Finite Group Databasebeta

← #49135354

Subgroups — #49135354

Order 36224.

992 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827193C211yesmaximal, characteristic, complemented
2#31827192C212yesmaximal, complemented
3#31827188C211yesmaximal, characteristic, complemented
4#31827187C212yesmaximal, complemented
5#31827156C212yesmaximal, complemented
6#31827147C214yesmaximal, complemented
7#31827130C212yesmaximal, complemented
8#31827130C211yesmaximal, characteristic, complemented
9#31827076C212yesmaximal, complemented
10#31827074C214yesmaximal, complemented
11#31827020C211yesmaximal, characteristic, complemented
12#31827019C212yesmaximal, complemented
13#31827002C211yesmaximal, characteristic
14#31827000C212yesmaximal
15#31826941C212yesmaximal, complemented
16C4×Q8×D283C211yesmaximal, characteristic, complemented
17#557×C283C211yesmaximal, characteristic, complemented
18C2×Q8×D283C2211yescharacteristic
19C2×Q8×D283C2212yes—
20#22583610C2211yescharacteristic
21#22583609C2212yes—
22#22583609C2214yes—
23#22583608C2212yes—
24#22583605C2214yes—
25#22583605C2212yes—
26#22583605C2214yes—
27#22583605C2214yes—
28#22583604C2214yes—
29#22583602C2212yes—
30#22583601C2211yescharacteristic
31#22583601C2212yes—
32#22583600C2214yes—
33#22583600C2214yes—
34#22583600C2214yes—
35#22583600C2214yes—
36#22583599C2211yescharacteristic
37#22583598C2212yes—
38#22583595C2214yes—
39#22583595C2212yes—
40#22583590C2212yes—
41#22583588C2214yes—
42#22583588C2214yes—
43#22583588C2214yes—
44#22583586C2211yescharacteristic
45#22583586C2212yes—
46#22583586C2212yes—
47#22583586C2212yes—
48#22583585C2211yescharacteristic
49Q8×Dic283C2212yes—
50Q8×Dic283C2211yescharacteristic
51C4×Dic566C2214yes—
52C4×Dic566C2212yes—
53C4×Dic566C2211yescharacteristic
54C4×Dic566C2212yes—
55C4×Dic566C2212yes—
56#22583575C2212yes—
57#22583575C2212yes—
58#22583575C2212yes—
59#22583575C2214yes—
60#22583575C2211yescharacteristic
61C4×D1132C2212yes—
62C4×D1132C2211yescharacteristic
63#22583572C2214yes—
64#22583572C2212yes—
65#22583568C2214yes—
66#22583568C2212yes—
67#22583568C2212yes—
68#22583567C2212yes—
69#22583567C2214yes—
70#22583567C2214yes—
71C4⋊C4×D283C2212yes—
72C4⋊C4×D283C2211yescharacteristic
73C42×D283C2212yes—
74C42×D283C2211yescharacteristic
75C4⋊Q8×C283C2211yescharacteristic
76C4⋊Q8×C283C2212yes—
77#125×C283C2212yes—
78#125×C283C2214yes—
79C4×Q8×C283C2211yescharacteristic
80C4×Q8×C283C2212yes—
81C4×Q8×C283C2212yes—
82C4×Q8×C283C2211yescharacteristic
83C2×Dic566C2312yes—
84C2×Dic566C2312yes—
85C2×Dic566C2314yes—
86C2×Dic566C2311yescharacteristic
