Finite Group Databasebeta

← #49131659

Subgroups — #49131659

Order 36224.

392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#31827011C211yesmaximal, characteristic
2C2×#22583601C211yesmaximal, characteristic, complemented
3#31826832C212yesmaximal, complemented
4#31826828C212yesmaximal, complemented
5#31826821C212yesmaximal, complemented
6#31826818C212yesmaximal, complemented
7#31826467C211yesmaximal, characteristic, complemented
8#31826466C211yesmaximal, characteristic, complemented
9#31826464C211yesmaximal, characteristic, complemented
10M16×Dic283C211yesmaximal, characteristic, complemented
11#417×C283C211yesmaximal, characteristic, complemented
12C2×#17985392C2211yescharacteristic
13C2×#17985376C2211yescharacteristic
14#22583608C2211yescharacteristic
15#22583606C2211yescharacteristic
16#22583601—21no—
17#22583601—21no—
18#22583601C2212yes—
19#22583601C2212yes—
20#22583599C2211yescharacteristic
21#22583597C2211yescharacteristic
22#22583582—21no—
23#22583582—21no—
24C4×C22⋊D283—21no—
25C4×C22⋊D283—21no—
26C2×C4⋊Dic283—21no—
27C2×C4⋊Dic283C2211yescharacteristic
28C2×Dic283⋊C4—21no—
29C2×C4×Dic283C2211yescharacteristic
30#22583535C2212yes—
31#22583534C2212yes—
32#22583533C2212yes—
33#22583532C2212yes—
34#22583531C2212yes—
35#22583530C2212yes—
36#22583520C2212yes—
37C8×Dic283C2212yes—
38C2×#41×C283C2211yescharacteristic
39C2×M16×C283C2211yescharacteristic
40C2×C4⋊C4×C283C2211yescharacteristic
41#103×C283C2212yes—
42#102×C283C2212yes—
43#17985392—22no—
44#17985392—22no—
45C2×Dic566C2311yescharacteristic
46C2×D1132C2311yescharacteristic
47C2×C22⋊D283—21no—
48C2×C22⋊D283—21no—
49C2×C22⋊D283—21no—
50C22×Dic283—21no—
51C22×Dic283D411yescharacteristic
52C22×Dic283D411yescharacteristic
53C2×C4×D283—21no—
54#17985376—22no—
55C2×C283⋊C8C2312yes—
56C4⋊Dic283—22no—
57C4⋊Dic283—22no—
58C4⋊Dic283—21no—
59C4⋊Dic283C2312yes—
60Dic283⋊C4—22no—
61Dic283⋊C4—22no—
62Dic283⋊C4—21no—
63Dic283⋊C4—21no—
64C22⋊Dic283—21no—
65C22⋊Dic283—21no—
66(C2×D283)⋊C4—21no—
67(C2×D283)⋊C4—21no—
68C4×Dic283D411yescharacteristic
69C4×Dic283D411yescharacteristic
70C4×Dic283C2311yescharacteristic
71C4×Dic283C2311yescharacteristic
72#41×C283—22no—
73#41×C283—22no—
74C2×Q8×C283C2311yescharacteristic
75C2×D4×C283—21no—
76C2×D4×C283C2311yescharacteristic
77C22×C4×C283—21no—
78C22×C4×C283—21no—
79C22×C4×C283C2311yescharacteristic
80M16×C283—22no—
81C2×C8×C283C2312yes—
82C4⋊C4×C283—21no—
83C4⋊C4×C283C2312yes—
84C22×D283—21no—
85Dic566—21no—
86Dic566—21no—
87D1132—21no—
88D1132—21no—
89C22⋊D283—42no—
90C22⋊D283—42no—
91C22⋊D283—42no—
92C2×Dic283—21no—
93C2×Dic283—22no—
94C2×Dic283—41no—
95C2×Dic283—41no—
96C2×Dic283—42no—
97C2×Dic283C2×D411yescharacteristic
98C2×Dic283C2×D411yescharacteristic
99C2×Dic283C2×D411yescharacteristic
100C2×Dic283C2×D411yescharacteristic
101C4×D283—42no—
102C283⋊C8—22no—
103C23×C283—21no—
104C23×C283C2×D411yescharacteristic
105Q8×C283—21no—
106Q8×C283—21no—
107D4×C283—21no—
108D4×C283—21no—
109D4×C283—42no—
110C2×C4×C283—22no—
111C2×C4×C283—21no—
112C2×C4×C283—42no—
113C2×C4×C283—42no—
114C2×C4×C283C2×D411yescharacteristic
115C2×C4×C283#4112yes—
116C2×C4×C283#4112yes—
117C2×C4×C283C2411yescharacteristic
118C8×C283—22no—
119C22×C283—21no—
120C22×C283—21no—
121C22×C283—41no—
122C22×C283—41no—
123C22×C283—41no—
124C22×C283#12412yes—
125C22×C283C22×D411yescharacteristic
126C2×D283—41no—
127C2×D283—41no—
128C2×D283—41no—
129C4×C283—41no—
130C4×C283—42no—
131C4×C283#12412yes—
132C4×C283C2×#4111yescharacteristic
133C4×C283C2×#4111yescharacteristic
134Dic283—21no—
135Dic283—21no—
136Dic283—21no—
137Dic283—21no—
138Dic283—41no—
139Dic283—42no—
140C2×C283—22no—
141C2×C283—81no—
