Finite Group Databasebeta

← C2×#31827169 #49134789

Subgroups — C2×#31827169 #49134789

Order 36224.

1200 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827169C2116yesmaximal, direct product factor
2C2×#22583608C211yesmaximal, characteristic, complemented
3C2×#22583600C211yesmaximal, characteristic, complemented
4C2×#22583600C211yesmaximal, characteristic, complemented
5C2×#22583588C211yesmaximal, characteristic, complemented
6C2×#22583588C211yesmaximal, characteristic, complemented
7C2×#22583587C211yesmaximal, characteristic, complemented
8C2×#22583587C211yesmaximal, characteristic, complemented
9C2×#22583586C211yesmaximal, characteristic, complemented
10C2×#22583575C211yesmaximal, characteristic, complemented
11C2×C4×C22⋊D283C211yesmaximal, characteristic
12C2×(C22⋊D283)⋊C4C211yesmaximal, characteristic
13C2×(C22⋊D283)⋊C4C211yesmaximal, characteristic
14C22×Dic283⋊C4C211yesmaximal, characteristic, complemented
15C2×C4⋊C4×D283C211yesmaximal, characteristic, complemented
16C2×C22⋊Q8×C283C211yesmaximal, characteristic, complemented
17C22×Dic566C2211yescharacteristic
18C22×Dic566C2211yescharacteristic
19C22×C22⋊D283C2211yescharacteristic
20C23×Dic283C2211yescharacteristic
21C23×Dic283C2211yescharacteristic
22C22×C4×D283C2211yescharacteristic
23C22×C4×D283C2211yescharacteristic
24C22×C4×D283C2211yescharacteristic
25#22583608C2218yescomplemented
26#22583600C2218yescomplemented
27#22583600C2218yescomplemented
28#22583588C2218yescomplemented
29#22583588C2218yescomplemented
30#22583587C2218yescomplemented
31#22583587C2218yescomplemented
32#22583586C2218yescomplemented
33#22583575C2218yescomplemented
34C4×C22⋊D283C2218yes—
35(C22⋊D283)⋊C4C2218yes—
36(C22⋊D283)⋊C4C2218yes—
37C2×C4⋊Dic283C2211yescharacteristic
38C2×C4⋊Dic283C2211yescharacteristic
39C2×Dic283⋊C4C2211yescharacteristic
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⋊C4C2211yescharacteristic
49C2×Dic283⋊C4C2211yescharacteristic
50C2×Dic283⋊C4C2218yescomplemented
51C4⋊C4×D283C2218yescomplemented
52C2×C22⋊Dic283C2211yescharacteristic
53C2×(C2×D283)⋊C4C2211yescharacteristic
54C2×(C2×D283)⋊C4C2211yescharacteristic
55C2×(C2×D283)⋊C4C2211yescharacteristic
56C2×C4×Dic283C2211yescharacteristic
57C2×C4×Dic283C2211yescharacteristic
58C2×C4×Dic283C2211yescharacteristic
59C22×Q8×C283C2211yescharacteristic
60C23×C4×C283C2211yescharacteristic
61C22⋊Q8×C283C2218yescomplemented
62C2×C4⋊C4×C283C2211yescharacteristic
63C2×C4⋊C4×C283C2211yescharacteristic
64C2×C4⋊C4×C283C2211yescharacteristic
65C2×C22⋊C4×C283C2211yescharacteristic
66C2×C22⋊C4×C283C2211yescharacteristic
67C23×D283C2311yescharacteristic
68C2×Dic566—24no—
69C2×Dic566—42no—
70C2×Dic566C2314yes—
71C2×Dic566C2314yes—
72C2×C22⋊D283Q818yescomplemented
73C2×C22⋊D283C2314yes—
74C22×Dic283—24no—
