Finite Group Databasebeta

← #49135349

Subgroups — #49135349

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#31827191C212yesmaximal, complemented
2#31827190C211yesmaximal, characteristic, complemented
3#31827190C211yesmaximal, characteristic, complemented
4#31827189C211yesmaximal, characteristic, complemented
5#31827187C212yesmaximal, complemented
6#31827157C212yesmaximal, complemented
7#31827148C214yesmaximal, complemented
8#31827130C211yesmaximal, characteristic, complemented
9#31827128C212yesmaximal, complemented
10#31827074C214yesmaximal, complemented
11#31827073C212yesmaximal, complemented
12#31827017C212yesmaximal, complemented
13#31827001C212yesmaximal
14#31826999C211yesmaximal, characteristic
15#31826941C212yesmaximal, complemented
16C2×C4×Dic566C211yesmaximal, characteristic, complemented
17#533×C283C211yesmaximal, characteristic, complemented
18C22×Dic566C2211yescharacteristic
19C2×Q8×D283C2212yes—
20#22583609C2212yes—
21#22583609C2214yes—
22#22583608C2212yes—
23#22583608C2212yes—
24#22583605C2214yes—
25#22583605C2214yes—
26#22583605C2214yes—
27#22583604C2212yes—
28#22583604C2214yes—
29#22583602C2211yescharacteristic
30#22583602C2214yes—
31#22583601C2212yes—
32#22583601C2212yes—
33#22583601C2214yes—
34#22583600C2214yes—
35#22583600C2214yes—
36#22583600C2214yes—
37#22583599C2212yes—
38#22583598C2211yescharacteristic
39#22583594C2214yes—
40#22583593C2212yes—
41#22583590C2212yes—
42#22583589C2211yescharacteristic
43#22583588C2214yes—
44#22583588C2214yes—
45#22583587C2214yes—
46#22583586C2212yes—
47#22583584C2211yescharacteristic
48Q8×Dic283C2212yes—
49Q8×Dic283C2212yes—
50Q8×Dic283C2212yes—
51C4×Dic566C2211yescharacteristic
52C4×Dic566C2211yescharacteristic
53C4×Dic566C2212yes—
54C4×Dic566C2211yescharacteristic
55C4×Dic566C2214yes—
56C4×Dic566C2212yes—
57C4×Dic566C2211yescharacteristic
58C4×Dic566C2211yescharacteristic
59#22583575C2214yes—
60#22583575C2212yes—
61#22583575C2212yes—
62C4×D1132C2211yescharacteristic
63#22583572C2212yes—
64C4×C22⋊D283C2212yes—
65(C22⋊D283)⋊C4C2214yes—
66#22583568C2214yes—
67#22583568C2212yes—
68#22583567C2212yes—
69#22583567C2214yes—
70#22583566C2214yes—
71C2×C4⋊Dic283C2211yescharacteristic
72C2×Dic283⋊C4C2212yes—
73C2×C4×Dic283C2212yes—
74C4⋊Q8×C283C2211yescharacteristic
75C4⋊Q8×C283C2211yescharacteristic
76#125×C283C2212yes—
77C22⋊Q8×C283C2212yes—
78C22⋊Q8×C283C2212yes—
79C4×Q8×C283C2212yes—
80C4×Q8×C283C2212yes—
81#117×C283C2212yes—
82C2×C42×C283C2211yescharacteristic
83C2×Dic566—24no—
84C2×Dic566C2311yescharacteristic
85C2×Dic566C2314yes—
86C2×Dic566C2312yes—
87C2×Dic566C2311yescharacteristic
88C2×Dic566C2311yescharacteristic