142C2×C283C2×#12411yescharacteristic
143C2×C283#57511yescharacteristic
144C2×C283#57511yescharacteristic
145D283—81no—
146C283#314211yescharacteristic, complemented, Sylow
147#3142—2831nomaximal, Sylow
148#532—2831no—
149C2×#125—2831no—
150#491—2832no—
151#490—2832no—
152#488—2832no—
153#488—2832no—
154#417—2831no—
155#417—2831no—
156#417—2831no—
157#417D28311yescharacteristic, complemented
158C4×M16—2831no—
159C2×#41—2831no—
160C2×#41C2×D28311yescharacteristic
161C2×M16—2831no—
162C2×M16C2×D28311yescharacteristic
163C4⋊Q8—2831no—
164#127—2831no—
165#125—2832no—
166#125—2832no—
167#125—5661no—
168#125—5661no—
169#124—2831no—
170#124—2831no—
171#121—5661no—
172#121—5661no—
173C4×D4—5661no—
174C4×D4—5661no—
175C2×C4⋊C4—2831no—
176C2×C4⋊C4—5661no—
177C2×C4⋊C4—5661no—
178C2×C4⋊C4C2×D28311yescharacteristic
179C2×C42—2831no—
180#103—2832no—
181#103—2832no—
182#103—2832no—
183#103C2×D28312yes—
184#102—2832no—
185#102—2832no—
186#102—2832no—
187#102C2×D28312yes—
188#97—2832no—
189C4×C8—2832no—
190#41—22no—
191#41—22no—
192#41—5662no—
193#41—5662no—
194C2×Q8—2831no—
195C2×Q8C22×D28311yescharacteristic
196C2×D4—21no—
197C2×D4—2831no—
198C2×D4—5661no—
199C2×D4—5661no—
200C2×D4—5661no—
201C2×D4C22×D28311yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—2831no—
205C22×C4—2831no—
206C22×C4—5661no—
207C22×C4—5661no—
208C22×C4C22×D28311yescharacteristic
209M16—22no—
210M16—5662no—
211C2×C8—2832no—
212C2×C8C22×D28312yes—
213C4⋊C4—21no—
214C4⋊C4—2832no—
215C4⋊C4—5662no—
216C4⋊C4—5662no—
217C4⋊C4—5661no—
218C4⋊C4—5661no—
219C4⋊C4—5662no—
220C4⋊C4—5662no—
221C4⋊C4—5661no—
222C4⋊C4C22×D28312yes—
223C22⋊C4—5661no—
224C22⋊C4—5661no—
225C22⋊C4—5661no—
226C22⋊C4—5661no—
227C42—2831no—
228C42—2831no—
229C42—2831no—
230C42—2831no—
231C23—21no—
232C23—5661no—
233C23D4×D28311yescharacteristic
234Q8—21no—
235Q8—21no—
236Q8—5661no—
237Q8—5661no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—5661no—
242D4—5661no—
243D4—11322no—
244D4—11322no—
245D4—11322no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—2831no—
251C2×C4—2831no—
252C2×C4—2831no—
253C2×C4—2831no—
254C2×C4—5661no—
255C2×C4—5662no—
256C2×C4—11321no—
257C2×C4—11321no—
258C2×C4—11322no—
259C2×C4—11322no—
260C2×C4D4×D28311yescharacteristic
261C2×C4#1798539112yes—
262C2×C4#1798539212yes—
263C2×C4C23×D28311yescharacteristic
264C8—22no—
265C8—5662no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—11321no—
272C22—11321no—
273C22—11321no—
274C22#2258359612yes—
275C22C2×D4×D28311yescharacteristic
276C4—41no—
277C4—42no—
278C4—5661no—
279C4—5661no—
280C4—5661no—
281C4—5661no—
282C4—11321no—
283C4—11322no—
284C4#2258359612yes—
285C4C2×#1798539111yescharacteristic
286C4C2×#1798539211yescharacteristic
287C2—22no—
288C2—81no—
289C2—22641no—
290C2C2×#2258359611yescharacteristic
291C2#3182726111yescharacteristic
292C2#3182726411yescharacteristic
subgroupG/H
centerC22C2×D4×D283
commutator, cosocleC2×C4×C283C24
FrattiniC2×C4C23×D283
Fitting#417×C283C2
socleC22×C283C22×D4
prime radical#31827011C2
hypercenter#417D283
hypocenterC283#3142
LCSG ▷ C2×C4×C283 ▷ C2×C283 ▷ C283
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#22583601C2semidirect
3. #31826832C2semidirect
4. #31826828C2semidirect
5. #31826821C2semidirect
6. #31826818C2semidirect
7. #31826467C2semidirect
8. #31826466C2semidirect
9. #31826464C2semidirect
10. M16×Dic283C2semidirect
11. #417×C283C2semidirect
146. C283#3142semidirect
157. #417D283semidirect

Maximal subgroups

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

subgroupindex
#318270112
C2×#225836012
#318268322
#318268282
#318268212
#318268182
#318264672
#318264662
#318264642
M16×Dic2832
#417×C2832
#3142283