Finite Group Databasebeta

← #46784051

Subgroups — #46784051

Order 33664.

264 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 63 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#30682736C211yesmaximal, characteristic
2#30681964C211yesmaximal, characteristic, complemented
3#30681935C211yesmaximal, characteristic, complemented
4#30681912C211yesmaximal, characteristic, complemented
5#30681911C211yesmaximal, characteristic, complemented
6#30681694C211yesmaximal, characteristic, complemented
7#416×C263C211yesmaximal, characteristic, complemented
8C2×#16862376C2211yescharacteristic
9#22554501—21no—
10#22554500—21no—
11#22554496—21no—
12#22554495—21no—
13#22554490—22no—
14C2×#16862360C2211yescharacteristic
15#22554429C2211yescharacteristic
16#22554429C2211yescharacteristic
17Dic263⋊C8—22no—
18#22554395—21no—
19#22554392—21no—
20#22554385—22no—
21#22554384—22no—
22C2×#41×C263C2211yescharacteristic
23C22×C8×C263C2211yescharacteristic
24#117×C263C2211yescharacteristic
25#103×C263—21no—
26#102×C263—21no—
27#16862376—21no—
28#16862376—21no—
29#16862376—21no—
30#16862376—21no—
31C2×Dic526D411yescharacteristic
32C2×D1052D411yescharacteristic
33C2×C22⋊D263—21no—
34C2×C4×D263—21no—
35#16862366—22no—
36#16862364—22no—
37#16862363—22no—
38#16862361—22no—
39#16862360—21no—
40#16862360—21no—
41C2×C263⋊C8D412yes—
42C4⋊Dic263—21no—
43C4⋊Dic263—21no—
44C22⋊Dic263—21no—
45C22⋊Dic263D412yes—
46C4×Dic263—21no—
47C4×Dic263—21no—
48#41×C263—21no—
49#41×C263—21no—
50#41×C263—21no—
51#41×C263—21no—
52C2×Q8×C263D411yescharacteristic
53C2×D4×C263—21no—
54C2×D4×C263D411yescharacteristic
55C22×C4×C263—21no—
56C22×C4×C263C2311yescharacteristic
57C2×C8×C263—21no—
58C2×C8×C263—21no—
59C2×C8×C263—21no—
60C4⋊C4×C263—21no—
61C22⋊C4×C263—21no—
62C42×C263—21no—
63C22×D263—21no—
64Dic526—21no—
65Dic526—21no—
66D1052—21no—
67D1052—21no—
68C22⋊D263—41no—
69C22⋊D263—41no—
70C2×Dic263—22no—
71C2×Dic263—21no—
72C2×Dic263—41no—
73C4×D263—41no—
74C4×D263—41no—
75C263⋊C8—22no—
76C23×C263—21no—
77C23×C263C2×D411yescharacteristic
78Q8×C263—21no—
79Q8×C263—21no—
80D4×C263—21no—
81D4×C263—21no—
82D4×C263—41no—
83D4×C263—41no—
84C2×C4×C263—21no—
85C2×C4×C263—41no—
86C2×C4×C263—41no—
87C2×C4×C263—41no—
88C2×C4×C263C2×D411yescharacteristic
89C2×C4×C263C2×D411yescharacteristic
90C2×C4×C263C2×D411yescharacteristic
91C2×C4×C263#4111yescharacteristic
92C2×C4×C263#4111yescharacteristic
93C2×C4×C263#4111yescharacteristic
94C8×C263—41no—
95C22×C263—41no—
96C22×C263—41no—
97C22×C263—81no—
98C22×C263#12111yescharacteristic
99C22×C263#12111yescharacteristic
100C22×C263#12111yescharacteristic
101C2×D263—41no—
102C2×D263—81no—
103C4×C263—41no—
104C4×C263—81no—
105C4×C263#12011yescharacteristic
106C4×C263#12311yescharacteristic
107C4×C263#12311yescharacteristic
108C4×C263#12411yescharacteristic
109Dic263—42no—
110Dic263—41no—
111Dic263—81no—
112C2×C263—21no—
113C2×C263—21no—
114C2×C263—21no—
115C2×C263—81no—
116C2×C263#39411yescharacteristic
