Finite Group Databasebeta

← #49135356

Subgroups — #49135356

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#31827191C212yesmaximal, complemented
3#31827190C211yesmaximal, characteristic, complemented
4#31827184C212yesmaximal, complemented
5#31827154C212yesmaximal, complemented
6#31827151C214yesmaximal, complemented
7D4×Dic566C211yesmaximal, characteristic, complemented
8#31827128C212yesmaximal, complemented
9#31827075C212yesmaximal, complemented
10#31827072C214yesmaximal, complemented
11#31827020C211yesmaximal, characteristic, complemented
12#31827019C212yesmaximal, complemented
13#31826997C211yesmaximal, characteristic
14#31826994C212yesmaximal
15#31826942C212yesmaximal, complemented
16C4×Q8×D283C211yesmaximal, characteristic, complemented
17#556×C283C211yesmaximal, characteristic, complemented
18C22×Dic566C2212yes—
19C2×Q8×D283C2211yescharacteristic
20#22583610C2211yescharacteristic
21#22583609C2212yes—
22#22583609C2214yes—
23#22583608C2212yes—
24#22583608C2212yes—
25#22583603C2214yes—
26#22583603C2212yes—
27#22583603C2214yes—
28#22583603C2214yes—
29#22583602C2212yes—
30#22583602C2214yes—
31#22583601C2211yescharacteristic
32#22583601C2212yes—
33#22583601C2214yes—
34#22583601C2212yes—
35#22583600C2214yes—
36#22583600C2214yes—
37#22583600C2214yes—
38#22583597C2211yescharacteristic
39#22583592C2214yes—
40#22583592C2212yes—
41#22583590C2211yescharacteristic
42#22583590C2212yes—
43#22583589C2212yes—
44#22583588C2214yes—
45#22583587C2214yes—
46#22583587C2214yes—
47#22583586C2212yes—
48#22583581C2211yescharacteristic
49Q8×Dic283C2212yes—
50Q8×Dic283C2212yes—
51Q8×Dic283C2211yescharacteristic
52C4×Dic566C2211yescharacteristic
53C4×Dic566C2214yes—
54C4×Dic566C2212yes—
55C4×Dic566C2211yescharacteristic
56C4×Dic566C2214yes—
57C4×Dic566C2212yes—
58#22583575C2212yes—
59#22583575C2212yes—
60#22583575C2212yes—
61#22583575C2212yes—
62#22583575C2211yescharacteristic
63D4×Dic283C2212yes—
64C4×C22⋊D283C2212yes—
65(C22⋊D283)⋊C4C2214yes—
66#22583569C2212yes—
67#22583568C2212yes—
68#22583567C2214yes—
69#22583566C2214yes—
70#22583566C2214yes—
71C4⋊C4×D283C2212yes—
72C4⋊C4×D283C2211yescharacteristic
73C42×D283C2212yes—
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—
83C2×Dic566—28no—
84C2×Dic566C2312yes—
85C2×Dic566C2312yes—
86C2×Dic566C2311yescharacteristic
87C2×Dic566C2314yes—
88C2×Dic566C2311yescharacteristic
89Q8×D283—24no—
90C2×C22⋊D283C2312yes—
91C22×Dic283C2314yes—
92C2×C4×D283C2311yescharacteristic
93C2×C4×D283C2312yes—
94C2×C4×D283C2311yescharacteristic
95C2×C4×D283C2312yes—
96C2×C4×D283C2311yescharacteristic
97C4⋊Dic283C2311yescharacteristic
98C4⋊Dic283C2312yes—
99C4⋊Dic283C2312yes—
100C4⋊Dic283C2312yes—
101C4⋊Dic283C2312yes—
102C4⋊Dic283C2312yes—
103C4⋊Dic283C2311yescharacteristic
104C4⋊Dic283C2314yes—
105C4⋊Dic283C2314yes—
106Dic283⋊C4C2312yes—
107Dic283⋊C4C2312yes—
108Dic283⋊C4C2312yes—
109Dic283⋊C4C2314yes—
110Dic283⋊C4C2314yes—
111Dic283⋊C4C2314yes—
112Dic283⋊C4C2314yes—
113Dic283⋊C4C2314yes—
114Dic283⋊C4C2314yes—
115Dic283⋊C4C2312yes—
116Dic283⋊C4C2314yes—
117Dic283⋊C4C2312yes—
118C22⋊Dic283C2312yes—
119C22⋊Dic283C2312yes—
120C22⋊Dic283C2312yes—
121C22⋊Dic283C2314yes—
122(C2×D283)⋊C4C2312yes—
123(C2×D283)⋊C4C2314yes—
124C4×Dic283C2312yes—
125C4×Dic283C2311yescharacteristic
126C4×Dic283C2312yes—
127C4×Dic283C2311yescharacteristic
