Finite Group Databasebeta

← #49134853

Subgroups — #49134853

Order 36224.

1112 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#31827197C212yesmaximal, complemented
3#31827190C211yesmaximal, characteristic, complemented
4#31827187C212yesmaximal, complemented
5#31827166C214yesmaximal, complemented
6#31827159C212yesmaximal, complemented
7#31827136C211yesmaximal, characteristic, complemented
8#31827125C212yesmaximal, complemented
9#31827016C211yesmaximal, characteristic, complemented
10#31827014C212yesmaximal, complemented
11#31827001C212yesmaximal, complemented
12#31826998C214yesmaximal, complemented
13C4×#17985393C212yesmaximal, complemented
14#31826933C212yesmaximal, complemented
15C4×#17985391C211yesmaximal, characteristic, complemented
16C4×Q8×D283C211yesmaximal, characteristic, complemented
17#556×C283C211yesmaximal, characteristic, complemented
18C2×#17985393C2212yes—
19C2×#17985392C2212yes—
20C2×#17985391C2211yescharacteristic
21C2×Q8×D283C2211yescharacteristic
22#22583608C2212yes—
23#22583607C2211yescharacteristic
24#22583606C2212yes—
25#22583605C2214yes—
26#22583604C2214yes—
27#22583601C2212yes—
28#22583600C2214yes—
29#22583599C2212yes—
30#22583599C2212yes—
31#22583598C2212yes—
32#22583596C2214yes—
33#22583595C2214yes—
34#22583594C2214yes—
35#22583592C2214yes—
36#22583588C2214yes—
37#22583585C2212yes—
38#22583585C2212yes—
39#22583584C2212yes—
40#22583583C2214yes—
41#22583583C2214yes—
42#22583583C2214yes—
43Q8×Dic283C2211yescharacteristic
44Q8×Dic283C2212yes—
45C4×Dic566C2211yescharacteristic
46C4×Dic566C2211yescharacteristic
47C4×Dic566C2212yes—
48#22583575C2212yes—
49#22583575C2212yes—
50#22583575C2211yescharacteristic
51#22583575C2211yescharacteristic
52C4×D1132C2212yes—
53C4×D1132C2214yes—
54D4×Dic283C2211yescharacteristic
55#22583572C2212yes—
56#22583572C2212yes—
57#22583572C2212yes—
58#22583572C2214yes—
59#22583572C2212yes—
60C4×C22⋊D283C2212yes—
61(C22⋊D283)⋊C4C2214yes—
62(C22⋊D283)⋊C4C2214yes—
63(C22⋊D283)⋊C4C2212yes—
64(C22⋊D283)⋊C4C2214yes—
65#22583568C2214yes—
66#22583568C2211yescharacteristic
67#22583567C2214yes—
68#22583566C2212yes—
69C4⋊C4×D283C2211yescharacteristic
70C4⋊C4×D283C2212yes—
71C2×C4×Dic283C2212yes—
72C42×D283C2214yes—
73C42×D283C2212yes—
74C42×D283C2211yescharacteristic
75C4⋊Q8×C283C2211yescharacteristic
76#126×C283C2214yes—
77#124×C283C2212yes—
78C22⋊Q8×C283C2212yes—
79C4×Q8×C283C2211yescharacteristic
80C4×Q8×C283C2212yes—
81C4×D4×C283C2211yescharacteristic
82#117×C283C2212yes—
83#17985393—28no—
84#17985392—28no—
85#17985391—24no—
86C2×Dic566C2312yes—
87C2×Dic566C2311yescharacteristic
88C2×Dic566C2311yescharacteristic
