Finite Group Databasebeta

← C2×#31827164 #49134783

Subgroups — C2×#31827164 #49134783

Order 36224.

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#31827164C2116yesmaximal, direct product factor
2C2×#22583604C211yesmaximal, characteristic, complemented
3C2×#22583603C211yesmaximal, characteristic, complemented
4C2×#22583600C211yesmaximal, characteristic, complemented
5C2×#22583595C211yesmaximal, characteristic, complemented
6C2×#22583593C211yesmaximal, characteristic, complemented
7C2×#22583584C211yesmaximal, characteristic, complemented
8C2×#22583583C211yesmaximal, characteristic, complemented
9C2×#22583582C211yesmaximal, characteristic, complemented
10C2×#22583582C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊D283C211yesmaximal, characteristic, complemented
12C2×(C22⋊D283)⋊C4C211yesmaximal, characteristic, complemented
13C2×(C22⋊D283)⋊C4C211yesmaximal, characteristic, complemented
14C2×#22583567C211yesmaximal, characteristic, complemented
15C22×Dic283⋊C4C211yesmaximal, characteristic, complemented
16C2×C4×D4×C283C211yesmaximal, characteristic, complemented
17C22×D1132C2211yescharacteristic
18C22×C22⋊D283C2211yescharacteristic
19C22×C22⋊D283C2211yescharacteristic
20C22×C22⋊D283C2211yescharacteristic
21C22×C22⋊D283C2211yescharacteristic
22C23×Dic283C2211yescharacteristic
23C23×Dic283C2211yescharacteristic
24C22×C4×D283C2211yescharacteristic
25C22×C4×D283C2211yescharacteristic
26#22583604C2218yescomplemented
27#22583603C2218yescomplemented
28#22583600C2218yescomplemented
29#22583595C2218yescomplemented
30#22583593C2218yescomplemented
31#22583584C2218yescomplemented
32#22583583C2218yescomplemented
33#22583582C2218yescomplemented
34#22583582C2218yescomplemented
35C4×C22⋊D283C2218yescomplemented
36(C22⋊D283)⋊C4C2218yescomplemented
37(C22⋊D283)⋊C4C2218yescomplemented
38#22583567C2218yescomplemented
39C2×C4⋊Dic283C2211yescharacteristic
40C2×Dic283⋊C4C2211yescharacteristic
41C2×Dic283⋊C4C2211yescharacteristic
42C2×Dic283⋊C4C2211yescharacteristic
43C2×Dic283⋊C4C2211yescharacteristic
44C2×Dic283⋊C4C2211yescharacteristic
45C2×Dic283⋊C4C2211yescharacteristic
46C2×Dic283⋊C4C2211yescharacteristic
47C2×Dic283⋊C4C2211yescharacteristic
48C2×Dic283⋊C4C2218yescomplemented
49C2×C22⋊Dic283C2211yescharacteristic
50C2×C22⋊Dic283C2211yescharacteristic
51C2×(C2×D283)⋊C4C2211yescharacteristic
52C2×(C2×D283)⋊C4C2211yescharacteristic
53C2×(C2×D283)⋊C4C2211yescharacteristic
54C2×(C2×D283)⋊C4C2211yescharacteristic
55C2×(C2×D283)⋊C4C2211yescharacteristic
56C2×(C2×D283)⋊C4C2211yescharacteristic
57C2×C4×Dic283C2211yescharacteristic
58C2×C4×Dic283C2211yescharacteristic
59C22×D4×C283C2211yescharacteristic
60C23×C4×C283C2211yescharacteristic
61C23×C4×C283C2211yescharacteristic
62C4×D4×C283C2218yescomplemented
63C2×C4⋊C4×C283C2211yescharacteristic
64C2×C22⋊C4×C283C2211yescharacteristic
65C2×C22⋊C4×C283C2211yescharacteristic
66C2×C42×C283C2211yescharacteristic
67C23×D283C2311yescharacteristic
68C23×D283C2311yescharacteristic
69C2×D1132—42no—
70C2×D1132C2314yes—
71C2×C22⋊D283—24no—