87Q8×D283—24no—
88Q8×D283—28no—
89C2×D1132C2312yes—
90C2×D1132C2311yescharacteristic
91C2×C4×D283C2311yescharacteristic
92C2×C4×D283C2312yes—
93C2×C4×D283C2311yescharacteristic
94C2×C4×D283C2312yes—
95C2×C4×D283C2311yescharacteristic
96C2×C4×D283C2314yes—
97C2×C4×D283C2312yes—
98C4⋊Dic283C2311yescharacteristic
99C4⋊Dic283C2312yes—
100C4⋊Dic283C2312yes—
101C4⋊Dic283C2314yes—
102C4⋊Dic283C2312yes—
103C4⋊Dic283C2311yescharacteristic
104C4⋊Dic283C2312yes—
105Dic283⋊C4C2312yes—
106Dic283⋊C4C2312yes—
107Dic283⋊C4C2312yes—
108Dic283⋊C4C2312yes—
109Dic283⋊C4C2312yes—
110Dic283⋊C4C2314yes—
111Dic283⋊C4C2314yes—
112Dic283⋊C4C2314yes—
113Dic283⋊C4C2314yes—
114Dic283⋊C4C2314yes—
115Dic283⋊C4C2314yes—
116(C2×D283)⋊C4C2312yes—
117(C2×D283)⋊C4C2312yes—
118(C2×D283)⋊C4C2312yes—
119(C2×D283)⋊C4C2314yes—
120(C2×D283)⋊C4C2314yes—
121(C2×D283)⋊C4C2314yes—
122(C2×D283)⋊C4C2314yes—
123(C2×D283)⋊C4C2312yes—
124(C2×D283)⋊C4C2312yes—
125C4×Dic283C2311yescharacteristic
126C4×Dic283C2312yes—
127C4×Dic283C2312yes—
128C4×Dic283C2311yescharacteristic
129C4×Dic283C2311yescharacteristic
130C4×Dic283C2312yes—
131C4×Dic283C2314yes—
132C4×Dic283C2314yes—
133C4×Dic283C2314yes—
134C2×Q8×C283C2312yes—
135C2×Q8×C283C2311yescharacteristic
136C2×Q8×C283C2311yescharacteristic
137C4⋊C4×C283C2311yescharacteristic
138C4⋊C4×C283C2312yes—
139C4⋊C4×C283C2314yes—
140C4⋊C4×C283C2312yes—
141C4⋊C4×C283C2312yes—
142C4⋊C4×C283C2314yes—
143C4⋊C4×C283C2311yescharacteristic
144C4⋊C4×C283C2314yes—
145C4⋊C4×C283C2312yes—
146C42×C283C2311yescharacteristic
147C42×C283C2312yes—
148C42×C283C2314yes—
149C42×C283C2312yes—
150C22×D283C2412yes—
151C22×D283C2411yescharacteristic
152Dic566—24no—
153Dic566—24no—
154Dic566—28no—
155Dic566#4114yes—
156D1132—24no—
157D1132—22no—
158C2×Dic283C2411yescharacteristic
159C2×Dic283C2411yescharacteristic
160C2×Dic283C2414yes—
161C2×Dic283C2412yes—
162C2×Dic283C2411yescharacteristic
163C2×Dic283C2412yes—
164C2×Dic283C2412yes—
165C4×D283—24no—
166C4×D283—22no—
167C4×D283—22no—
168C4×D283—24no—
169C4×D283—22no—
170C4×D283—24no—
171C4×D283—28no—
172Q8×C283—24no—
173Q8×C283—22no—
174Q8×C283—22no—
175C2×C4×C283C2412yes—
176C2×C4×C283C2412yes—
177C2×C4×C283C2412yes—
178C2×C4×C283C2411yescharacteristic
179C2×C4×C283C2412yes—
180C2×C4×C283C2414yes—
181C2×C4×C283C2411yescharacteristic
182C2×C4×C283C2411yescharacteristic
183C22×C283C2511yescharacteristic
184C2×D283—21no—
185C2×D283—21no—
186C2×D283—22no—
187C2×D283—22no—
188C2×D283—22no—
189C2×D2832−1+412yes—
190C4×C283—22no—
191C4×C283—24no—
192C4×C283—22no—
193C4×C283—22no—
194C4×C283—21no—
195C4×C283—22no—