89Q8×D283—28no—
90C2×D1132C2311yescharacteristic
91C2×C22⋊D283C2312yes—
92C22×Dic283C2312yes—
93C22×Dic283C2312yes—
94C2×C4×D283C2314yes—
95C2×C4×D283C2312yes—
96C4⋊Dic283C2311yescharacteristic
97C4⋊Dic283C2311yescharacteristic
98C4⋊Dic283C2312yes—
99C4⋊Dic283C2314yes—
100C4⋊Dic283C2311yescharacteristic
101C4⋊Dic283C2311yescharacteristic
102C4⋊Dic283C2314yes—
103C4⋊Dic283C2314yes—
104Dic283⋊C4C2312yes—
105Dic283⋊C4C2312yes—
106Dic283⋊C4C2312yes—
107Dic283⋊C4C2312yes—
108Dic283⋊C4C2314yes—
109Dic283⋊C4C2314yes—
110Dic283⋊C4C2314yes—
111Dic283⋊C4C2312yes—
112Dic283⋊C4C2312yes—
113Dic283⋊C4C2314yes—
114Dic283⋊C4C2312yes—
115Dic283⋊C4C2314yes—
116Dic283⋊C4C2312yes—
117C22⋊Dic283C2312yes—
118(C2×D283)⋊C4C2312yes—
119(C2×D283)⋊C4C2312yes—
120(C2×D283)⋊C4C2314yes—
121(C2×D283)⋊C4C2314yes—
122(C2×D283)⋊C4C2312yes—
123(C2×D283)⋊C4C2314yes—
124(C2×D283)⋊C4C2312yes—
125C4×Dic283C2312yes—
126C4×Dic283C2312yes—
127C4×Dic283C2312yes—
128C4×Dic283C2312yes—
129C4×Dic283C2314yes—
130C4×Dic283C2314yes—
131C4×Dic283C2312yes—
132C4×Dic283C2314yes—
133C2×Q8×C283C2312yes—
134C2×Q8×C283C2312yes—
135C22×C4×C283C2311yescharacteristic
136C22×C4×C283C2311yescharacteristic
137C22×C4×C283C2311yescharacteristic
138C4⋊C4×C283C2312yes—
139C4⋊C4×C283C2314yes—
140C4⋊C4×C283C2312yes—
141C4⋊C4×C283C2312yes—
142C4⋊C4×C283C2314yes—
143C4⋊C4×C283C2312yes—
144C22⋊C4×C283C2314yes—
145C42×C283C2311yescharacteristic
146C42×C283C2314yes—
147C42×C283C2311yescharacteristic
148C42×C283C2311yescharacteristic
149C42×C283C2311yescharacteristic
150C22×D283C2412yes—
151Dic566—22no—
152Dic566—22no—
153Dic566—28no—
154Dic566—24no—
155Dic566#4114yes—
156D1132—22no—
157C22⋊D283—24no—
158C2×Dic283—24no—
159C2×Dic283—24no—
160C2×Dic283C2412yes—
161C2×Dic283C2412yes—
162C2×Dic283C2412yes—
163C2×Dic283C2412yes—
164C2×Dic283C2412yes—
165C2×Dic283C2412yes—
166C2×Dic283C2412yes—
167C4×D283—24no—
168C4×D283—28no—
169C23×C283C2411yescharacteristic
170Q8×C283—24no—
171Q8×C283—24no—
172C2×C4×C283—22no—
173C2×C4×C283—22no—
174C2×C4×C283—22no—
175C2×C4×C283C2411yescharacteristic
176C2×C4×C283C2411yescharacteristic
177C2×C4×C283C2411yescharacteristic
178C2×C4×C283C2411yescharacteristic
179C2×C4×C283C2414yes—
180C2×C4×C283C2414yes—
181C2×C4×C283C2411yescharacteristic
182C2×C4×C283C2411yescharacteristic
183C22×C283—21no—
184C22×C283—21no—
185C22×C2832−1+412yes—
186C22×C283C2511yescharacteristic
187C2×D283—22no—
188C2×D283—22no—
189C2×D283—22no—
190C4×C283—21no—
191C4×C283—24no—
192C4×C283—21no—
193C4×C283—21no—
194C4×C283—21no—
195C4×C283—24no—
196C4×C283C2×#4112yes—
197C4×C2832−1+412yes—
198Dic283—22no—
199Dic283—22no—