89Q8×D283—24no—
90C2×D1132C2312yes—
91C2×D1132C2314yes—
92C2×D1132C2312yes—
93C2×D1132C2312yes—
94C2×C22⋊D283C2314yes—
95C2×C22⋊D283C2312yes—
96C2×C22⋊D283C2314yes—
97C22×Dic283C2312yes—
98C22×Dic283C2312yes—
99C2×C4×D283C2311yescharacteristic
100C2×C4×D283C2312yes—
101C2×C4×D283C2314yes—
102C2×C4×D283C2314yes—
103C2×C4×D283C2311yescharacteristic
104C2×C4×D283C2312yes—
105C2×C4×D283C2314yes—
106C2×C4×D283C2311yescharacteristic
107C4⋊Dic283C2311yescharacteristic
108C4⋊Dic283C2312yes—
109C4⋊Dic283C2311yescharacteristic
110C4⋊Dic283C2314yes—
111Dic283⋊C4C2312yes—
112Dic283⋊C4C2312yes—
113Dic283⋊C4C2312yes—
114Dic283⋊C4C2312yes—
115Dic283⋊C4C2312yes—
116Dic283⋊C4C2314yes—
117Dic283⋊C4C2314yes—
118C22⋊Dic283C2312yes—
119(C2×D283)⋊C4C2314yes—
120(C2×D283)⋊C4C2314yes—
121(C2×D283)⋊C4C2312yes—
122(C2×D283)⋊C4C2314yes—
123(C2×D283)⋊C4C2314yes—
124(C2×D283)⋊C4C2314yes—
125(C2×D283)⋊C4C2314yes—
126C4×Dic283C2312yes—
127C4×Dic283C2312yes—
128C4×Dic283C2311yescharacteristic
129C4×Dic283C2311yescharacteristic
130C4×Dic283C2312yes—
131C4×Dic283C2314yes—
132C4×Dic283C2312yes—
133C4×Dic283C2311yescharacteristic
134C4×Dic283C2312yes—
135C4×Dic283C2312yes—
136C2×Q8×C283C2311yescharacteristic
137C2×Q8×C283C2312yes—
138C2×D4×C283C2311yescharacteristic
139C22×C4×C283C2312yes—
140C4⋊C4×C283C2311yescharacteristic
141C4⋊C4×C283C2312yes—
142C4⋊C4×C283C2311yescharacteristic
143C4⋊C4×C283C2314yes—
144C4⋊C4×C283C2312yes—
145C4⋊C4×C283C2312yes—
146C22⋊C4×C283C2312yes—
147C22⋊C4×C283C2314yes—
148C22⋊C4×C283C2314yes—
149C42×C283C2314yes—
150C42×C283C2312yes—
151C42×C283C2311yescharacteristic
152C22×D283C2414yes—
153C22×D283C2411yescharacteristic
154Dic566—22no—
155Dic566—24no—
156Dic566#4114yescomplemented
157D1132—28no—
158D1132—24no—
159D1132—24no—
160D1132—42no—
161C22⋊D283—24no—
162C22⋊D283—28no—
163C22⋊D283—44no—
164C2×Dic283—24no—
165C2×Dic283—42no—
166C2×Dic283C2411yescharacteristic
167C2×Dic283C2412yes—
168C2×Dic283C2411yescharacteristic
169C2×Dic283C2412yes—
170C2×Dic283C2412yes—
171C2×Dic283C2411yescharacteristic
172C2×Dic283C2412yes—
173C4×D283—22no—
174C4×D283—24no—
175C4×D283—28no—
176C4×D283—28no—
177C4×D283—44no—
178C4×D283—41no—
179C4×D283—42no—
180C4×D283#4114yescomplemented
181C23×C283C2412yes—
182Q8×C283—22no—
183Q8×C283—24no—
184D4×C283—22no—
185C2×C4×C283—24no—
186C2×C4×C283C2412yes—
187C2×C4×C283C2412yes—
188C2×C4×C283C2411yescharacteristic
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×D283C2×#4112yes—
203C4×C283—22no—
204C4×C283—24no—
205C4×C283—22no—
206C4×C283—22no—
207C4×C283—21no—
208C4×C283C2×#4112yes—
209C4×C283C2×#4112yes—
