Finite Group Databasebeta

← C2×#31827083 #49134636

Subgroups — C2×#31827083 #49134636

Order 36224.

1132 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827083C2116yesmaximal, direct product factor
2C2×#22583609C211yesmaximal, characteristic, complemented
3C2×#22583605C211yesmaximal, characteristic, complemented
4C2×#22583603C211yesmaximal, characteristic, complemented
5C2×#22583597C211yesmaximal, characteristic, complemented
6C2×#22583596C211yesmaximal, characteristic, complemented
7C2×#22583596C211yesmaximal, characteristic, complemented
8C2×#22583592C211yesmaximal, characteristic, complemented
9C2×#22583589C211yesmaximal, characteristic, complemented
10C2×#22583588C211yesmaximal, characteristic, complemented
11C2×#22583587C211yesmaximal, characteristic, complemented
12C2×#22583582C211yesmaximal, characteristic
13C2×#22583575C211yesmaximal, characteristic, complemented
14C2×C4×C22⋊D283C211yesmaximal, characteristic, complemented
15C2×#22583566C211yesmaximal, characteristic, complemented
16C2×#123×C283C211yesmaximal, characteristic, complemented
17C22×Dic566C2211yescharacteristic
18C22×Dic566C2211yescharacteristic
19C22×Dic566C2211yescharacteristic
20C22×C22⋊D283C2211yescharacteristic
21C22×C22⋊D283C2211yescharacteristic
22C23×Dic283C2211yescharacteristic
23C22×C4×D283C2211yescharacteristic
24#22583609C2218yescomplemented
25#22583605C2218yescomplemented
26#22583603C2218yescomplemented
27#22583597C2218yescomplemented
28#22583596C2218yescomplemented
29#22583596C2218yescomplemented
30#22583592C2218yescomplemented
31#22583589C2218yescomplemented
32#22583588C2218yescomplemented
33#22583587C2218yescomplemented
34#22583582C2218yes—
35#22583575C2218yescomplemented
36C4×C22⋊D283C2218yescomplemented
37#22583566C2218yescomplemented
38C2×C4⋊Dic283C2211yescharacteristic
39C2×C4⋊Dic283C2211yescharacteristic
40C2×Dic283⋊C4C2211yescharacteristic
41C2×Dic283⋊C4C2211yescharacteristic
42C2×Dic283⋊C4C2211yescharacteristic
43C2×Dic283⋊C4C2211yescharacteristic
44C2×Dic283⋊C4C2211yescharacteristic
45C2×Dic283⋊C4C2211yescharacteristic
46C2×C22⋊Dic283C2211yescharacteristic
47C2×C22⋊Dic283C2211yescharacteristic
48C2×C22⋊Dic283C2211yescharacteristic
49C2×C22⋊Dic283C2211yescharacteristic
50C2×C22⋊Dic283C2211yescharacteristic
51C2×(C2×D283)⋊C4C2211yescharacteristic
52C2×(C2×D283)⋊C4C2211yescharacteristic
53C2×(C2×D283)⋊C4C2211yescharacteristic
54C2×(C2×D283)⋊C4C2211yescharacteristic
55C2×C4×Dic283C2211yescharacteristic
56C2×C4×Dic283C2211yescharacteristic
57C2×C4×Dic283C2211yescharacteristic
58C2×C4×Dic283C2211yescharacteristic
59C22×D4×C283C2211yescharacteristic
60C23×C4×C283C2211yescharacteristic
61#123×C283C2218yescomplemented
62C2×C4⋊C4×C283C2211yescharacteristic
63C2×C4⋊C4×C283C2211yescharacteristic
64C2×C22⋊C4×C283C2211yescharacteristic
65C2×C22⋊C4×C283C2211yescharacteristic
66C2×C22⋊C4×C283C2211yescharacteristic
