Finite Group Databasebeta

← #49134880

Subgroups — #49134880

Order 36224.

1094 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827193C211yesmaximal, characteristic, complemented
2#31827183C212yesmaximal, complemented
3#31827156C212yesmaximal, complemented
4#31827151C214yesmaximal, complemented
5#31827130C211yesmaximal, characteristic, complemented
6#31827123C212yesmaximal, complemented
7#31827120C211yesmaximal, characteristic, complemented
8#31827117C212yesmaximal, complemented
9#31827020C211yesmaximal, characteristic, complemented
10#31827017C212yesmaximal, complemented
11#31826998C214yesmaximal, complemented
12#31826994C212yesmaximal, complemented
13C4×#17985393C211yesmaximal, characteristic, complemented
14#31826933C212yesmaximal, complemented
15C4×#17985391C211yesmaximal, characteristic, complemented
16C4×Q8×D283C212yesmaximal, complemented
17#556×C283C211yesmaximal, characteristic, complemented
18C2×#17985393C2211yescharacteristic
19C2×#17985392C2212yes—
20C2×#17985391C2211yescharacteristic
21C2×Q8×D283C2212yes—
22#22583610C2211yescharacteristic
23#22583609C2212yes—
24#22583608C2212yes—
25#22583604C2214yes—
26#22583603C2214yes—
27#22583602C2212yes—
28#22583601C2212yes—
29#22583600C2214yes—
30#22583600C2214yes—
31#22583596C2214yes—
32#22583595C2214yes—
33#22583592C2214yes—
34#22583592C2214yes—
35#22583590C2212yes—
36#22583589C2212yes—
37#22583588C2214yes—
38#22583587C2214yes—
39#22583586C2212yes—
40#22583586C2212yes—
41#22583583C2214yes—
42#22583581C2212yes—
43Q8×Dic283C2212yes—
44Q8×Dic283C2211yescharacteristic
45C4×Dic566C2212yes—
46C4×Dic566C2211yescharacteristic
47C4×Dic566C2214yes—
48#22583575C2214yes—
49#22583575C2212yes—
50#22583575C2212yes—
51#22583575C2211yescharacteristic
52C4×D1132C2211yescharacteristic
53C4×D1132C2212yes—
54D4×Dic283C2211yescharacteristic
55#22583572C2212yes—
56#22583572C2212yes—
57#22583572C2211yescharacteristic
58C4×C22⋊D283C2212yes—
59(C22⋊D283)⋊C4C2214yes—
60(C22⋊D283)⋊C4C2212yes—
61(C22⋊D283)⋊C4C2214yes—
62#22583568C2212yes—
63#22583568C2212yes—
64#22583568C2211yescharacteristic
65#22583568C2211yescharacteristic
66#22583567C2214yes—
67#22583566C2214yes—
68#22583566C2212yes—
69C4⋊C4×D283C2214yes—
70C4⋊C4×D283C2212yes—
71C2×C4×Dic283C2212yes—
72C42×D283C2212yes—
73C42×D283C2214yes—
74C42×D283C2211yescharacteristic
75C4⋊Q8×C283C2211yescharacteristic
76#126×C283C2214yes—
77#124×C283C2212yes—
78C22⋊Q8×C283C2212yes—
79C4×Q8×C283C2212yes—
80C4×Q8×C283C2211yescharacteristic
81C4×D4×C283C2211yescharacteristic
82#117×C283C2212yes—
83#17985393—24no—
84#17985392—28no—
85#17985391—24no—
86C2×Dic566C2312yes—
87C2×Dic566C2314yes—
88C2×Dic566C2311yescharacteristic
89Q8×D283—28no—
90C2×D1132C2312yes—