117D263—81no—
118C263#205311yescharacteristic, complemented, Sylow
119#2053—2631nomaximal, Sylow
120#575—2631no—
121#424—2631no—
122#416—2631no—
123#416D26311yescharacteristic, complemented
124#410—2631no—
125#410—2631no—
126H(3,ℤ/4)—2631no—
127C2×#41—2631no—
128C2×#41C2×D26311yescharacteristic
129#137—5261no—
130#137—5261no—
131AGL(1,ℤ/8)—5261no—
132AGL(1,ℤ/8)—5261no—
133#135—5262no—
134C2×M16—2631no—
135C22×C8C2×D26311yescharacteristic
136#117—2631no—
137#117—2631no—
138#117C2×D26311yescharacteristic
139C4⋊C8—5262no—
140#103—21no—
141#103—5261no—
142#102—21no—
143#102—5261no—
144#99—5262no—
145#99—5262no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—5261no—
151#41—5261no—
152#41—5261no—
153#41—5261no—
154C2×Q8—2631no—
155C2×Q8C22⋊D26311yescharacteristic
156C2×D4—21no—
157C2×D4—2631no—
158C2×D4—5261no—
159C2×D4C22⋊D26311yescharacteristic
160C22×C4—21no—
161C22×C4—5261no—
162C22×C4C22×D26311yescharacteristic
163Q16—5262no—
164SD16—5262no—
165SD16—5262no—
166D8—5262no—
167M16—5261no—
168M16—5261no—
169C2×C8—21no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—2632no—
173C4⋊C4—21no—
174C4⋊C4—5261no—
175C4⋊C4—5261no—
176C22⋊C4—21no—
177C22⋊C4—2632no—
178C22⋊C4—5261no—
179C42—21no—
180C42—5261no—
181C42—5261no—
182C23—21no—
183C23—5261no—
184C23D4×D26311yescharacteristic
185Q8—21no—
186Q8—21no—
187Q8—5261no—
188Q8—5261no—
189D4—21no—
190D4—21no—
191D4—41no—
192D4—41no—
193D4—5261no—
194D4—5261no—
195D4—10521no—
196D4—10521no—
197C2×C4—21no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—5262no—
202C2×C4—5261no—
203C2×C4—10521no—
204C2×C4—10521no—
205C2×C4—10521no—
206C2×C4D4×D26311yescharacteristic
207C2×C4D4×D26311yescharacteristic
208C2×C4C2×C22⋊D26311yescharacteristic
209C2×C4#1686237511yescharacteristic
210C2×C4#1686237611yescharacteristic
211C2×C4#1686237611yescharacteristic
212C8—41no—
213C8—5262no—
214C22—41no—
215C22—41no—
216C22—81no—
217C22—10521no—
218C22—21041no—
219C22#2255444211yescharacteristic
220C22#2255444311yescharacteristic
221C22#2255444311yescharacteristic
222C4—41no—
223C4—81no—
224C4—10522no—
225C4—10521no—
226C4—21041no—
227C4#2255443811yescharacteristic
228C4#2255445211yescharacteristic
229C4#2255445311yescharacteristic
230C4#2255445611yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—21041no—
236C2#3068185311yescharacteristic
subgroupG/H
centerC4#22554456
commutator, cosocleC22×C4×C263C23
FrattiniC22×C4C22×D263
Fitting#416×C263C2
socleC2×C263#394
prime radical#30682736C2
hypercenter#416D263
hypocenterC263#2053
LCSG ▷ C22×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. #30681964C2semidirect
3. #30681935C2semidirect
4. #30681912C2semidirect
5. #30681911C2semidirect
6. #30681694C2semidirect
7. #416×C263C2semidirect
118. C263#2053semidirect
123. #416D263semidirect

Maximal subgroups

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

subgroupindex
#306827362
#306819642
#306819352
#306819122
#306819112
#306816942
#416×C2632
#2053263