72C2×C22⋊D283—24no—
73C2×C22⋊D283—24no—
74C2×C22⋊D283—42no—
75C2×C22⋊D283C2314yes—
76C2×C22⋊D283C2314yes—
77C2×C22⋊D283C2314yes—
78C2×C22⋊D283C2314yes—
79C22×Dic283—24no—
80C22×Dic283—42no—
81C22×Dic283C2311yescharacteristic
82C22×Dic283C2311yescharacteristic
83C22×Dic283C2311yescharacteristic
84C22×Dic283C2311yescharacteristic
85C22×Dic283C2311yescharacteristic
86C22×Dic283C2314yes—
87C22×Dic283C2314yes—
88C22×Dic283C2311yescharacteristic
89C2×C4×D283—24no—
90C2×C4×D283—42no—
91C2×C4×D283C2314yes—
92C2×C4×D283C2314yes—
93C4⋊Dic283C2314yes—
94Dic283⋊C4C2314yes—
95Dic283⋊C4C2314yes—
96Dic283⋊C4C2314yes—
97Dic283⋊C4C2314yes—
98Dic283⋊C4C2314yes—
99Dic283⋊C4C2314yes—
100Dic283⋊C4C2314yes—
101Dic283⋊C4C2314yes—
102C22⋊Dic283C2314yes—
103C22⋊Dic283C2314yes—
104(C2×D283)⋊C4C2314yes—
105(C2×D283)⋊C4C2314yes—
106(C2×D283)⋊C4C2314yes—
107(C2×D283)⋊C4C2314yes—
108(C2×D283)⋊C4C2314yes—
109(C2×D283)⋊C4C2314yes—
110C4×Dic283C2314yes—
111C4×Dic283C2314yes—
112C24×C283C2311yescharacteristic
113C24×C283C2311yescharacteristic
114C2×D4×C283—24no—
115C2×D4×C283C2314yes—
116C22×C4×C283—42no—
117C22×C4×C283—42no—
118C22×C4×C283C2311yescharacteristic
119C22×C4×C283C2311yescharacteristic
120C22×C4×C283C2311yescharacteristic
121C22×C4×C283C2311yescharacteristic
122C22×C4×C283C2311yescharacteristic
123C22×C4×C283C2314yes—
124C22×C4×C283C2314yes—
125C4⋊C4×C283C2314yes—
126C22⋊C4×C283C2314yes—
127C22⋊C4×C283C2314yes—
128C42×C283C2314yes—
129C22×D283—22no—
130C22×D283—22no—
131C22×D283—22no—
132C22×D283—22no—
133C22×D283—22no—
134C22×D283—22no—
135C22×D283C2412yes—
136C22×D283C2412yes—
137D1132—44no—
138C22⋊D283—28no—
139C22⋊D283—28no—
140C22⋊D283—28no—
141C22⋊D283—44no—
142C2×Dic283—22no—
143C2×Dic283—22no—
144C2×Dic283—22no—
145C2×Dic283—22no—
146C2×Dic283—22no—
147C2×Dic283—28no—
148C2×Dic283—44no—
149C2×Dic283#4114yes—
150C2×Dic283C2412yes—
151C2×Dic283C2412yes—
152C2×Dic283C2412yes—
153C2×Dic283C2412yes—
154C2×Dic283C2412yes—
155C2×Dic283C2412yes—
156C4×D283—28no—
157C4×D283—44no—
158C23×C283—22no—
159C23×C283—22no—
160C23×C283—22no—
161C23×C283—22no—
162C23×C283—22no—
163C23×C283#4114yes—
164C23×C283C2412yes—
165C23×C283C2412yes—
166C23×C283C2411yescharacteristic
167D4×C283—28no—
168C2×C4×C283—22no—
169C2×C4×C283—22no—
170C2×C4×C283—22no—
171C2×C4×C283—22no—
172C2×C4×C283—22no—
173C2×C4×C283—44no—
174C2×C4×C283—44no—
175C2×C4×C283C2412yes—
176C2×C4×C283C2412yes—
177C2×C4×C283C2412yes—
178C2×C4×C283C2412yes—
179C2×C4×C283C2412yes—
180C22×C283—28no—
181C22×C283—22no—
182C22×C283—22no—
183C22×C283—22no—
184C22×C283—22no—
185C22×C283—22no—
186C22×C283—44no—
187C22×C283C2×#4112yes—
188C22×C283C2×#4114yes—
189C22×C283C2×#4112yes—
190C22×C283H(5,𝔽2)12yes—
191C22×C283C2511yescharacteristic
192C2×D283—22no—
193C2×D283—22no—
194C2×D283—22no—
195C2×D283—22no—
196C2×D283—22no—
197C2×D283—22no—
198C2×D283—44no—
199C2×D283—44no—
200C4×C283—22no—
201C4×C283—22no—
202C4×C283—22no—
203C4×C283—22no—
204C4×C283—22no—
205Dic283—22no—
206Dic283—22no—
207Dic283—22no—
208Dic283—22no—