200Dic283—22no—
201Dic283—22no—
202Dic283—22no—
203Dic283—22no—
204Dic283C2×#4114yes—
205C2×C283—22no—
206C2×C283C22×#4111yescharacteristic
207C2×C283C2×2−1+411yescharacteristic
208C2×C283C2×2−1+411yescharacteristic
209D283—42no—
210C283#352311yescharacteristic, complemented, Sylow
211#3523—2831nomaximal, Sylow
212#557—2831no—
213#557—2831no—
214#557—2832no—
215#556—2831no—
216#556—2832no—
217#552—2832no—
218#552—2834no—
219D4×Q8—2831no—
220D4×Q8—2832no—
221#541—2834no—
222#541—2832no—
223#533—2832no—
224#533D28311yescharacteristic, complemented
225#529—2832no—
226#529—2831no—
227#519—2832no—
228C2×C4×Q8—2831no—
229C22×Q8—2832no—
230C22×Q8—2831no—
231C4⋊Q8—2832no—
232C4⋊Q8—2832no—
233C4⋊Q8—2832no—
234C4⋊Q8—2834no—
235C4⋊Q8C2×D28311yescharacteristic
236C4⋊Q8C2×D28311yescharacteristic
237#126—2834no—
238#126—2834no—
239#126—2834no—
240#126—2832no—
241#126—2834no—
242#125—2832no—
243#125—2834no—
244#125—2834no—
245#125—2834no—
246#125—2834no—
247#125—2832no—
248#125—2831no—
249#125—2834no—
250#125C2×D28312yes—
251#124—2832no—
252#124—2831no—
253#123—2832no—
254#123—2834no—
255C22⋊Q8—2832no—
256C22⋊Q8—2831no—
257C22⋊Q8—2834no—
258C22⋊Q8—2834no—
259C22⋊Q8—2832no—
260C22⋊Q8—2834no—
261C22⋊Q8C2×D28312yes—
262C22⋊Q8C2×D28312yes—
263#121—2831no—
264C4×Q8—2832no—
265C4×Q8—2832no—
266C4×Q8—2832no—
267C4×Q8—2834no—
268C4×Q8—2832no—
269C4×Q8—2832no—
270C4×Q8—2831no—
271C4×Q8—2832no—
272C4×Q8—2831no—
273C4×Q8—2831no—
274C4×Q8—2832no—
275C4×Q8—2831no—
276C4×Q8—2834no—
277C4×Q8—2831no—
278C4×Q8C2×D28312yes—
279C4×Q8C2×D28312yes—
280C4×D4—2832no—
281C4×D4—2832no—
282C4×D4—2834no—
283C4×D4—2831no—
284#117—2834no—
285#117—2832no—
286#117—2834no—
287#117—2832no—
288#117—2834no—
289#117C2×D28312yes—
290C2×C4⋊C4—2832no—
291C2×C4⋊C4—2831no—
292C2×C42—2832no—
293C2×C42C2×D28311yescharacteristic
294C2×Q8—2831no—
295C2×Q8—2831no—
296C2×Q8—2834no—
297C2×Q8—2832no—
298C2×Q8—2831no—
299C2×Q8—5664no—
300C2×Q8—5668no—
301C2×Q8C22×D28312yes—
302C2×Q8C22×D28312yes—
303C2×D4—2832no—
304C2×D4—2831no—
305C22×C4—2832no—
306C22×C4—2832no—
307C22×C4—2834no—
308C22×C4—2832no—
309C22×C4C22×D28311yescharacteristic
310C22×C4C22×D28311yescharacteristic
311C22×C4C22×D28311yescharacteristic
312C4⋊C4—2832no—
313C4⋊C4—2832no—
314C4⋊C4—2832no—
315C4⋊C4—2832no—
316C4⋊C4—2834no—
317C4⋊C4—2832no—
318C4⋊C4—2832no—
319C4⋊C4—2834no—
320C4⋊C4—2834no—
321C4⋊C4—2834no—
322C4⋊C4—2832no—
323C4⋊C4—2834no—
324C4⋊C4—2832no—
325C4⋊C4—2831no—
326C4⋊C4—2831no—
327C4⋊C4—2832no—
328C4⋊C4—2834no—
329C4⋊C4—2831no—
330C4⋊C4—2831no—
331C4⋊C4—2834no—
332C4⋊C4—2834no—
333C4⋊C4C22×D28312yes—
334C4⋊C4C22×D28314yes—
335C4⋊C4C22×D28312yes—
