Finite Group Databasebeta

← #49135345

Subgroups — #49135345

Order 36224.

1104 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827200C211yesmaximal, characteristic, complemented
2#31827198C212yesmaximal, complemented
3#31827195C211yesmaximal, characteristic, complemented
4#31827171C212yesmaximal, complemented
5#31827136C211yesmaximal, characteristic, complemented
6#31827125C212yesmaximal, complemented
7#31827117C214yesmaximal, complemented
8#31827116C212yesmaximal, complemented
9#31827088C212yesmaximal, complemented
10#31827085C214yesmaximal, complemented
11#31827016C211yesmaximal, characteristic, complemented
12#31827012C212yesmaximal, complemented
13#31827000C212yesmaximal, complemented
14#31826997C211yesmaximal, characteristic, complemented
15#31826946C212yesmaximal, complemented
16C4×Q8×D283C211yesmaximal, characteristic, complemented
17#554×C283C211yesmaximal, characteristic, complemented
18C2×#17985393C2212yes—
19C2×#17985392C2212yes—
20C2×#17985392C2214yes—
21C2×Q8×D283C2211yescharacteristic
22#22583609C2212yes—
23#22583609C2212yes—
24#22583608C2214yes—
25#22583607C2211yescharacteristic
26#22583606C2212yes—
27#22583605C2214yes—
28#22583603C2212yes—
29#22583601C2211yescharacteristic
30#22583599C2212yes—
31#22583599C2212yes—
32#22583598C2212yes—
33#22583597C2211yescharacteristic
34#22583596C2214yes—
35#22583596C2214yes—
36#22583595C2214yes—
37#22583595C2214yes—
38#22583593C2214yes—
39#22583592C2212yes—
40#22583592C2214yes—
41#22583590C2211yescharacteristic
42#22583590C2214yes—
43#22583588C2214yes—
44#22583588C2214yes—
45#22583585C2212yes—
46#22583585C2212yes—
47#22583584C2212yes—
48#22583583C2214yes—
49#22583583C2214yes—
50#22583581C2211yescharacteristic
51Q8×Dic283C2211yescharacteristic
52C4×Dic566C2211yescharacteristic
53C4×Dic566C2211yescharacteristic
54C4×Dic566C2212yes—
55#22583575C2211yescharacteristic
56#22583575C2212yes—
57#22583575C2212yes—
58#22583575C2212yes—
59C4×D1132C2212yes—
60#22583572C2212yes—
61#22583572C2212yes—
62#22583572C2214yes—
63#22583572C2212yes—
64#22583572C2212yes—
65C4×C22⋊D283C2214yes—
66C4×C22⋊D283C2214yes—
67C4×C22⋊D283C2212yes—
68#22583569C2212yes—
69#22583568C2214yes—
70C4⋊C4×D283C2214yes—
71C4⋊C4×D283C2211yescharacteristic
72C4⋊C4×D283C2212yes—
73C42×D283C2211yescharacteristic
74C42×D283C2212yes—
75C4⋊Q8×C283C2211yescharacteristic
76#125×C283C2212yes—
77C22⋊Q8×C283C2214yes—
78C22⋊Q8×C283C2212yes—
79C4×Q8×C283C2211yescharacteristic
80C4×D4×C283C2211yescharacteristic
81C4×D4×C283C2212yes—
82C2×C4⋊C4×C283C2212yes—
83#17985393—44no—
84#17985392—28no—
85#17985392—48no—
86C2×Dic566C2311yescharacteristic
87C2×Dic566C2312yes—
88C2×Dic566C2314yes—
89C2×Dic566C2311yescharacteristic
90Q8×D283—24no—
91C2×D1132C2312yes—
