacteristic
67C2×C22⋊C4×C97C2211yescharacteristic
68C2×C22⋊C4×C97C2211yescharacteristic
69C23×D97C2311yescharacteristic
70C23×D97C2311yescharacteristic
71#13095511—232no—
72#13095510—232no—
73C2×Dic194—24no—
74C2×Dic194—24no—
75C2×Dic194C2314yes—
76C2×Dic194C2314yes—
77C2×D388—24no—
78C2×D388C2314yes—
79C2×C22⋊D97—24no—
80C2×C22⋊D97—24no—
81C2×C22⋊D97—24no—
82C2×C22⋊D97D418yescomplemented
83C2×C22⋊D97C2314yes—
84C2×C22⋊D97C2314yes—
85C2×C22⋊D97C2314yes—
86C2×C22⋊D97C2314yes—
87C22×Dic97—24no—
88C22×Dic97—24no—
89C22×Dic97—24no—
90C22×Dic97C2311yescharacteristic
91C22×Dic97C2311yescharacteristic
92C22×Dic97C2311yescharacteristic
93C22×Dic97C2311yescharacteristic
94C22×Dic97C2314yes—
95C22×Dic97C2314yes—
96C22×Dic97C2314yes—
97C22×Dic97C2311yescharacteristic
98C22×Dic97C2311yescharacteristic
99C2×C4×D97—24no—
100C2×C4×D97—24no—
101C2×C4×D97—24no—
102C2×C4×D97—24no—
103C2×C4×D97C2314yes—
104C2×C4×D97C2314yes—
105C2×C4×D97C2314yes—
106C2×C4×D97C2314yes—
107C4⋊Dic97C2314yes—
108C4⋊Dic97C2314yes—
109#13095490C2314yes—
110#13095490C2314yes—
111#13095490C2314yes—
112#13095490C2314yes—
113#13095490C2314yes—
114#13095490C2314yes—
115C22⋊Dic97C2314yes—
116#13095488C2314yes—
117#13095488C2314yes—
118#13095488C2314yes—
119#13095488C2314yes—
120C4×Dic97C2314yes—
121C24×C97C2311yescharacteristic
122C24×C97C2311yescharacteristic
123C2×D4×C97—24no—
124C2×D4×C97C2314yes—
125C22×C4×C97—24no—
126C22×C4×C97C2311yescharacteristic
127C22×C4×C97C2311yescharacteristic
128C22×C4×C97C2311yescharacteristic
129C22×C4×C97C2311yescharacteristic
130C22×C4×C97C2311yescharacteristic
131C22×C4×C97C2314yes—
132C4⋊C4×C97C2314yes—
133C4⋊C4×C97C2314yes—
134C22⋊C4×C97C2314yes—
135C22⋊C4×C97C2314yes—
136C22⋊C4×C97C2314yes—
137C23×C97—22no—
138C23×C97—22no—
139C23×C97—22no—
140C23×C97—22no—
141C23×C97—22no—
142C23×C97C2×D414yes—
143C23×C97C2412yes—
144C23×C97C2412yes—
145C23×C97C2411yescharacteristic
146C22×D97—22no—
147C22×D97—22no—
148C22×D97—22no—
149C22×D97—22no—
150C22×D97—22no—
151C22×D97C2×D414yes—
152C22×D97C2412yes—
153C22×D97C2412yes—
154D4×C97—28no—
155C2×C4×C97—22no—
156C2×C4×C97—22no—
157C2×C4×C97—28no—
158C2×C4×C97—22no—
159C2×C4×C97—22no—
160C2×C4×C97—22no—
161C2×C4×C97C2412yes—
162C2×C4×C97C2412yes—
163C2×C4×C97C2412yes—
164C2×C4×C97C2412yes—
165C2×C4×C97C2412yes—
166C22⋊D97—28no—
167C22⋊D97—28no—
168C22⋊D97—28no—
169C22⋊D97C2×D4116yescomplemented
170C2×Dic97—22no—
171C2×Dic97—22no—
172C2×Dic97—22no—
173C2×Dic97—28no—
174C2×Dic97—22no—
175C2×Dic97—28no—
176C2×Dic97—28no—
177C2×Dic97C2×D414yes—
178C2×Dic97#4114yescomplemented
179C2×Dic97C2412yes—
180C2×Dic97C2412yes—
181C2×Dic97C2412yes—
182C2×Dic97C2412yes—
183C2×Dic97C2412yes—
184C2×Dic97C2412yes—
185D388—28no—
186C4×D97—28no—
187C4×D97—28no—
188C4×D97—28no—
189C4×D97—28no—
190Dic194—28no—
191Dic194—28no—
192C22×C97—28no—
193C22×C97—22no—
194C22×C97—22no—
195C22×C97—22no—
196C22×C97—22no—
197C22×C97—22no—
198C22×C97—44no—
199C22×C97C22×D414yes—
200C22×C97C22×D412yes—
201C22×C97C2×#4112yes—
202C22×C972−1+412yes—
203C22×C97C2511yescharacteristic
204C2×D97—28no—
205C2×D97—22no—
206C2×D97—22no—
207C2×D97—22no—
208C2×D97—22no—
209C2×D97—22no—
210C2×D97—44no—