128C4×Dic283C2312yes—
129C4×Dic283C2311yescharacteristic
130C4×Dic283C2314yes—
131C4×Dic283C2314yes—
132C4×Dic283C2314yes—
133C4×Dic283C2312yes—
134C2×Q8×C283C2312yes—
135C2×Q8×C283C2311yescharacteristic
136C2×D4×C283C2311yescharacteristic
137C22×C4×C283C2312yes—
138C4⋊C4×C283C2311yescharacteristic
139C4⋊C4×C283C2312yes—
140C4⋊C4×C283C2312yes—
141C4⋊C4×C283C2312yes—
142C4⋊C4×C283C2314yes—
143C4⋊C4×C283C2311yescharacteristic
144C22⋊C4×C283C2314yes—
145C22⋊C4×C283C2314yes—
146C22⋊C4×C283C2312yes—
147C42×C283C2312yes—
148C42×C283C2311yescharacteristic
149C42×C283C2314yes—
150C22×D283C2411yescharacteristic
151Dic566—24no—
152Dic566—22no—
153Dic566—24no—
154Dic566—28no—
155Dic566#4114yes—
156C22⋊D283—24no—
157C2×Dic283—28no—
158C2×Dic283C2411yescharacteristic
159C2×Dic283C2411yescharacteristic
160C2×Dic283C2412yes—
161C2×Dic283C2412yes—
162C2×Dic283C2414yes—
163C2×Dic283C2412yes—
164C2×Dic283C2411yescharacteristic
165C2×Dic283C2412yes—
166C4×D283—24no—
167C4×D283—22no—
168C4×D283—22no—
169C4×D283—22no—
170C4×D283—24no—
171C23×C283C2412yes—
172Q8×C283—22no—
173Q8×C283—24no—
174D4×C283—22no—
175C2×C4×C283—24no—
176C2×C4×C283C2412yes—
177C2×C4×C283C2412yes—
178C2×C4×C283C2411yescharacteristic
179C2×C4×C283C2414yes—
180C2×C4×C283C2412yes—
181C2×C4×C283C2411yescharacteristic
182C2×C4×C283C2411yescharacteristic
183C22×C283—22no—
184C22×C283—22no—
185C22×C283—22no—
186C22×C283C2511yescharacteristic
187C2×D283—21no—
188C2×D283—21no—
189C2×D2832−1+412yes—
190C4×C283—22no—
191C4×C283—22no—
192C4×C283—24no—
193C4×C283—22no—
194C4×C283—21no—
195C4×C283C2×#4112yes—
196C4×C2832−1+412yes—
197Dic283—21no—
198Dic283—24no—
199Dic283—21no—
200Dic283—22no—
201Dic283—21no—
202Dic283—22no—
203Dic283—22no—
204Dic283C2×#4114yes—
205C2×C283—42no—
206C2×C283C22×#4111yescharacteristic
207C2×C283C2×2−1+411yescharacteristic
208C2×C283C2×2−1+411yescharacteristic
209D283—22no—
210C283#352311yescharacteristic, complemented, Sylow
211#3523—2831nomaximal, Sylow
212#557—2831no—
213#557—2832no—
214#557—2831no—
215#556—2832no—
216#556D28311yescharacteristic, complemented
217#552—2834no—
218#552—2832no—
219D4×Q8—2832no—
220D4×Q8—2831no—
221#541—2834no—
222#541—2832no—
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—2832no—
234C4⋊Q8—2834no—
235C4⋊Q8—2831no—
236C4⋊Q8C2×D28311yescharacteristic
237#126—2834no—
238#126—2832no—
239#126—2834no—
240#126—2834no—
241#126C2×D28314yes—
242#125—2831no—
243#125—2834no—
244#125—2832no—
245#125—2834no—
246#125—2834no—
247#125—2834no—
248#125—2832no—
249#125—2832no—
250#125—2834no—
251#124—2831no—
252#124C2×D28312yes—
253#123—2832no—
254#123—2834no—
255C22⋊Q8—2832no—
256C22⋊Q8—2832no—
257C22⋊Q8—2831no—
258C22⋊Q8—2834no—
259C22⋊Q8—2832no—
260C22⋊Q8—2834no—
261C22⋊Q8—2834no—
262C22⋊Q8C2×D28312yes—
263#121—2831no—
264C4×Q8—2831no—
265C4×Q8—2832no—
266C4×Q8—2832no—
267C4×Q8—2831no—
268C4×Q8—2832no—
269C4×Q8—2832no—
270C4×Q8—2832no—
271C4×Q8—2832no—
272C4×Q8—2831no—
273C4×Q8—2832no—
274C4×Q8—2834no—
275C4×Q8—2834no—
276C4×Q8—2831no—
277C4×Q8—2832no—
278C4×Q8C2×D28311yescharacteristic
279C4×Q8C2×D28312yes—
280C4×D4—2832no—
281C4×D4—2832no—
282C4×D4—2834no—
283C4×D4C2×D28311yescharacteristic
284#117—2832no—
285#117—2832no—
286#117—2834no—
287#117—2834no—
288#117—2834no—
289#117C2×D28312yes—
290C2×C4⋊C4—2832no—
291C2×C4⋊C4—2831no—