209Dic283—22no—
210Dic283C2×#4114yes—
211C2×C283—24no—
212C2×C283—42no—
213C2×C283#55314yescomplemented
214C2×C283C22×#4111yescharacteristic
215C2×C283C22×#4111yescharacteristic
216C2×C283C2×H(5,𝔽2)11yescharacteristic
217D283—42no—
218D283—42no—
219C283C2×#55311yescharacteristic, complemented, Sylow
220C2×#553—2831nomaximal, Sylow
221#553—28316no—
222C2×#126—2831no—
223C2×#126—2831no—
224C2×#125—2831no—
225C2×#123—2831no—
226C2×#123—2831no—
227C2×#121—2831no—
228C2×#121—2831no—
229C2×#121—2831no—
230C2×#121—2831no—
231C2×C4×D4—2831no—
232C2×C4×D4—2831no—
233C2×C4×D4—2831no—
234C2×C4×D4D28311yescharacteristic, complemented
235C2×#117—2831no—
236C22×C4⋊C4—2831no—
237C22×D4—2831no—
238C22×D4—2831no—
239C22×D4—2831no—
240C22×D4—2831no—
241C22×D4—2831no—
242C22×D4C2×D28311yescharacteristic
243C23×C4—2831no—
244C23×C4—2831no—
245C23×C4—2831no—
246C23×C4—2831no—
247C23×C4C2×D28311yescharacteristic
248C23×C4C2×D28311yescharacteristic
249#126—2838no—
250#126—2838no—
251#125—2838no—
252#123—2838no—
253#123—2838no—
254#121—2838no—
255#121—2838no—
256#121—2838no—
257#121—2838no—
258C4×D4—2838no—
259C4×D4—2838no—
260C4×D4—2838no—
261C4×D4C2×D28318yescomplemented
262#117—2838no—
263C2×C4⋊C4—2831no—
264C2×C4⋊C4—2831no—
265C2×C4⋊C4—2831no—
266C2×C4⋊C4—2831no—
267C2×C4⋊C4—2838no—
268C2×C4⋊C4—2831no—
269C2×C4⋊C4—2831no—
270C2×C4⋊C4—2831no—
271C2×C4⋊C4—2831no—
272C2×C4⋊C4—2831no—
273C2×C4⋊C4C2×D28311yescharacteristic
274C2×C22⋊C4—2831no—
275C2×C22⋊C4—2831no—
276C2×C22⋊C4—2831no—
277C2×C22⋊C4—2831no—
278C2×C22⋊C4—2831no—
279C2×C22⋊C4—2831no—
280C2×C22⋊C4—2831no—
281C2×C22⋊C4—2831no—
282C2×C22⋊C4C2×D28311yescharacteristic
283C2×C22⋊C4C2×D28311yescharacteristic
284C2×C42—2831no—
285C2×C42—2831no—
286C2×C42C2×D28311yescharacteristic
287C24—2831no—
288C24—2831no—
289C24C22×D28311yescharacteristic
290C24C22×D28311yescharacteristic
291C2×D4—24no—
292C2×D4—2834no—
293C2×D4—2834no—
294C2×D4—2834no—
295C2×D4—2834no—
296C2×D4—2834no—
297C2×D4—5664no—
298C2×D4—5664no—
299C2×D4—5664no—
300C2×D4—11322no—
301C2×D4—11322no—
302C2×D4C22×D28314yes—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—2831no—
306C22×C4—2834no—
307C22×C4—2831no—
308C22×C4—2831no—
309C22×C4—2831no—
310C22×C4—2831no—
311C22×C4—2831no—
312C22×C4—2834no—
313C22×C4—2834no—
314C22×C4—2834no—
315C22×C4—5664no—
316C22×C4—5664no—
317C22×C4—11322no—
318C22×C4—11322no—
319C22×C4C22×D28311yescharacteristic
320C22×C4C22×D28311yescharacteristic
321C22×C4C22×D28311yescharacteristic
322C22×C4C22×D28311yescharacteristic
323C22×C4C22×D28311yescharacteristic
324C22×C4C22×D28314yes—
325C22×C4C22×D28314yes—
326C4⋊C4—2834no—
327C4⋊C4—2834no—
328C4⋊C4—2834no—
329C4⋊C4—2834no—
330C4⋊C4—2834no—
331C4⋊C4—2834no—
332C4⋊C4—2834no—
333C4⋊C4—2834no—
334C4⋊C4—2834no—
335C4⋊C4C22×D28314yes—
336C22⋊C4—2834no—
337C22⋊C4—2834no—
338C22⋊C4—2834no—
339C22⋊C4—2834no—
340C22⋊C4—2834no—
341C22⋊C4—2834no—
342C22⋊C4—2834no—
343C22⋊C4—2834no—
344C22⋊C4C22×D28314yes—
345C22⋊C4C22×D28314yes—
346C42—2834no—
347C42—2834no—