75C22×Dic283—42no—
76C22×Dic283C2311yescharacteristic
77C22×Dic283C2311yescharacteristic
78C22×Dic283C2311yescharacteristic
79C22×Dic283C2311yescharacteristic
80C22×Dic283C2311yescharacteristic
81C22×Dic283C2314yes—
82C22×Dic283C2314yes—
83C22×Dic283C2311yescharacteristic
84C22×Dic283C2311yescharacteristic
85C2×C4×D283—24no—
86C2×C4×D283—42no—
87C2×C4×D283—42no—
88C2×C4×D283C2314yes—
89C2×C4×D283C2314yes—
90C2×C4×D283C2314yes—
91C4⋊Dic283C2314yes—
92C4⋊Dic283C2314yes—
93Dic283⋊C4C2314yes—
94Dic283⋊C4C2314yes—
95Dic283⋊C4C2314yes—
96Dic283⋊C4C2314yes—
97Dic283⋊C4C2314yes—
98Dic283⋊C4C2314yes—
99Dic283⋊C4C2314yes—
100Dic283⋊C4C2314yes—
101Dic283⋊C4C2314yes—
102Dic283⋊C4C2314yes—
103Dic283⋊C4C2314yes—
104C22⋊Dic283C2314yes—
105(C2×D283)⋊C4C2314yes—
106(C2×D283)⋊C4C2314yes—
107(C2×D283)⋊C4C2314yes—
108C4×Dic283C2314yes—
109C4×Dic283C2314yes—
110C4×Dic283C2314yes—
111C24×C283C2311yescharacteristic
112C2×Q8×C283—42no—
113C2×Q8×C283C2314yes—
114C22×C4×C283—42no—
115C22×C4×C283C2311yescharacteristic
116C22×C4×C283C2311yescharacteristic
117C22×C4×C283C2311yescharacteristic
118C22×C4×C283C2311yescharacteristic
119C22×C4×C283C2311yescharacteristic
120C22×C4×C283C2311yescharacteristic
121C22×C4×C283C2314yes—
122C4⋊C4×C283C2314yes—
123C4⋊C4×C283C2314yes—
124C4⋊C4×C283C2314yes—
125C22⋊C4×C283C2314yes—
126C22⋊C4×C283C2314yes—
127C22×D283—22no—
128C22×D283—22no—
129C22×D283C2×Q814yes—
130C22×D283C2412yes—
131Dic566—28no—
132Dic566—44no—
133C22⋊D283C2×Q8116yescomplemented
134C2×Dic283—22no—
135C2×Dic283—22no—
136C2×Dic283—22no—
137C2×Dic283—22no—
138C2×Dic283—22no—
139C2×Dic283—28no—
140C2×Dic283—44no—
141C2×Dic283C2×Q814yes—
142C2×Dic283#4114yes—
143C2×Dic283C2412yes—
144C2×Dic283C2412yes—
145C2×Dic283C2412yes—
146C2×Dic283C2412yes—
147C2×Dic283C2412yes—
148C2×Dic283C2412yes—
149C2×Dic283C2412yes—
150C4×D283—28no—
151C4×D283—44no—
152C4×D283—44no—
153C23×C283—22no—
154C23×C283—22no—
155C23×C283C2×Q814yes—
156C23×C283C2412yes—
157C23×C283C2411yescharacteristic
158Q8×C283—44no—
159C2×C4×C283—22no—
160C2×C4×C283—22no—
161C2×C4×C283—22no—
162C2×C4×C283—22no—
163C2×C4×C283—22no—
164C2×C4×C283—22no—
165C2×C4×C283—44no—
166C2×C4×C283C2412yes—
167C2×C4×C283C2412yes—
168C2×C4×C283C2412yes—
169C2×C4×C283C2412yes—
170C2×C4×C283C2412yes—
171C2×C4×C283C2412yes—
172C22×C283—28no—
173C22×C283—22no—
174C22×C283—22no—
175C22×C283C22×Q814yes—
176C22×C283C22×Q812yes—
177C22×C283C2×#4112yes—
178C22×C283H(5,𝔽2)12yes—
179C22×C283C2511yescharacteristic
180C2×D283—28no—
181C2×D283—22no—
182C2×D283—22no—
183C2×D283C22×Q814yes—
184C4×C283—22no—
185C4×C283—22no—
186C4×C283—22no—
187C4×C283—22no—
188C4×C283—22no—
189C4×C283—22no—
190Dic283—22no—
191Dic283—22no—
192Dic283—22no—
193Dic283—22no—
194Dic283—22no—
195Dic283C22×Q814yes—