196C4×C283C2×#4112yes—
197C4×C2832−1+412yes—
198Dic283—21no—
199Dic283—22no—
200Dic283—21no—
201Dic283—22no—
202Dic283—21no—
203Dic283—24no—
204Dic283C2×#4114yes—
205C2×C283C22×#4111yescharacteristic
206C2×C283C2×2−1+411yescharacteristic
207C2×C283C2×2−1+411yescharacteristic
208D283—22no—
209D283—42no—
210C283#352311yescharacteristic, complemented, Sylow
211#3523—2831nomaximal, Sylow
212#557—2832no—
213#557—2831no—
214#557D28311yescharacteristic, complemented
215#556—2832no—
216#556—2831no—
217#552—2834no—
218#552—2832no—
219D4×Q8—2832no—
220D4×Q8—2831no—
221#541—2832no—
222#541—2834no—
223#533—2831no—
224#533—2832no—
225#529—2832no—
226#529—2831no—
227#519—2832no—
228C2×C4×Q8—2831no—
229C22×Q8—2831no—
230C22×Q8—2832no—
231C4⋊Q8—2832no—
232C4⋊Q8—2832no—
233C4⋊Q8—2834no—
234C4⋊Q8—2831no—
235C4⋊Q8C2×D28311yescharacteristic
236C4⋊Q8C2×D28312yes—
237#126—2832no—
238#126—2834no—
239#126—2834no—
240#126—2834no—
241#126—2834no—
242#125—2831no—
243#125—2834no—
244#125—2832no—
245#125—2834no—
246#125—2834no—
247#125—2834no—
248#125—2832no—
249#125C2×D28312yes—
250#125C2×D28314yes—
251#124—2831no—
252#124—2832no—
253#123—2832no—
254#123—2834no—
255C22⋊Q8—2831no—
256C22⋊Q8—2832no—
257C22⋊Q8—2832no—
258C22⋊Q8—2832no—
259C22⋊Q8—2832no—
260C22⋊Q8—2834no—
261C22⋊Q8—2834no—
262C22⋊Q8—2834no—
263#121—2831no—
264C4×Q8—2831no—
265C4×Q8—2832no—
266C4×Q8—2831no—
267C4×Q8—2832no—
268C4×Q8—2834no—
269C4×Q8—2832no—
270C4×Q8—2832no—
271C4×Q8—2831no—
272C4×Q8—2832no—
273C4×Q8—2834no—
274C4×Q8—2832no—
275C4×Q8—2832no—
276C4×Q8C2×D28311yescharacteristic
277C4×Q8C2×D28312yes—
278C4×Q8C2×D28311yescharacteristic
279C4×Q8C2×D28312yes—
280C4×D4—2834no—
281C4×D4—2832no—
282C4×D4—2832no—
283C4×D4—2831no—
284#117—2834no—
285#117—2832no—
286#117—2832no—
287#117—2832no—
288#117—2834no—
289#117—2834no—
290C2×C4⋊C4—2832no—
291C2×C4⋊C4—2831no—
292C2×C42—2832no—
293C2×C42—2831no—
294C2×Q8—2831no—
295C2×Q8—2832no—
296C2×Q8—2832no—
297C2×Q8—2834no—
298C2×Q8—5664no—
299C2×Q8—5668no—
300C2×Q8C22×D28311yescharacteristic
301C2×Q8C22×D28312yes—
302C2×Q8C22×D28311yescharacteristic
303C2×D4—2832no—
304C2×D4—2831no—
305C22×C4—2831no—
306C22×C4—2831no—
307C22×C4—2832no—
308C22×C4—2832no—
309C22×C4—2831no—
310C22×C4—2834no—
311C22×C4—2832no—
312C4⋊C4—2832no—
313C4⋊C4—2832no—
314C4⋊C4—2832no—
315C4⋊C4—2834no—
316C4⋊C4—2832no—
317C4⋊C4—2832no—
318C4⋊C4—2834no—
319C4⋊C4—2834no—
320C4⋊C4—2834no—
321C4⋊C4—2834no—
322C4⋊C4—2834no—
323C4⋊C4—2831no—
324C4⋊C4—2832no—
325C4⋊C4—2832no—
326C4⋊C4—2834no—
327C4⋊C4—2832no—
328C4⋊C4—2831no—
329C4⋊C4—2832no—
330C4⋊C4C22×D28311yescharacteristic