91C2×D1132C2311yescharacteristic
92C2×C22⋊D283C2314yes—
93C2×C22⋊D283C2312yes—
94C22×Dic283C2312yes—
95C22×Dic283C2312yes—
96C2×C4×D283C2312yes—
97C2×C4×D283C2314yes—
98C2×C4×D283C2312yes—
99C2×C4×D283C2314yes—
100C2×C4×D283C2312yes—
101C2×C4×D283C2311yescharacteristic
102C2×C4×D283C2311yescharacteristic
103C2×C4×D283C2311yescharacteristic
104C4⋊Dic283C2312yes—
105C4⋊Dic283C2314yes—
106C4⋊Dic283C2311yescharacteristic
107C4⋊Dic283C2312yes—
108C4⋊Dic283C2311yescharacteristic
109C4⋊Dic283C2312yes—
110Dic283⋊C4C2312yes—
111Dic283⋊C4C2314yes—
112Dic283⋊C4C2314yes—
113Dic283⋊C4C2312yes—
114Dic283⋊C4C2314yes—
115Dic283⋊C4C2314yes—
116Dic283⋊C4C2314yes—
117Dic283⋊C4C2314yes—
118C22⋊Dic283C2312yes—
119C22⋊Dic283C2314yes—
120(C2×D283)⋊C4C2312yes—
121(C2×D283)⋊C4C2314yes—
122(C2×D283)⋊C4C2312yes—
123(C2×D283)⋊C4C2312yes—
124(C2×D283)⋊C4C2312yes—
125(C2×D283)⋊C4C2314yes—
126C4×Dic283C2311yescharacteristic
127C4×Dic283C2312yes—
128C4×Dic283C2311yescharacteristic
129C4×Dic283C2311yescharacteristic
130C4×Dic283C2312yes—
131C4×Dic283C2314yes—
132C4×Dic283C2312yes—
133C4×Dic283C2314yes—
134C4×Dic283C2312yes—
135C4×Dic283C2312yes—
136C2×Q8×C283C2312yes—
137C2×Q8×C283C2311yescharacteristic
138C2×D4×C283C2311yescharacteristic
139C22×C4×C283C2312yes—
140C4⋊C4×C283C2311yescharacteristic
141C4⋊C4×C283C2314yes—
142C4⋊C4×C283C2311yescharacteristic
143C4⋊C4×C283C2312yes—
144C4⋊C4×C283C2312yes—
145C4⋊C4×C283C2312yes—
146C22⋊C4×C283C2312yes—
147C22⋊C4×C283C2314yes—
148C22⋊C4×C283C2314yes—
149C42×C283C2314yes—
150C42×C283C2312yes—
151C42×C283C2311yescharacteristic
152C22×D283C2411yescharacteristic
153C22×D283C2412yes—
154Dic566—24no—
155Dic566—28no—
156Dic566#4114yescomplemented
157D1132—24no—
158D1132#4114yescomplemented
159C22⋊D283—28no—
160C22⋊D283—24no—
161C2×Dic283—24no—
162C2×Dic283—24no—
163C2×Dic283C2411yescharacteristic
164C2×Dic283C2411yescharacteristic
165C2×Dic283C2412yes—
166C2×Dic283C2412yes—
167C2×Dic283C2414yes—
168C2×Dic283C2412yes—
169C2×Dic283C2411yescharacteristic
170C4×D283—28no—
171C4×D283—24no—
172C4×D283—24no—
173C4×D283—22no—
174C4×D283—22no—
175C4×D283—28no—
176C4×D283—24no—
177C4×D283—22no—
178C23×C283C2412yes—
179Q8×C283—24no—
180Q8×C283—22no—
181D4×C283—22no—
182C2×C4×C283—24no—
183C2×C4×C283C2412yes—
184C2×C4×C283C2412yes—
185C2×C4×C283C2411yescharacteristic
186C2×C4×C283C2414yes—
187C2×C4×C283C2412yes—
188C2×C4×C283C2411yescharacteristic
189C2×C4×C283C2411yescharacteristic
190C22×C283—22no—
191C22×C283—22no—
192C22×C283—22no—
193C22×C283C2511yescharacteristic
194C2×D283—22no—
195C2×D283—22no—
196C2×D283—21no—
197C2×D283—21no—
198C2×D283—21no—
199C2×D283C2×#4114yes—
200C4×C283—22no—
201C4×C283—22no—
202C4×C283—24no—
203C4×C283—22no—
204C4×C283—21no—