211C2×D97C22×D414yes—
212C4×C97—22no—
213C4×C97—22no—
214C4×C97—22no—
215C4×C97—22no—
216C4×C97—22no—
217Dic97—22no—
218Dic97—22no—
219Dic97—22no—
220Dic97—22no—
221Dic97C22×D414yes—
222Dic97C2×#4114yescomplemented
223C2×C97—24no—
224C2×C97—42no—
225C2×C97#54714yescomplemented
226C2×C97C23×D411yescharacteristic
227C2×C97C22×#4111yescharacteristic
228C2×C97C2×2−1+411yescharacteristic
229D97—24no—
230D97—42no—
231C2×#547—971nomaximal, Sylow
232C97C2×#54711yescharacteristic, complemented, Sylow
233C22×#41—971no—
234C22×#41—971no—
235#547—9716no—
236C2×C4⋊Q8—971no—
237C2×#123—971no—
238C2×#123—971no—
239C2×#123—971no—
240C2×#123D9711yescharacteristic, complemented
241C2×C22⋊Q8—971no—
242C2×C22⋊Q8—971no—
243C2×#121—971no—
244C2×#121—971no—
245C2×C4×D4—971no—
246C2×C4×D4—971no—
247C22×C4⋊C4—971no—
248C22×C4⋊C4—971no—
249C2×#41—978no—
250C2×#41—978no—
251C2×#41—1948no—
252C2×#41—1948no—
253C22×Q8—971no—
254C22×Q8—971no—
255C22×D4—971no—
256C22×D4—971no—
257C22×D4—971no—
258C22×D4—971no—
259C22×D4—971no—
260C22×D4C2×D9711yescharacteristic
261C23×C4—971no—
262C23×C4—971no—
263C23×C4—971no—
264C23×C4—971no—
265C23×C4—971no—
266C23×C4—971no—
267C23×C4—971no—
268C23×C4C2×D9711yescharacteristic
269C4⋊Q8—978no—
270#123—978no—
271#123—978no—
272#123—978no—
273#123C2×D9718yescomplemented
274C22⋊Q8—978no—
275C22⋊Q8—978no—
276#121—978no—
277#121—978no—
278C4×D4—978no—
279C4×D4—978no—
280C2×C4⋊C4—971no—
281C2×C4⋊C4—971no—
282C2×C4⋊C4—971no—
283C2×C4⋊C4—971no—
284C2×C4⋊C4—971no—
285C2×C4⋊C4—971no—
286C2×C4⋊C4—978no—
287C2×C4⋊C4—971no—
288C2×C4⋊C4—971no—
289C2×C4⋊C4—978no—
290C2×C4⋊C4C2×D9711yescharacteristic
291C2×C4⋊C4C2×D9711yescharacteristic
292C2×C22⋊C4—971no—
293C2×C22⋊C4—971no—
294C2×C22⋊C4—971no—
295C2×C22⋊C4—971no—
296C2×C22⋊C4—971no—
297C2×C22⋊C4C2×D9711yescharacteristic
298C2×C22⋊C4C2×D9711yescharacteristic
299C2×C22⋊C4C2×D9711yescharacteristic
300C2×C42—971no—
301C24—971no—
302C24—971no—
303C24C22×D9711yescharacteristic
304C24C22×D9711yescharacteristic
305#41—19432no—
306#41—19432no—
307C2×Q8—974no—
308C2×Q8—974no—
309C2×Q8—1944no—
310C2×Q8—1944no—
311C2×D4—24no—
312C2×D4—978no—
313C2×D4—974no—
314C2×D4—974no—
315C2×D4—974no—
316C2×D4—974no—
317C2×D4—974no—
318C2×D4—1944no—
319C2×D4—1944no—
320C2×D4—1944no—
321C2×D4—1944no—
322C2×D4C22×D9714yes—
323C22×C4—24no—
324C22×C4—971no—
325C22×C4—971no—
326C22×C4—974no—
327C22×C4—974no—
328C22×C4—974no—
329C22×C4—971no—
330C22×C4—971no—
331C22×C4—971no—
332C22×C4—971no—
333C22×C4—974no—
334C22×C4—974no—
335C22×C4—974no—
336C22×C4—974no—
337C22×C4—1944no—
338C22×C4—1944no—
339C22×C4—1944no—
340C22×C4—1944no—
341C22×C4—1944no—
342C22×C4—1944no—
343C22×C4—1944no—
344C22×C4C22×D9714yes—
345C22×C4C22×D9711yescharacteristic
346C22×C4C22×D9711yescharacteristic
347C22×C4C22×D9711yescharacteristic
348C22×C4C22×D9711yescharacteristic
349C22×C4C22×D9711yescharacteristic
350C4⋊C4—974no—
351C4⋊C4—974no—
352C4⋊C4—974no—
353C4⋊C4—974no—
354C4⋊C4—974no—
355C4⋊C4—974no—
356C4⋊C4—974no—
357C4⋊C4—974no—
358C4⋊C4C22×D9714yes—
359C4⋊C4C22×D9714yes—
360C22⋊C4—974no—
361C22⋊C4—974no—
362C22⋊C4—974no—
363C22⋊C4—974no—
364C22⋊C4—974no—
365C22⋊C4C22×D9714yes—
366C22⋊C4C22×D9714yes—