92C2×D1132C2312yes—
93C2×D1132C2314yes—
94C2×D1132C2312yes—
95C2×C22⋊D283C2314yes—
96C2×C22⋊D283C2312yes—
97C2×C22⋊D283C2314yes—
98C2×C4×D283C2311yescharacteristic
99C2×C4×D283C2311yescharacteristic
100C2×C4×D283C2312yes—
101C2×C4×D283C2311yescharacteristic
102C2×C4×D283C2312yes—
103C2×C4×D283C2314yes—
104C2×C4×D283C2314yes—
105C2×C4×D283C2314yes—
106C4⋊Dic283C2312yes—
107C4⋊Dic283C2311yescharacteristic
108C4⋊Dic283C2312yes—
109C4⋊Dic283C2312yes—
110C4⋊Dic283C2311yescharacteristic
111Dic283⋊C4C2312yes—
112Dic283⋊C4C2314yes—
113Dic283⋊C4C2312yes—
114Dic283⋊C4C2312yes—
115Dic283⋊C4C2312yes—
116Dic283⋊C4C2314yes—
117Dic283⋊C4C2312yes—
118C22⋊Dic283C2312yes—
119C22⋊Dic283C2312yes—
120C22⋊Dic283C2312yes—
121(C2×D283)⋊C4C2314yes—
122(C2×D283)⋊C4C2314yes—
123(C2×D283)⋊C4C2312yes—
124(C2×D283)⋊C4C2314yes—
125(C2×D283)⋊C4C2314yes—
126(C2×D283)⋊C4C2314yes—
127(C2×D283)⋊C4C2314yes—
128C4×Dic283C2312yes—
129C4×Dic283C2311yescharacteristic
130C4×Dic283C2311yescharacteristic
131C4×Dic283C2312yes—
132C4×Dic283C2311yescharacteristic
133C4×Dic283C2312yes—
134C4×Dic283C2314yes—
135C4×Dic283C2312yes—
136C2×Q8×C283C2311yescharacteristic
137C2×Q8×C283C2312yes—
138C2×D4×C283C2311yescharacteristic
139C22×C4×C283C2312yes—
140C22×C4×C283C2314yes—
141C4⋊C4×C283C2314yes—
142C4⋊C4×C283C2311yescharacteristic
143C4⋊C4×C283C2312yes—
144C4⋊C4×C283C2311yescharacteristic
145C4⋊C4×C283C2312yes—
146C4⋊C4×C283C2312yes—
147C4⋊C4×C283C2314yes—
148C22⋊C4×C283C2312yes—
149C22⋊C4×C283C2314yes—
150C42×C283C2312yes—
151C42×C283C2311yescharacteristic
152C22×D283C2414yes—
153C22×D283C2411yescharacteristic
154Dic566—22no—
155Dic566—24no—
156Dic566—44no—
157Dic566#4114yescomplemented
158D1132—28no—
159D1132—24no—
160D1132—24no—
161D1132—42no—
162C22⋊D283—24no—
163C22⋊D283—28no—
164C22⋊D283—44no—
165C2×Dic283C2411yescharacteristic
166C2×Dic283C2412yes—
167C2×Dic283C2411yescharacteristic
168C2×Dic283C2412yes—
169C2×Dic283C2411yescharacteristic
170C2×Dic283C2412yes—
171C2×Dic283C2412yes—
172C4×D283—24no—
173C4×D283—22no—
174C4×D283—22no—
175C4×D283—22no—
176C4×D283—24no—
177C4×D283—28no—
178C4×D283—44no—
179C4×D283—44no—
180C23×C283C2412yes—
181Q8×C283—22no—
182Q8×C283—42no—
183D4×C283—22no—
184C2×C4×C283—24no—
185C2×C4×C283—44no—
186C2×C4×C283C2412yes—
187C2×C4×C283C2411yescharacteristic
188C2×C4×C283C2412yes—
189C2×C4×C283C2412yes—
190C2×C4×C283C2414yes—
191C2×C4×C283C2411yescharacteristic
192C2×C4×C283C2411yescharacteristic
193C22×C283—22no—
194C22×C283—22no—
195C22×C283—22no—
196C22×C283C2511yescharacteristic
197C2×D283—21no—
198C2×D283—21no—
199C2×D283—24no—
200C2×D283—24no—
201C2×D283—24no—
202C2×D283H(5,𝔽2)12yes—
203C4×C283—22no—
204C4×C283—22no—
205C4×C283—22no—
206C4×C283—24no—
207C4×C283—21no—