336C4⋊C4C22×D28312yes—
337C4⋊C4C22×D28312yes—
338C4⋊C4C22×D28314yes—
339C22⋊C4—2832no—
340C22⋊C4—2832no—
341C22⋊C4—2834no—
342C22⋊C4—2832no—
343C22⋊C4—2834no—
344C22⋊C4—2832no—
345C22⋊C4—2832no—
346C22⋊C4—2834no—
347C22⋊C4C22×D28314yes—
348C42—2832no—
349C42—2832no—
350C42—2832no—
351C42—2832no—
352C42—2834no—
353C42—2834no—
354C42—2834no—
355C42—2832no—
356C42C22×D28311yescharacteristic
357C42C22×D28311yescharacteristic
358C42C22×D28311yescharacteristic
359C42C22×D28311yescharacteristic
360C42C22×D28314yes—
361C23—2832no—
362C23C23×D28311yescharacteristic
363Q8—24no—
364Q8—24no—
365Q8—2834no—
366Q8—5664no—
367Q8—5668no—
368Q8—5662no—
369Q8—5662no—
370D4—5664no—
371D4—5662no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—2832no—
376C2×C4—2832no—
377C2×C4—2832no—
378C2×C4—2832no—
379C2×C4—2832no—
380C2×C4—2832no—
381C2×C4—2832no—
382C2×C4—5668no—
383C2×C4—5664no—
384C2×C4—5664no—
385C2×C4—5664no—
386C2×C4C23×D28311yescharacteristic
387C2×C4C23×D28314yes—
388C2×C4C23×D28314yes—
389C2×C4C23×D28311yescharacteristic
390C2×C4C23×D28311yescharacteristic
391C2×C4C23×D28311yescharacteristic
392C2×C4C23×D28311yescharacteristic
393C2×C4C23×D28311yescharacteristic
394C22—21no—
395C22—21no—
396C22—5662no—
397C22—5662no—
398C22—5662no—
399C22#2258365812yes—
400C22C24×D28311yescharacteristic
401C4—21no—
402C4—24no—
403C4—21no—
404C4—24no—
405C4—21no—
406C4—21no—
407C4—2834no—
408C4—5662no—
409C4—5662no—
410C4—5662no—
411C4—5662no—
412C4—5662no—
413C4—5662no—
414C4#41×D28312yes—
415C4#2258365712yes—
416C2—22no—
417C2—11322no—
418C2C2×#41×D28311yescharacteristic
419C2C2×#2258365711yescharacteristic
420C2C2×#2258365811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D283
commutator, socle, cosocleC22×C283C25
Fitting#533×C283C2
prime radical#22583584C22
hypercenter#533D283
hypocenterC283#3523
LCSG ▷ C22×C283 ▷ C283
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. #31827191C2semidirect
2. #31827190C2semidirect
3. #31827190C2semidirect
4. #31827189C2semidirect
5. #31827187C2semidirect
6. #31827157C2semidirect
7. #31827148C2semidirect
8. #31827130C2semidirect
9. #31827128C2semidirect
10. #31827074C2semidirect
11. #31827073C2semidirect
12. #31827017C2semidirect
15. #31826941C2semidirect
16. C2×C4×Dic566C2semidirect
17. #533×C283C2semidirect
210. C283#3523semidirect
224. #533D283semidirect

Maximal subgroups

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

subgroupindex
#318271912
#318271902
#318271892
#318271872
#318271572
#318271482
#318271302
#318271282
#318270742
#318270732
#318270172
#318270012
#318269992
#318269412
C2×C4×Dic5662
#533×C2832
#3523283