67C23×D283C2311yescharacteristic
68C2×Dic566—24no—
69C2×Dic566—42no—
70C2×Dic566—42no—
71C2×Dic566C2314yes—
72C2×Dic566C2314yes—
73C2×Dic566C2314yes—
74C2×C22⋊D283—24no—
75C2×C22⋊D283—42no—
76C2×C22⋊D283C2314yes—
77C2×C22⋊D283C2314yes—
78C22×Dic283—24no—
79C22×Dic283C2311yescharacteristic
80C22×Dic283C2311yescharacteristic
81C22×Dic283C2311yescharacteristic
82C22×Dic283C2311yescharacteristic
83C22×Dic283C2311yescharacteristic
84C22×Dic283C2311yescharacteristic
85C22×Dic283C2314yes—
86C22×Dic283C2311yescharacteristic
87C2×C4×D283—42no—
88C2×C4×D283C2314yes—
89C4⋊Dic283C2314yes—
90C4⋊Dic283C2314yes—
91Dic283⋊C4C2314yes—
92Dic283⋊C4C2314yes—
93Dic283⋊C4C2314yes—
94Dic283⋊C4C2314yes—
95Dic283⋊C4C2314yes—
96Dic283⋊C4C2314yes—
97C22⋊Dic283C2314yes—
98C22⋊Dic283C2314yes—
99C22⋊Dic283C2314yes—
100C22⋊Dic283C2314yes—
101C22⋊Dic283C2314yes—
102(C2×D283)⋊C4C2314yes—
103(C2×D283)⋊C4C2314yes—
104(C2×D283)⋊C4C2314yes—
105(C2×D283)⋊C4C2314yes—
106C4×Dic283C2314yes—
107C4×Dic283C2314yes—
108C4×Dic283C2314yes—
109C4×Dic283C2314yes—
110C24×C283C2311yescharacteristic
111C24×C283C2311yescharacteristic
112C2×D4×C283—42no—
113C2×D4×C283C2314yes—
114C22×C4×C283—42no—
115C22×C4×C283C2311yescharacteristic
116C22×C4×C283C2311yescharacteristic
117C22×C4×C283C2311yescharacteristic
118C22×C4×C283C2311yescharacteristic
119C22×C4×C283C2311yescharacteristic
120C22×C4×C283C2314yes—
121C4⋊C4×C283C2314yes—
122C4⋊C4×C283C2314yes—
123C22⋊C4×C283C2314yes—
124C22⋊C4×C283C2314yes—
125C22⋊C4×C283C2314yes—
126C22×D283—22no—
127C22×D283—22no—
128C22×D283—22no—
129C22×D283C2412yes—
130Dic566—28no—
131Dic566—44no—
132Dic566—44no—
133C22⋊D283—28no—
134C22⋊D283—44no—
135C2×Dic283—22no—
136C2×Dic283—22no—
137C2×Dic283—22no—
138C2×Dic283—22no—
139C2×Dic283—22no—
140C2×Dic283—22no—
141C2×Dic283—28no—
142C2×Dic283#4114yes—
143C2×Dic283C2412yes—
144C2×Dic283C2412yes—
145C2×Dic283C2412yes—
146C2×Dic283C2412yes—
147C2×Dic283C2412yes—
148C2×Dic283C2412yes—
149C2×Dic283C2412yes—
150C4×D283—44no—
151C23×C283—22no—
152C23×C283—22no—
153C23×C283—22no—
154C23×C283—22no—
155C23×C283—22no—
156C23×C283—22no—
157C23×C283C2412yes—
158C23×C283C2412yes—
159C23×C283C2411yescharacteristic
160D4×C283—44no—
161C2×C4×C283—22no—
162C2×C4×C283—22no—
163C2×C4×C283—22no—
164C2×C4×C283—22no—
165C2×C4×C283—22no—
166C2×C4×C283—44no—
167C2×C4×C283C2412yes—
168C2×C4×C283C2412yes—
169C2×C4×C283C2412yes—
170C2×C4×C283C2412yes—
171C2×C4×C283C2412yes—
172C22×C283—22no—
173C22×C283—22no—
174C22×C283—22no—
175C22×C283—22no—
176C22×C283—22no—
177C22×C283—22no—
178C22×C283—44no—
179C22×C283—44no—
180C22×C283C2×#4112yes—
181C22×C283H(5,𝔽2)12yes—
182C22×C2832−1+412yes—
183C22×C283C2511yescharacteristic
184C2×D283—22no—
185C2×D283—22no—
186C2×D283—22no—
187C2×D283—44no—
188C4×C283—22no—
189C4×C283—22no—
190C4×C283—22no—
191C4×C283—22no—
192C4×C283—22no—
193Dic283—22no—
194Dic283—22no—
195Dic283—22no—