208C4×C283C2×#4112yes—
209C4×C2832−1+412yes—
210Dic283—21no—
211Dic283—21no—
212Dic283—22no—
213Dic283—22no—
214Dic283—21no—
215Dic283—22no—
216Dic283C2×#4114yes—
217C2×C283—42no—
218C2×C283C22×#4111yescharacteristic
219C2×C283C2×H(5,𝔽2)11yescharacteristic
220C2×C283C2×2−1+411yescharacteristic
221D283—22no—
222D283—44no—
223C283#352211yescharacteristic, complemented, Sylow
224#3522—2831nomaximal, Sylow
225#559—2831no—
226#559—2832no—
227Q82—2831no—
228#554—2832no—
229#554D28311yescharacteristic, complemented
230#550—2831no—
231#548—2832no—
232#547—2834no—
233#547—2832no—
234#542—2834no—
235#542—2832no—
236#532—2831no—
237#532—2832no—
238#529—2832no—
239#529—2831no—
240#520—2832no—
241C2×C4×Q8—2831no—
242C2×#41—2832no—
243C2×#41—2832no—
244C2×#41—2834no—
245C22×Q8—2831no—
246C4⋊Q8—2834no—
247C4⋊Q8—2832no—
248C4⋊Q8—2832no—
249C4⋊Q8C2×D28311yescharacteristic
250#127—2832no—
251#127—2831no—
252#126—2834no—
253#126—2832no—
254#125—2831no—
255#125C2×D28312yes—
256#124—2832no—
257#124—2832no—
258#124—2831no—
259#124—2834no—
260#124—2834no—
261#124—2832no—
262#123—2834no—
263#123—2834no—
264#123—2834no—
265#123—2832no—
266#123—2834no—
267C22⋊Q8—2831no—
268C22⋊Q8—2834no—
269C22⋊Q8—2834no—
270C22⋊Q8—2834no—
271C22⋊Q8C2×D28314yes—
272C22⋊Q8C2×D28312yes—
273#121—2831no—
274#121—2832no—
275#121—2832no—
276#121—2832no—
277#121—2834no—
278#121—2834no—
279C4×Q8—2831no—
280C4×Q8—2831no—
281C4×Q8—2832no—
282C4×Q8—2832no—
283C4×Q8—2832no—
284C4×Q8—2831no—
285C4×Q8—2832no—
286C4×Q8—2831no—
287C4×Q8C2×D28311yescharacteristic
288C4×D4—2834no—
289C4×D4—2834no—
290C4×D4—2832no—
291C4×D4—2832no—
292C4×D4—2832no—
293C4×D4—2834no—
294C4×D4—2832no—
295C4×D4—2832no—
296C4×D4—2832no—
297C4×D4C2×D28311yescharacteristic
298C4×D4C2×D28312yes—
299#117—2832no—
300#117—2834no—
301C2×C4⋊C4—2832no—
302C2×C4⋊C4—2831no—
303C2×C4⋊C4—2834no—
304C2×C4⋊C4C2×D28312yes—
305C2×C42—2831no—
306C2×C42—2832no—
307#41—5668no—
308#41—11328no—
309#41—11324no—
310C2×Q8—2831no—
311C2×Q8—2831no—
312C2×Q8—2832no—
313C2×Q8—2834no—
314C2×Q8—5664no—
315C2×Q8C22×D28311yescharacteristic
316C2×Q8C22×D28312yes—
317C2×D4—2834no—
318C2×D4—2832no—
319C2×D4—2834no—
320C2×D4—2832no—
321C2×D4—2832no—
322C2×D4—2834no—
323C2×D4—2832no—
324C2×D4C22×D28311yescharacteristic
325C22×C4—2831no—
326C22×C4—2831no—
327C22×C4—2832no—
328C22×C4—2832no—
329C22×C4—2831no—
330C22×C4—2834no—
331C22×C4—2834no—
332C22×C4—2834no—
333C22×C4C22×D28312yes—
334C22×C4C22×D28314yes—
335C4⋊C4—2832no—
336C4⋊C4—2834no—
337C4⋊C4—2832no—
338C4⋊C4—2832no—
339C4⋊C4—2832no—
340C4⋊C4—2834no—
341C4⋊C4—2832no—
342C4⋊C4—2832no—
343C4⋊C4—2831no—
344C4⋊C4—2832no—
345C4⋊C4—2832no—
346C4⋊C4—2831no—
347C4⋊C4C22×D28311yescharacteristic
348C4⋊C4C22×D28314yes—