348C42C22×D28314yes—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—2832no—
355C23—2832no—
356C23—5662no—
357C23—5662no—
358C23—5662no—
359C23—5662no—
360C23—5662no—
361C23—5662no—
362C23#1798539214yes—
363C23C23×D28312yes—
364C23C23×D28312yes—
365C23C23×D28311yescharacteristic
366D4—28no—
367D4—5668no—
368D4—5668no—
369D4—5668no—
370D4—11324no—
371D4—11324no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—2834no—
380C2×C4—2832no—
381C2×C4—2832no—
382C2×C4—2832no—
383C2×C4—2832no—
384C2×C4—2832no—
385C2×C4—2832no—
386C2×C4—5668no—
387C2×C4—5668no—
388C2×C4—5662no—
389C2×C4—5662no—
390C2×C4—5662no—
391C2×C4—5662no—
392C2×C4—5662no—
393C2×C4—11324no—
394C2×C4—11324no—
395C2×C4C23×D28312yes—
396C2×C4C23×D28312yes—
397C2×C4C23×D28312yes—
398C2×C4C23×D28312yes—
399C2×C4C23×D28312yes—
400C22—22no—
401C22—22no—
402C22—28no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—44no—
407C22—5662no—
408C22—5662no—
409C22—5662no—
410C22—5662no—
411C22—5662no—
412C22—5662no—
413C22—11324no—
414C22—11324no—
415C22#41×D28312yes—
416C22C2×#1798539214yes—
417C22C2×#1798539212yes—
418C22#2258365512yes—
419C22C24×D28311yescharacteristic
420C4—22no—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—2834no—
426C4—5662no—
427C4—5662no—
428C4—5662no—
429C4—5662no—
430C4—5662no—
431C2—24no—
432C2—42no—
433C2—11322no—
434C2—11322no—
435C2#3182716414yesdirect product factor
436C2C2×#41×D28311yescharacteristic
437C2C22×#1798539211yescharacteristic
438C2C2×#2258365511yescharacteristic
subgroupG/H
centerC23C23×D283
commutator, cosocleC22×C283C25
FrattiniC22C24×D283
FittingC2×C4×D4×C283C2
socleC23×C283C24
hypercenterC2×C4×D4D283
hypocenterC283C2×#553
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #31827164C2direct (N × H)
2. C2×#22583604C2semidirect
3. C2×#22583603C2semidirect
4. C2×#22583600C2semidirect
5. C2×#22583595C2semidirect
6. C2×#22583593C2semidirect
7. C2×#22583584C2semidirect
8. C2×#22583583C2semidirect
9. C2×#22583582C2semidirect
10. C2×#22583582C2semidirect
11. C2×C4×C22⋊D283C2semidirect
12. C2×(C22⋊D283)⋊C4C2semidirect
13. C2×(C22⋊D283)⋊C4C2semidirect
14. C2×#22583567C2semidirect
15. C22×Dic283⋊C4C2semidirect
16. C2×C4×D4×C283C2semidirect
26. #22583604C22semidirect
27. #22583603C22semidirect
28. #22583600C22semidirect
29. #22583595C22semidirect
30. #22583593C22semidirect
31. #22583584C22semidirect
32. #22583583C22semidirect
33. #22583582C22semidirect
34. #22583582C22semidirect
35. C4×C22⋊D283C22semidirect
36. (C22⋊D283)⋊C4C22semidirect
37. (C22⋊D283)⋊C4C22semidirect
38. #22583567C22semidirect
48. C2×Dic283⋊C4C22semidirect
62. C4×D4×C283C22semidirect
213. C2×C283#553semidirect
219. C283C2×#553semidirect
234. C2×C4×D4D283semidirect
261. C4×D4C2×D283semidirect
435. C2#31827164direct (N × H)

Maximal subgroups

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

subgroupindex
#318271642
C2×#225836042
C2×#225836032
C2×#225836002
C2×#225835952
C2×#225835932
C2×#225835842
C2×#225835832
C2×#225835822
C2×C4×C22⋊D2832
C2×(C22⋊D283)⋊C42
C2×#225835672
C22×Dic283⋊C42
C2×C4×D4×C2832
C2×#553283