205C4×C283C2×#4112yes—
206C4×C283C2×#4112yes—
207Dic283—21no—
208Dic283—21no—
209Dic283—24no—
210Dic283—22no—
211Dic283—22no—
212Dic283C2×#4114yes—
213Dic2832−1+412yes—
214C2×C283—42no—
215C2×C283C22×#4111yescharacteristic
216C2×C283C22×#4111yescharacteristic
217C2×C283C2×2−1+411yescharacteristic
218D283—24no—
219D283—41no—
220C283#348211yescharacteristic, complemented, Sylow
221#3482—2831nomaximal, Sylow
222#557—2831no—
223#556—2832no—
224#556D28311yescharacteristic, complemented
225#552—2834no—
226#552—2832no—
227D4×Q8—2831no—
228#548—2832no—
229#547—2832no—
230#547—2831no—
231#533—2831no—
232#533—2832no—
233#529—2834no—
234#529—2832no—
235C4×#41—2831no—
236C4×#41—2831no—
237C4×#41—2832no—
238C2×C4×Q8—2832no—
239C2×#41—2831no—
240C2×#41—2831no—
241C2×#41—2832no—
242C22×Q8—2832no—
243C4⋊Q8—2832no—
244C4⋊Q8—2832no—
245C4⋊Q8—2831no—
246C4⋊Q8C2×D28311yescharacteristic
247#126—2834no—
248#126—2834no—
249#126C2×D28314yes—
250#125—2834no—
251#125—2834no—
252#125—2832no—
253#125—2832no—
254#124—2834no—
255#124C2×D28312yes—
256#123—2834no—
257#123—2834no—
258#123—2834no—
259C22⋊Q8—2832no—
260C22⋊Q8—2832no—
261C22⋊Q8—2832no—
262C22⋊Q8—2834no—
263C22⋊Q8—2832no—
264C22⋊Q8—2834no—
265C22⋊Q8C2×D28312yes—
266#121—2832no—
267#121—2834no—
268C4×Q8—2832no—
269C4×Q8—2831no—
270C4×Q8—2831no—
271C4×Q8—2834no—
272C4×Q8—2832no—
273C4×Q8—2832no—
274C4×Q8—2832no—
275C4×Q8—2834no—
276C4×Q8—2831no—
277C4×Q8C2×D28312yes—
278C4×Q8C2×D28311yescharacteristic
279C4×D4—2831no—
280C4×D4—2832no—
281C4×D4—2831no—
282C4×D4—2832no—
283C4×D4—2832no—
284C4×D4—2832no—
285C4×D4—2834no—
286C4×D4—2834no—
287C4×D4—2831no—
288C4×D4—2832no—
289C4×D4C2×D28311yescharacteristic
290#117—2831no—
291#117—2832no—
292#117—2831no—
293#117—2832no—
294#117—2832no—
295#117—2834no—
296#117—2834no—
297#117C2×D28312yes—
298C2×C4⋊C4—2834no—
299C2×C4⋊C4—2832no—
300C2×C42—2832no—
301C2×C42—2832no—
302C2×C42—2834no—
303C2×C42—2831no—
304#41—5668no—
305#41—5664no—
306#41—5664no—
307C2×Q8—2831no—
308C2×Q8—2834no—
309C2×Q8—2832no—
310C2×Q8—5668no—
311C2×Q8C22×D28312yes—
312C2×Q8C22×D28311yescharacteristic
313C2×D4—2834no—
314C2×D4—2832no—
315C2×D4—2831no—
316C2×D4—2832no—
317C2×D4C22×D28311yescharacteristic
318C22×C4—2832no—
319C22×C4—2832no—
320C22×C4—2832no—
321C22×C4—2832no—
322C22×C4—2834no—
323C22×C4—2834no—
324C22×C4—2832no—
325C22×C4—2831no—
326C22×C4—2831no—
327C22×C4—2831no—
328C22×C4C22×D28312yes—
329C4⋊C4—2832no—
330C4⋊C4—2834no—
331C4⋊C4—2834no—
332C4⋊C4—2832no—
333C4⋊C4—2834no—
334C4⋊C4—2834no—
335C4⋊C4—2834no—
336C4⋊C4—2834no—
337C4⋊C4—2832no—
338C4⋊C4—2834no—
339C4⋊C4—2832no—
340C4⋊C4—2831no—
341C4⋊C4—2831no—
342C4⋊C4—2832no—
343C4⋊C4C22×D28311yescharacteristic
344C4⋊C4C22×D28314yes—
345C4⋊C4C22×D28311yescharacteristic