349C4⋊C4C22×D28312yes—
350C4⋊C4C22×D28312yes—
351C4⋊C4C22×D28311yescharacteristic
352C4⋊C4C22×D28312yes—
353C4⋊C4C22×D28314yes—
354C22⋊C4—2832no—
355C22⋊C4—2832no—
356C22⋊C4—2832no—
357C22⋊C4—2834no—
358C22⋊C4—2834no—
359C22⋊C4—2832no—
360C22⋊C4—2834no—
361C22⋊C4—2834no—
362C22⋊C4—2834no—
363C22⋊C4—2834no—
364C22⋊C4C22×D28312yes—
365C22⋊C4C22×D28314yes—
366C42—2832no—
367C42—2831no—
368C42—2831no—
369C42—2832no—
370C42—2831no—
371C42—2832no—
372C42—2834no—
373C42—2832no—
374C42C22×D28312yes—
375C42C22×D28311yescharacteristic
376C23—2831no—
377C23—2834no—
378C23C23×D28312yes—
379Q8—22no—
380Q8—42no—
381Q8—2834no—
382Q8—5664no—
383Q8—5662no—
384Q8—11324no—
385D4—22no—
386D4—5668no—
387D4—5668no—
388D4—5664no—
389D4—5664no—
390D4—5664no—
391D4—11322no—
392D4—11324no—
393C2×C4—24no—
394C2×C4—44no—
395C2×C4—2832no—
396C2×C4—2831no—
397C2×C4—2832no—
398C2×C4—2831no—
399C2×C4—2831no—
400C2×C4—2832no—
401C2×C4—2832no—
402C2×C4—5664no—
403C2×C4—5668no—
404C2×C4—5664no—
405C2×C4—5662no—
406C2×C4—5662no—
407C2×C4—5662no—
408C2×C4—11324no—
409C2×C4—11324no—
410C2×C4C23×D28312yes—
411C2×C4C23×D28312yes—
412C2×C4C23×D28311yescharacteristic
413C2×C4C23×D28314yes—
414C2×C4C23×D28312yes—
415C2×C4C23×D28311yescharacteristic
416C2×C4C23×D28311yescharacteristic
417C22—22no—
418C22—22no—
419C22—22no—
420C22—2832no—
421C22—5661no—
422C22—5661no—
423C22—5664no—
424C22—5664no—
425C22—5664no—
426C22C24×D28311yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—24no—
431C4—21no—
432C4—2834no—
433C4—5661no—
434C4—5661no—
435C4—5662no—
436C4—5662no—
437C4—5661no—
438C4—5662no—
439C4#41×D28312yes—
440C4#2258365812yes—
441C2—42no—
442C2—5662no—
443C2—11324no—
444C2C2×#41×D28311yescharacteristic
445C2H(5,𝔽2)×D28311yescharacteristic
446C2C2×#2258365811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#554×C283C2
hypercenter#554D283
hypocenterC283#3522
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #31827200C2semidirect
2. #31827198C2semidirect
3. #31827195C2semidirect
4. #31827171C2semidirect
5. #31827136C2semidirect
6. #31827125C2semidirect
7. #31827117C2semidirect
8. #31827116C2semidirect
9. #31827088C2semidirect
10. #31827085C2semidirect
11. #31827016C2semidirect
12. #31827012C2semidirect
13. #31827000C2semidirect
14. #31826997C2semidirect
15. #31826946C2semidirect
16. C4×Q8×D283C2semidirect
17. #554×C283C2semidirect
157. Dic566#41semidirect
223. C283#3522semidirect
229. #554D283semidirect

Maximal subgroups

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

subgroupindex
#318272002
#318271982
#318271952
#318271712
#318271362
#318271252
#318271172
#318271162
#318270882
#318270852
#318270162
#318270122
#318270002
#318269972
#318269462
C4×Q8×D2832
#554×C2832
#3522283