196Dic283—22no—
197Dic283—22no—
198Dic283—22no—
199Dic283C2×#4114yes—
200C2×C283—42no—
201C2×C283—42no—
202C2×C283#54214yescomplemented
203C2×C283C22×#4111yescharacteristic
204C2×C283C2×H(5,𝔽2)11yescharacteristic
205C2×C283C2×2−1+411yescharacteristic
206D283—42no—
207C283C2×#54211yescharacteristic, complemented, Sylow
208C2×#542—2831nomaximal, Sylow
209#542—28316no—
210C2×C4⋊Q8—2831no—
211C2×#126—2831no—
212C2×#126—2831no—
213C2×#124—2831no—
214C2×#124—2831no—
215C2×#124—2831no—
216C2×#123—2831no—
217C2×#123D28311yescharacteristic, complemented
218C2×C22⋊Q8—2831no—
219C2×C22⋊Q8—2831no—
220C2×C22⋊Q8—2831no—
221C2×#121—2831no—
222C2×C4×Q8—2831no—
223C2×C4×D4—2831no—
224C2×#117—2831no—
225C22×Q8—2831no—
226C22×Q8—2831no—
227C22×Q8—2831no—
228C22×D4—2831no—
229C22×D4—2831no—
230C22×D4C2×D28311yescharacteristic
231C23×C4—2831no—
232C23×C4—2831no—
233C23×C4C2×D28311yescharacteristic
234C4⋊Q8—2838no—
235#126—2838no—
236#126—2838no—
237#124—2838no—
238#124—2838no—
239#124—2838no—
240#123—2838no—
241#123C2×D28318yescomplemented
242C22⋊Q8—2838no—
243C22⋊Q8—2838no—
244C22⋊Q8—2838no—
245#121—2838no—
246C4×Q8—2838no—
247C4×D4—2838no—
248#117—2838no—
249C2×C4⋊C4—2831no—
250C2×C4⋊C4—2831no—
251C2×C4⋊C4—2831no—
252C2×C4⋊C4—2831no—
253C2×C4⋊C4—2831no—
254C2×C4⋊C4—2831no—
255C2×C4⋊C4—2831no—
256C2×C4⋊C4—2831no—
257C2×C4⋊C4C2×D28311yescharacteristic
258C2×C4⋊C4C2×D28311yescharacteristic
259C2×C22⋊C4—2831no—
260C2×C22⋊C4—2831no—
261C2×C22⋊C4—2831no—
262C2×C22⋊C4—2831no—
263C2×C22⋊C4—2831no—
264C2×C22⋊C4—2831no—
265C2×C22⋊C4—2831no—
266C2×C22⋊C4—2831no—
267C2×C22⋊C4—2831no—
268C2×C22⋊C4C2×D28311yescharacteristic
269C2×C22⋊C4C2×D28311yescharacteristic
270C2×C22⋊C4C2×D28311yescharacteristic
271C2×C42—2831no—
272C2×C42—2831no—
273C2×C42—2831no—
274C2×C42—2831no—
275C24—2831no—
276C24C22×D28311yescharacteristic
277C24C22×D28311yescharacteristic
278C2×Q8—2834no—
279C2×Q8—2834no—
280C2×Q8—2834no—
281C2×Q8—5664no—
282C2×Q8—11322no—
283C2×Q8—11322no—
284C2×D4—42no—
285C2×D4—2834no—
286C2×D4—2834no—
287C2×D4—5664no—
288C2×D4—11322no—
289C2×D4C22×D28314yes—
290C22×C4—42no—
291C22×C4—2831no—
292C22×C4—2834no—
293C22×C4—2831no—
294C22×C4—2831no—
295C22×C4—2831no—
296C22×C4—2831no—
297C22×C4—2831no—
298C22×C4—2831no—
299C22×C4—2834no—
300C22×C4—5664no—
301C22×C4—11322no—
302C22×C4C22×D28311yescharacteristic
303C22×C4C22×D28311yescharacteristic
304C22×C4C22×D28311yescharacteristic
305C22×C4C22×D28311yescharacteristic
306C22×C4C22×D28311yescharacteristic
307C22×C4C22×D28314yes—
308C4⋊C4—2834no—
309C4⋊C4—2834no—
310C4⋊C4—2834no—
311C4⋊C4—2834no—
312C4⋊C4—2834no—
313C4⋊C4—2834no—
314C4⋊C4—2834no—
315C4⋊C4—2834no—
316C4⋊C4C22×D28314yes—
317C4⋊C4C22×D28314yes—
318C22⋊C4—2834no—
319C22⋊C4—2834no—
320C22⋊C4—2834no—
321C22⋊C4—2834no—
322C22⋊C4—2834no—
323C22⋊C4—2834no—
324C22⋊C4—2834no—
325C22⋊C4—2834no—
326C22⋊C4—2834no—
327C22⋊C4C22×D28314yes—