367C22⋊C4C22×D9714yes—
368C42—974no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—974no—
375C23—972no—
376C23—972no—
377C23—1942no—
378C23—1942no—
379C23—1942no—
380C23—1942no—
381C23—1942no—
382C23D4×D9714yes—
383C23C23×D9712yes—
384C23C23×D9712yes—
385C23C23×D9711yescharacteristic
386Q8—1948no—
387Q8—1948no—
388D4—28no—
389D4—9716no—
390D4—1948no—
391D4—1948no—
392D4—1948no—
393D4—1948no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—28no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—974no—
401C2×C4—974no—
402C2×C4—972no—
403C2×C4—972no—
404C2×C4—972no—
405C2×C4—972no—
406C2×C4—972no—
407C2×C4—972no—
408C2×C4—1948no—
409C2×C4—1948no—
410C2×C4—1948no—
411C2×C4—1948no—
412C2×C4—1948no—
413C2×C4—1948no—
414C2×C4—1942no—
415C2×C4—1942no—
416C2×C4—1942no—
417C2×C4—1948no—
418C2×C4—1942no—
419C2×C4C23×D9712yes—
420C2×C4C23×D9712yes—
421C2×C4C23×D9712yes—
422C2×C4C23×D9712yes—
423C2×C4C23×D9712yes—
424C22—28no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—44no—
431C22—974no—
432C22—3884no—
433C22—1942no—
434C22—1942no—
435C22—1942no—
436C22—1948no—
437C22—1942no—
438C22—1942no—
439C22C2×D4×D9714yes—
440C22C2×D4×D9712yes—
441C22#41×D9712yes—
442C22#1569085712yes—
443C22C24×D9711yescharacteristic
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—974no—
450C4—974no—
451C4—1942no—
452C4—1942no—
453C4—1942no—
454C4—1942no—
455C2—24no—
456C2—42no—
457C2—3882no—
458C2—1944no—
459C2#2026138314yesdirect product factor
460C2C22×D4×D9711yescharacteristic
461C2C2×#41×D9711yescharacteristic
462C2C2×#1569085711yescharacteristic
subgroupG/H
centerC23C23×D97
commutator, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC2×#123×C97C2
socleC23×C97C24
hypercenterC2×#123D97
hypocenterC97C2×#547
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C22×#13095511C2semidirect
2. C22×#13095510C2semidirect
3. #20261383C2direct (N × H)
4. C2×#15690809C2semidirect
5. C2×#15690795C2semidirect
6. C2×#15690794C2semidirect
7. C2×#15690792C2semidirect
8. C2×#15690788C2semidirect
9. C2×#15690787C2semidirect
10. C2×#15690783C2semidirect
11. C2×#15690782C2semidirect
12. C2×#15690770C2semidirect
13. C2×#15690770C2semidirect
14. C22×#13095490C2semidirect
15. C2×C4⋊C4×D97C2semidirect
16. C2×#123×C97C2semidirect
18. C2×#13095511C22semidirect
20. C2×#13095510C22semidirect
35. #15690809C22semidirect
36. #15690795C22semidirect
37. #15690794C22semidirect
38. #15690792C22semidirect
39. #15690788C22semidirect
40. #15690787C22semidirect
41. #15690783C22semidirect
42. #15690782C22semidirect
43. #15690770C22semidirect
44. #15690770C22semidirect
53. C2×#13095490C22semidirect
54. C4⋊C4×D97C22semidirect
63. #123×C97C22semidirect
82. C2×C22⋊D97D4semidirect
169. C22⋊D97C2×D4semidirect
178. C2×Dic97#41semidirect
222. Dic97C2×#41semidirect
225. C2×C97#547semidirect
232. C97C2×#547semidirect
240. C2×#123D97semidirect
273. #123C2×D97semidirect
459. C2#20261383direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130955112
C22×#130955102
#202613832
C2×#156908092
C2×#156907952
C2×#156907942
C2×#156907922
C2×#156907882
C2×#156907872
C2×#156907832
C2×#156907822
C2×#156907702
C22×#130954902
C2×C4⋊C4×D972
C2×#123×C972
C2×#54797