331C4⋊C4C22×D28312yes—
332C4⋊C4C22×D28314yes—
333C4⋊C4C22×D28312yes—
334C4⋊C4C22×D28312yes—
335C4⋊C4C22×D28314yes—
336C4⋊C4C22×D28311yescharacteristic
337C4⋊C4C22×D28314yes—
338C4⋊C4C22×D28312yes—
339C22⋊C4—2832no—
340C22⋊C4—2832no—
341C22⋊C4—2834no—
342C22⋊C4—2834no—
343C22⋊C4—2832no—
344C22⋊C4—2834no—
345C22⋊C4—2832no—
346C22⋊C4—2834no—
347C22⋊C4—2832no—
348C42—2832no—
349C42—2831no—
350C42—2832no—
351C42—2831no—
352C42—2832no—
353C42—2831no—
354C42—2834no—
355C42—2834no—
356C42—2834no—
357C42C22×D28311yescharacteristic
358C42C22×D28312yes—
359C42C22×D28314yes—
360C42C22×D28312yes—
361C23—2831no—
362C23—2832no—
363Q8—22no—
364Q8—24no—
365Q8—22no—
366Q8—2834no—
367Q8—5664no—
368Q8—5668no—
369Q8—5664no—
370D4—5664no—
371D4—5662no—
372C2×C4—2832no—
373C2×C4—2831no—
374C2×C4—2831no—
375C2×C4—2834no—
376C2×C4—2832no—
377C2×C4—2831no—
378C2×C4—2832no—
379C2×C4—5664no—
380C2×C4—5668no—
381C2×C4—5664no—
382C2×C4—5662no—
383C2×C4—5662no—
384C2×C4—5664no—
385C2×C4—5662no—
386C2×C4C23×D28312yes—
387C2×C4C23×D28312yes—
388C2×C4C23×D28311yescharacteristic
389C2×C4C23×D28312yes—
390C2×C4C23×D28314yes—
391C2×C4C23×D28312yes—
392C2×C4C23×D28311yescharacteristic
393C2×C4C23×D28311yescharacteristic
394C22—2832no—
395C22—5661no—
396C22—5661no—
397C22—5662no—
398C22—5662no—
399C22—5662no—
400C22C24×D28311yescharacteristic
401C4—22no—
402C4—24no—
403C4—22no—
404C4—21no—
405C4—22no—
406C4—22no—
407C4—2834no—
408C4—5661no—
409C4—5662no—
410C4—5661no—
411C4—5662no—
412C4—5661no—
413C4—5664no—
414C4#41×D28312yes—
415C4#2258365812yes—
416C2—5662no—
417C2—11322no—
418C2C2×#41×D28311yescharacteristic
419C22−1+4×D28311yescharacteristic
420C2C2×#2258365811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#557×C283C2
prime radical#22583585C22
hypercenter#557D283
hypocenterC283#3523
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #31827193C2semidirect
2. #31827192C2semidirect
3. #31827188C2semidirect
4. #31827187C2semidirect
5. #31827156C2semidirect
6. #31827147C2semidirect
7. #31827130C2semidirect
8. #31827130C2semidirect
9. #31827076C2semidirect
10. #31827074C2semidirect
11. #31827020C2semidirect
12. #31827019C2semidirect
15. #31826941C2semidirect
16. C4×Q8×D283C2semidirect
17. #557×C283C2semidirect
210. C283#3523semidirect
214. #557D283semidirect

Maximal subgroups

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

subgroupindex
#318271932
#318271922
#318271882
#318271872
#318271562
#318271472
#318271302
#318270762
#318270742
#318270202
#318270192
#318270022
#318270002
#318269412
C4×Q8×D2832
#557×C2832
#3523283