328C22⋊C4C22×D28314yes—
329C22⋊C4C22×D28314yes—
330C42—2834no—
331C42—2834no—
332C42—2834no—
333C42—2834no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—2832no—
341C23—5662no—
342C23—5662no—
343C23—5662no—
344C23C23×D28312yes—
345C23C23×D28312yes—
346C23C23×D28311yescharacteristic
347Q8—5668no—
348Q8—11324no—
349Q8—11324no—
350D4—44no—
351D4—5668no—
352D4—11324no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—44no—
359C2×C4—2834no—
360C2×C4—2832no—
361C2×C4—2832no—
362C2×C4—2832no—
363C2×C4—2832no—
364C2×C4—2832no—
365C2×C4—2832no—
366C2×C4—2832no—
367C2×C4—5668no—
368C2×C4—5662no—
369C2×C4—5662no—
370C2×C4—5662no—
371C2×C4—5662no—
372C2×C4—5662no—
373C2×C4—5662no—
374C2×C4—11324no—
375C2×C4C23×D28312yes—
376C2×C4C23×D28312yes—
377C2×C4C23×D28312yes—
378C2×C4C23×D28312yes—
379C2×C4C23×D28312yes—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—44no—
387C22—44no—
388C22—5662no—
389C22—5662no—
390C22—5662no—
391C22—11324no—
392C22#41×D28312yes—
393C22#2258365512yes—
394C22#2258365712yes—
395C22C24×D28311yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—2834no—
402C4—5662no—
403C4—5662no—
404C4—5662no—
405C4—5662no—
406C4—5662no—
407C4—5662no—
408C2—42no—
409C2—42no—
410C2—11322no—
411C2#3182708314yesdirect product factor
412C2C2×#41×D28311yescharacteristic
413C2C2×#2258365511yescharacteristic
414C2C2×#2258365711yescharacteristic
subgroupG/H
centerC23C23×D283
commutator, cosocleC22×C283C25
FrattiniC22C24×D283
FittingC2×#123×C283C2
socleC23×C283C24
prime radicalC2×#22583582C2
hypercenterC2×#123D283
hypocenterC283C2×#542
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. #31827083C2direct (N × H)
2. C2×#22583609C2semidirect
3. C2×#22583605C2semidirect
4. C2×#22583603C2semidirect
5. C2×#22583597C2semidirect
6. C2×#22583596C2semidirect
7. C2×#22583596C2semidirect
8. C2×#22583592C2semidirect
9. C2×#22583589C2semidirect
10. C2×#22583588C2semidirect
11. C2×#22583587C2semidirect
13. C2×#22583575C2semidirect
14. C2×C4×C22⋊D283C2semidirect
15. C2×#22583566C2semidirect
16. C2×#123×C283C2semidirect
24. #22583609C22semidirect
25. #22583605C22semidirect
26. #22583603C22semidirect
27. #22583597C22semidirect
28. #22583596C22semidirect
29. #22583596C22semidirect
30. #22583592C22semidirect
31. #22583589C22semidirect
32. #22583588C22semidirect
33. #22583587C22semidirect
35. #22583575C22semidirect
36. C4×C22⋊D283C22semidirect
37. #22583566C22semidirect
61. #123×C283C22semidirect
202. C2×C283#542semidirect
207. C283C2×#542semidirect
217. C2×#123D283semidirect
241. #123C2×D283semidirect
411. C2#31827083direct (N × H)

Maximal subgroups

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

subgroupindex
#318270832
C2×#225836092
C2×#225836052
C2×#225836032
C2×#225835972
C2×#225835962
C2×#225835922
C2×#225835892
C2×#225835882
C2×#225835872
C2×#225835822
C2×#225835752
C2×C4×C22⋊D2832
C2×#225835662
C2×#123×C2832
C2×#542283