196Dic283C2×#4114yes—
197C2×C283—24no—
198C2×C283#55414yescomplemented
199C2×C283C23×Q811yescharacteristic
200C2×C283C22×#4111yescharacteristic
201C2×C283C2×H(5,𝔽2)11yescharacteristic
202D283—24no—
203C283C2×#55411yescharacteristic, complemented, Sylow
204C2×#554—2831nomaximal, Sylow
205#554—28316no—
206C2×C4⋊Q8—2831no—
207C2×#125—2831no—
208C2×#125—2831no—
209C2×C22⋊Q8—2831no—
210C2×C22⋊Q8—2831no—
211C2×C22⋊Q8—2831no—
212C2×C22⋊Q8—2831no—
213C2×C22⋊Q8—2831no—
214C2×C22⋊Q8D28311yescharacteristic, complemented
215C2×C4×Q8—2831no—
216C2×C4×D4—2831no—
217C2×C4×D4—2831no—
218C2×C4×D4—2831no—
219C22×C4⋊C4—2831no—
220C22×C4⋊C4—2831no—
221C22×Q8—2831no—
222C22×Q8—2831no—
223C22×Q8C2×D28311yescharacteristic
224C22×D4—2831no—
225C23×C4—2831no—
226C23×C4—2831no—
227C23×C4—2831no—
228C23×C4—2831no—
229C23×C4—2831no—
230C23×C4C2×D28311yescharacteristic
231C4⋊Q8—2838no—
232#125—2838no—
233#125—2838no—
234C22⋊Q8—2838no—
235C22⋊Q8—2838no—
236C22⋊Q8—2838no—
237C22⋊Q8—2838no—
238C22⋊Q8—2838no—
239C22⋊Q8C2×D28318yescomplemented
240C4×Q8—2838no—
241C4×D4—2838no—
242C4×D4—2838no—
243C4×D4—2838no—
244C2×C4⋊C4—2831no—
245C2×C4⋊C4—2831no—
246C2×C4⋊C4—2831no—
247C2×C4⋊C4—2831no—
248C2×C4⋊C4—2831no—
249C2×C4⋊C4—2831no—
250C2×C4⋊C4—2838no—
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⋊C4—2831no—
258C2×C4⋊C4—2838no—
259C2×C4⋊C4C2×D28311yescharacteristic
260C2×C4⋊C4C2×D28311yescharacteristic
261C2×C4⋊C4C2×D28311yescharacteristic
262C2×C22⋊C4—2831no—
263C2×C22⋊C4—2831no—
264C2×C22⋊C4—2831no—
265C2×C22⋊C4—2831no—
266C2×C22⋊C4C2×D28311yescharacteristic
267C2×C22⋊C4C2×D28311yescharacteristic
268C2×C42—2831no—
269C2×C42—2831no—
270C2×C42—2831no—
271C24—2831no—
272C24C22×D28311yescharacteristic
273C2×Q8—42no—
274C2×Q8—2834no—
275C2×Q8—2834no—
276C2×Q8—5664no—
277C2×Q8—11322no—
278C2×Q8C22×D28314yes—
279C2×D4—2838no—
280C2×D4—2834no—
281C22×C4—42no—
282C22×C4—2831no—
283C22×C4—2831no—
284C22×C4—2834no—
285C22×C4—2831no—
286C22×C4—2831no—
287C22×C4—2831no—
288C22×C4—2831no—
289C22×C4—2831no—
290C22×C4—2834no—
291C22×C4—2834no—
292C22×C4—2834no—
293C22×C4—2834no—
294C22×C4—5664no—
295C22×C4—5664no—
296C22×C4—11322no—
297C22×C4—11322no—
298C22×C4—11322no—
299C22×C4C22×D28311yescharacteristic
300C22×C4C22×D28311yescharacteristic
301C22×C4C22×D28311yescharacteristic
302C22×C4C22×D28311yescharacteristic
303C22×C4C22×D28311yescharacteristic
304C22×C4C22×D28311yescharacteristic
305C22×C4C22×D28314yes—
306C4⋊C4—2834no—
307C4⋊C4—2834no—
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⋊C4—2834no—
317C4⋊C4—2834no—
318C4⋊C4—2834no—
319C4⋊C4C22×D28314yes—
320C4⋊C4C22×D28314yes—
321C4⋊C4C22×D28314yes—
322C22⋊C4—2834no—
323C22⋊C4—2834no—
324C22⋊C4—2834no—
325C22⋊C4—2834no—
326C22⋊C4C22×D28314yes—
327C22⋊C4C22×D28314yes—
328C42—2834no—