292C2×C42—2832no—
293C2×C42—2831no—
294C2×Q8—2831no—
295C2×Q8—2831no—
296C2×Q8—2832no—
297C2×Q8—2834no—
298C2×Q8—2832no—
299C2×Q8—5668no—
300C2×Q8—5664no—
301C2×Q8C22×D28312yes—
302C2×Q8C22×D28311yescharacteristic
303C2×D4—2832no—
304C2×D4C22×D28311yescharacteristic
305C22×C4—2834no—
306C22×C4—2831no—
307C22×C4—2831no—
308C22×C4—2832no—
309C22×C4—2832no—
310C22×C4—2831no—
311C22×C4C22×D28312yes—
312C4⋊C4—2832no—
313C4⋊C4—2832no—
314C4⋊C4—2832no—
315C4⋊C4—2834no—
316C4⋊C4—2834no—
317C4⋊C4—2834no—
318C4⋊C4—2834no—
319C4⋊C4—2834no—
320C4⋊C4—2834no—
321C4⋊C4—2832no—
322C4⋊C4—2834no—
323C4⋊C4—2832no—
324C4⋊C4—2831no—
325C4⋊C4—2832no—
326C4⋊C4—2832no—
327C4⋊C4—2832no—
328C4⋊C4—2832no—
329C4⋊C4—2832no—
330C4⋊C4—2831no—
331C4⋊C4—2834no—
332C4⋊C4—2834no—
333C4⋊C4C22×D28311yescharacteristic
334C4⋊C4C22×D28312yes—
335C4⋊C4C22×D28312yes—
336C4⋊C4C22×D28312yes—
337C4⋊C4C22×D28311yescharacteristic
338C4⋊C4C22×D28314yes—
339C22⋊C4—2832no—
340C22⋊C4—2832no—
341C22⋊C4—2834no—
342C22⋊C4—2832no—
343C22⋊C4—2832no—
344C22⋊C4—2834no—
345C22⋊C4C22×D28314yes—
346C22⋊C4C22×D28314yes—
347C22⋊C4C22×D28312yes—
348C42—2832no—
349C42—2831no—
350C42—2832no—
351C42—2831no—
352C42—2832no—
353C42—2831no—
354C42—2834no—
355C42—2834no—
356C42—2834no—
357C42—2832no—
358C42C22×D28311yescharacteristic
359C42C22×D28312yes—
360C42C22×D28314yes—
361C23—2831no—
362C23C23×D28312yes—
363Q8—22no—
364Q8—24no—
365Q8—2834no—
366Q8—5668no—
367Q8—5664no—
368Q8—5664no—
369Q8—5662no—
370D4—22no—
371D4—5664no—
372C2×C4—24no—
373C2×C4—2832no—
374C2×C4—2831no—
375C2×C4—2831no—
376C2×C4—2832no—
377C2×C4—2832no—
378C2×C4—2832no—
379C2×C4—2831no—
380C2×C4—2834no—
381C2×C4—5664no—
382C2×C4—5664no—
383C2×C4—5662no—
384C2×C4—5668no—
385C2×C4—5662no—
386C2×C4—5662no—
387C2×C4C23×D28312yes—
388C2×C4C23×D28314yes—
389C2×C4C23×D28311yescharacteristic
390C2×C4C23×D28312yes—
391C2×C4C23×D28312yes—
392C2×C4C23×D28311yescharacteristic
393C2×C4C23×D28311yescharacteristic
394C22—22no—
395C22—22no—
396C22—22no—
397C22—2832no—
398C22—5661no—
399C22—5661no—
400C22C24×D28311yescharacteristic
401C4—22no—
402C4—22no—
403C4—22no—
404C4—21no—
405C4—24no—
406C4—2834no—
407C4—5664no—
408C4—5661no—
409C4—5661no—
410C4—5662no—
411C4—5662no—
412C4—5661no—
413C4—5662no—
414C4#41×D28312yes—
415C4#2258365712yes—
416C2—42no—
417C2—5662no—
418C2C2×#41×D28311yescharacteristic
419C22−1+4×D28311yescharacteristic
420C2C2×#2258365711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#556×C283C2
prime radical#22583581C22
hypercenter#556D283
hypocenterC283#3523
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #31827193C2semidirect
2. #31827191C2semidirect
3. #31827190C2semidirect
4. #31827184C2semidirect
5. #31827154C2semidirect
6. #31827151C2semidirect
7. D4×Dic566C2semidirect
8. #31827128C2semidirect
9. #31827075C2semidirect
10. #31827072C2semidirect
11. #31827020C2semidirect
12. #31827019C2semidirect
15. #31826942C2semidirect
16. C4×Q8×D283C2semidirect
17. #556×C283C2semidirect
210. C283#3523semidirect
216. #556D283semidirect

Maximal subgroups

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

subgroupindex
#318271932
#318271912
#318271902
#318271842
#318271542
#318271512
D4×Dic5662
#318271282
#318270752
#318270722
#318270202
#318270192
#318269972
#318269942
#318269422
C4×Q8×D2832
#556×C2832
#3523283