346C4⋊C4C22×D28312yes—
347C4⋊C4C22×D28312yes—
348C4⋊C4C22×D28312yes—
349C22⋊C4—2832no—
350C22⋊C4—2834no—
351C22⋊C4—2832no—
352C22⋊C4—2834no—
353C22⋊C4—2832no—
354C22⋊C4—2832no—
355C22⋊C4—2832no—
356C22⋊C4—2834no—
357C22⋊C4C22×D28312yes—
358C22⋊C4C22×D28314yes—
359C22⋊C4C22×D28314yes—
360C42—2832no—
361C42—2831no—
362C42—2831no—
363C42—2831no—
364C42—2832no—
365C42—2832no—
366C42—2834no—
367C42—2832no—
368C42—2832no—
369C42—2834no—
370C42C22×D28314yes—
371C42C22×D28311yescharacteristic
372C42C22×D28312yes—
373C23—2832no—
374C23—2831no—
375C23C23×D28312yes—
376Q8—24no—
377Q8—22no—
378Q8—2834no—
379Q8—5668no—
380Q8—5664no—
381D4—22no—
382D4—2834no—
383D4—5664no—
384D4—5664no—
385D4—5668no—
386C2×C4—24no—
387C2×C4—2831no—
388C2×C4—2832no—
389C2×C4—2831no—
390C2×C4—2831no—
391C2×C4—2832no—
392C2×C4—2832no—
393C2×C4—2834no—
394C2×C4—5668no—
395C2×C4—5664no—
396C2×C4—5662no—
397C2×C4—5668no—
398C2×C4—5664no—
399C2×C4—5664no—
400C2×C4—5664no—
401C2×C4—5664no—
402C2×C4—5662no—
403C2×C4—5662no—
404C2×C4C23×D28312yes—
405C2×C4C23×D28312yes—
406C2×C4C23×D28311yescharacteristic
407C2×C4C23×D28314yes—
408C2×C4C23×D28312yes—
409C2×C4C23×D28311yescharacteristic
410C2×C4C23×D28311yescharacteristic
411C22—22no—
412C22—22no—
413C22—22no—
414C22—2834no—
415C22—5662no—
416C22—5662no—
417C22—5661no—
418C22—5661no—
419C22—5661no—
420C22C24×D28311yescharacteristic
421C4—22no—
422C4—22no—
423C4—24no—
424C4—22no—
425C4—21no—
426C4—2834no—
427C4—2832no—
428C4—5661no—
429C4—5661no—
430C4—5664no—
431C4—5662no—
432C4—5662no—
433C4#41×D28312yes—
434C4#41×D28312yes—
435C2—42no—
436C2—5664no—
437C2—11321no—
438C2C2×#41×D28311yescharacteristic
439C2C2×#41×D28311yescharacteristic
440C22−1+4×D28311yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#556×C283C2
hypercenter#556D283
hypocenterC283#3482
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #31827193C2semidirect
2. #31827183C2semidirect
3. #31827156C2semidirect
4. #31827151C2semidirect
5. #31827130C2semidirect
6. #31827123C2semidirect
7. #31827120C2semidirect
8. #31827117C2semidirect
9. #31827020C2semidirect
10. #31827017C2semidirect
11. #31826998C2semidirect
12. #31826994C2semidirect
13. C4×#17985393C2semidirect
14. #31826933C2semidirect
15. C4×#17985391C2semidirect
16. C4×Q8×D283C2semidirect
17. #556×C283C2semidirect
156. Dic566#41semidirect
158. D1132#41semidirect
220. C283#3482semidirect
224. #556D283semidirect

Maximal subgroups

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

subgroupindex
#318271932
#318271832
#318271562
#318271512
#318271302
#318271232
#318271202
#318271172
#318270202
#318270172
#318269982
#318269942
C4×#179853932
#318269332
C4×#179853912
C4×Q8×D2832
#556×C2832
#3482283