329C42—2834no—
330C42—2834no—
331C23—22no—
332C23—22no—
333C23—2834no—
334C23—2832no—
335C23—5662no—
336C23—5662no—
337C23Q8×D28314yes—
338C23C23×D28312yes—
339C23C23×D28311yescharacteristic
340Q8—44no—
341Q8—5668no—
342Q8—11324no—
343D4—28316no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—44no—
351C2×C4—2834no—
352C2×C4—2834no—
353C2×C4—2832no—
354C2×C4—2832no—
355C2×C4—2832no—
356C2×C4—2832no—
357C2×C4—2832no—
358C2×C4—2832no—
359C2×C4—2832no—
360C2×C4—5668no—
361C2×C4—5668no—
362C2×C4—5662no—
363C2×C4—5662no—
364C2×C4—5662no—
365C2×C4—5662no—
366C2×C4—5662no—
367C2×C4—11324no—
368C2×C4—11324no—
369C2×C4—11324no—
370C2×C4C23×D28312yes—
371C2×C4C23×D28312yes—
372C2×C4C23×D28312yes—
373C2×C4C23×D28312yes—
374C2×C4C23×D28312yes—
375C2×C4C23×D28312yes—
376C22—22no—
377C22—22no—
378C22—28no—
379C22—2834no—
380C22—5662no—
381C22—5662no—
382C22—5668no—
383C22C2×Q8×D28314yes—
384C22C2×Q8×D28312yes—
385C22#41×D28312yes—
386C22#2258365512yes—
387C22C24×D28311yescharacteristic
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—2834no—
395C4—2834no—
396C4—5662no—
397C4—5662no—
398C4—5662no—
399C4—5662no—
400C4—5662no—
401C2—24no—
402C2—5664no—
403C2#3182716914yesdirect product factor
404C2C22×Q8×D28311yescharacteristic
405C2C2×#41×D28311yescharacteristic
406C2C2×#2258365511yescharacteristic
subgroupG/H
centerC23C23×D283
commutator, cosocleC22×C283C25
FrattiniC22C24×D283
FittingC2×C22⋊Q8×C283C2
socleC23×C283C24
prime radicalC22×C22⋊D283C22
hypercenterC2×C22⋊Q8D283
hypocenterC283C2×#554
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. #31827169C2direct (N × H)
2. C2×#22583608C2semidirect
3. C2×#22583600C2semidirect
4. C2×#22583600C2semidirect
5. C2×#22583588C2semidirect
6. C2×#22583588C2semidirect
7. C2×#22583587C2semidirect
8. C2×#22583587C2semidirect
9. C2×#22583586C2semidirect
10. C2×#22583575C2semidirect
14. C22×Dic283⋊C4C2semidirect
15. C2×C4⋊C4×D283C2semidirect
16. C2×C22⋊Q8×C283C2semidirect
25. #22583608C22semidirect
26. #22583600C22semidirect
27. #22583600C22semidirect
28. #22583588C22semidirect
29. #22583588C22semidirect
30. #22583587C22semidirect
31. #22583587C22semidirect
32. #22583586C22semidirect
33. #22583575C22semidirect
50. C2×Dic283⋊C4C22semidirect
51. C4⋊C4×D283C22semidirect
61. C22⋊Q8×C283C22semidirect
72. C2×C22⋊D283Q8semidirect
133. C22⋊D283C2×Q8semidirect
198. C2×C283#554semidirect
203. C283C2×#554semidirect
214. C2×C22⋊Q8D283semidirect
239. C22⋊Q8C2×D283semidirect
403. C2#31827169direct (N × H)

Maximal subgroups

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

subgroupindex
#318271692
C2×#225836082
C2×#225836002
C2×#225835882
C2×#225835872
C2×#225835862
C2×#225835752
C2×C4×C22⋊D2832
C2×(C22⋊D283)⋊C42
C22×Dic283⋊C42
C2×C4⋊C4×D2832
C2×C22⋊Q8×C2832
C2×#554283