210Dic283—21no—
211Dic283—22no—
212Dic283—22no—
213Dic283—21no—
214Dic283—22no—
215Dic283C2×#4112yes—
216Dic283C2×#4114yes—
217C2×C283—42no—
218C2×C283C22×#4111yescharacteristic
219C2×C283C22×#4111yescharacteristic
220C2×C283#58511yescharacteristic
221D283—22no—
222D283—44no—
223C283#348011yescharacteristic, complemented, Sylow
224#3480—2831nomaximal, Sylow
225#559—2832no—
226#559—2831no—
227#557—2831no—
228#556—2832no—
229#556D28311yescharacteristic, complemented
230#553—2834no—
231#553—2832no—
232#550—2831no—
233#548—2832no—
234#532—2831no—
235#532—2832no—
236#529—2834no—
237#529—2832no—
238C4×#41—2832no—
239C4×#41—2831no—
240C4×#41—2832no—
241C2×C4×Q8—2831no—
242C2×#41—2832no—
243C2×#41—2831no—
244C2×#41—2832no—
245C22×Q8—2831no—
246C4⋊Q8—2832no—
247C4⋊Q8C2×D28311yescharacteristic
248#127—2832no—
249#127—2831no—
250#126—2834no—
251#126—2834no—
252#126C2×D28314yes—
253#125—2834no—
254#125—2832no—
255#124—2832no—
256#124—2832no—
257#124—2834no—
258#124—2832no—
259#124C2×D28312yes—
260#123—2834no—
261#123—2834no—
262#123—2834no—
263C22⋊Q8—2834no—
264C22⋊Q8C2×D28312yes—
265#121—2832no—
266#121—2832no—
267#121—2832no—
268#121—2834no—
269#121—2834no—
270#121—2834no—
271C4×Q8—2831no—
272C4×Q8—2832no—
273C4×Q8—2831no—
274C4×Q8—2832no—
275C4×Q8—2832no—
276C4×Q8—2831no—
277C4×Q8—2831no—
278C4×Q8—2832no—
279C4×Q8—2831no—
280C4×Q8C2×D28311yescharacteristic
281C4×Q8C2×D28312yes—
282C4×D4—2831no—
283C4×D4—2832no—
284C4×D4—2832no—
285C4×D4—2834no—
286C4×D4—2832no—
287C4×D4—2832no—
288C4×D4—2832no—
289C4×D4—2834no—
290C4×D4—2832no—
291C4×D4—2834no—
292C4×D4—2834no—
293C4×D4—2834no—
294C4×D4—2832no—
295C4×D4C2×D28311yescharacteristic
296#117—2831no—
297#117—2834no—
298#117—2832no—
299#117—2834no—
300#117C2×D28312yes—
301C2×C4⋊C4—2832no—
302C2×C4⋊C4—2831no—
303C2×C42—2832no—
304C2×C42—2831no—
305C2×C42—2832no—
306C2×C42—2834no—
307#41—5668no—
308#41—5664no—
309#41—5668no—
310C2×Q8—2831no—
311C2×Q8—2832no—
312C2×Q8—2831no—
313C2×Q8—5664no—
314C2×Q8C22×D28311yescharacteristic
315C2×Q8C22×D28312yes—
316C2×D4—2832no—
317C2×D4—2834no—
318C2×D4—2834no—
319C2×D4—2834no—
320C2×D4—2832no—
321C2×D4—2832no—
322C2×D4—2832no—
323C2×D4C22×D28311yescharacteristic
324C22×C4—2831no—
325C22×C4—2832no—
326C22×C4—2832no—
327C22×C4—2832no—
328C22×C4—2831no—
329C22×C4—2834no—
330C22×C4—2834no—
331C22×C4—2831no—
332C22×C4—2832no—
333C22×C4—2834no—
334C22×C4C22×D28312yes—
335C4⋊C4—2832no—
336C4⋊C4—2832no—
337C4⋊C4—2832no—
338C4⋊C4—2834no—
339C4⋊C4—2832no—
340C4⋊C4—2832no—
341C4⋊C4—2834no—
342C4⋊C4—2831no—
343C4⋊C4—2832no—
344C4⋊C4—2831no—
345C4⋊C4—2834no—
346C4⋊C4C22×D28311yescharacteristic
347C4⋊C4C22×D28312yes—
348C4⋊C4C22×D28311yescharacteristic
349C4⋊C4C22×D28314yes—
350C4⋊C4C22×D28312yes—
351C4⋊C4C22×D28312yes—
352C22⋊C4—2832no—
353C22⋊C4—2832no—
354C22⋊C4—2834no—
355C22⋊C4—2834no—
356C22⋊C4—2834no—
357C22⋊C4—2834no—
358C22⋊C4—2834no—
359C22⋊C4—2834no—
360C22⋊C4C22×D28312yes—
361C22⋊C4C22×D28314yes—
362C22⋊C4C22×D28314yes—
363C42—2832no—
364C42—2832no—
365C42—2831no—
366C42—2831no—
367C42—2832no—
368C42—2832no—
369C42—2834no—
370C42—2832no—
371C42—2831no—
372C42—2832no—
373C42C22×D28314yes—
374C42C22×D28312yes—
375C42C22×D28311yescharacteristic
376C23—2831no—
377C23—2834no—
378C23C23×D28312yes—
379Q8—22no—
380Q8—24no—
381Q8—2834no—
382Q8—5664no—
383Q8—5662no—
384D4—22no—
385D4—5668no—
386D4—5668no—
387D4—5664no—
388D4—5664no—
389D4—5664no—
390D4—11322no—
391D4—11324no—
392C2×C4—24no—
393C2×C4—2831no—
394C2×C4—2832no—
395C2×C4—2834no—
396C2×C4—2832no—
397C2×C4—2831no—
398C2×C4—2831no—
399C2×C4—2832no—
400C2×C4—2832no—
401C2×C4—5664no—
402C2×C4—5668no—
403C2×C4—5668no—
404C2×C4—5664no—
405C2×C4—5662no—
406C2×C4—11324no—
407C2×C4—11322no—
408C2×C4—11321no—
409C2×C4—11322no—
410C2×C4C23×D28312yes—
411C2×C4C23×D28312yes—
412C2×C4C23×D28312yes—
413C2×C4C23×D28314yes—
414C2×C4C23×D28311yescharacteristic
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—24no—
429C4—22no—
430C4—22no—
431C4—21no—
432C4—2832no—
433C4—2834no—
434C4—5662no—
435C4—5661no—
436C4—5662no—
437C4—5661no—
438C4—5662no—
439C4#41×D28312yes—
440C4#41×D28312yes—
441C2—42no—
442C2—5662no—
443C2—11324no—
444C2C2×#41×D28311yescharacteristic
445C2C2×#41×D28311yescharacteristic
446C2#3182730211yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#556×C283C2
hypercenter#556D283
hypocenterC283#3480
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #31827200C2semidirect
2. #31827197C2semidirect
3. #31827190C2semidirect
4. #31827187C2semidirect
5. #31827166C2semidirect
6. #31827159C2semidirect
7. #31827136C2semidirect
8. #31827125C2semidirect
9. #31827016C2semidirect
10. #31827014C2semidirect
11. #31827001C2semidirect
12. #31826998C2semidirect
13. C4×#17985393C2semidirect
14. #31826933C2semidirect
15. C4×#17985391C2semidirect
16. C4×Q8×D283C2semidirect
17. #556×C283C2semidirect
156. Dic566#41semidirect
180. C4×D283#41semidirect
223. C283#3480semidirect
229. #556D283semidirect

Maximal subgroups

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

subgroupindex
#318272002
#318271972
#318271902
#318271872
#318271662
#318271592
#318271362
#318271252
#318270162
#318270142
#318270012
#318269982
C4×#179853932
#318269332
C4×#179853912
C4×Q8×D2832
#556×C2832
#3480283