Finite Group Databasebeta

← #46784017

Subgroups — #46784017

Order 33664.

274 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 67 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#30681986C211yesmaximal, characteristic, complemented
2#30681955C211yesmaximal, characteristic, complemented
3#30681943C211yesmaximal, characteristic, complemented
4#30681922C211yesmaximal, characteristic, complemented
5C2×#22554387C211yesmaximal, characteristic, complemented
6#30681710C211yesmaximal, characteristic, complemented
7C2×AGL(1,ℤ/8)×C263C211yesmaximal, characteristic
8C2×#16862360C2211yescharacteristic
9C2×#16862360C2211yescharacteristic
10C2×#16862360C2211yescharacteristic
11#22554429C2211yescharacteristic
12#22554405—22no—
13#22554397—22no—
14#22554391—22no—
15#22554389—21no—
16#22554387—22no—
17#22554387—21no—
18C2×#41×C263C2211yescharacteristic
19C22×D4×C263C2211yescharacteristic
20AGL(1,ℤ/8)×C263—22no—
21AGL(1,ℤ/8)×C263—22no—
22C2×SD16×C263—21no—
23C2×D8×C263—21no—
24C2×M16×C263C2211yescharacteristic
25#16862360—21no—
26#16862360—21no—
27#16862360—21no—
28#16862360—21no—
29#16862360—21no—
30#16862360D412yes—
31C2×C263⋊C8—21no—
32C2×C263⋊C8—21no—
33C2×C263⋊C8—21no—
34C4⋊Dic263—21no—
35C22⋊Dic263—21no—
36C4×Dic263—21no—
37C24×C263—21no—
38#41×C263—21no—
39#41×C263—21no—
40#41×C263—21no—
41#41×C263—21no—
42C2×Q8×C263D411yescharacteristic
43C2×D4×C263—21no—
44C2×D4×C263—21no—
45C2×D4×C263—21no—
46C2×D4×C263—21no—
47C2×D4×C263—21no—
48C2×D4×C263—21no—
49C2×D4×C263D412yes—
50C2×D4×C263D411yescharacteristic
51C22×C4×C263—21no—
52C22×C4×C263C2311yescharacteristic
53SD16×C263—24no—
54D8×C263—24no—
55M16×C263—21no—
56M16×C263D412yes—
57C2×C8×C263—21no—
58C2×Dic263—41no—
59C263⋊C8—22no—
60C263⋊C8—41no—
61C263⋊C8—41no—
62C23×C263—22no—
63C23×C263—21no—
64C23×C263—41no—
65C23×C263—41no—
66C23×C263—82no—
67C23×C263#4111yescharacteristic
68Q8×C263—21no—
69Q8×C263—21no—
70D4×C263—21no—
71D4×C263—22no—
72D4×C263—21no—
73D4×C263—22no—
74D4×C263—41no—
75D4×C263—41no—
76D4×C263—41no—
77D4×C263—41no—
78C2×C4×C263—21no—
79C2×C4×C263—41no—
80C2×C4×C263—41no—
81C2×C4×C263C2×D411yescharacteristic
82C2×C4×C263C2×D411yescharacteristic
83C2×C4×C263C2×D411yescharacteristic
84C2×C4×C263C2×D411yescharacteristic
85C2×C4×C263#4111yescharacteristic
86C2×C4×C263#4111yescharacteristic
87C8×C263—22no—
88C22×C263—41no—
89C22×C263—41no—
90C22×C263—42no—
91C22×C263—81no—
92C22×C263—82no—
93C22×C263—82no—
94C22×C263—82no—
95C22×C263#12111yescharacteristic
96C22×C263#12311yescharacteristic
97C22×C263#12411yescharacteristic
98C4×C263—41no—
99C4×C263#12011yescharacteristic
100C4×C263#12111yescharacteristic
101C4×C263#12111yescharacteristic
102C4×C263#12311yescharacteristic
103Dic263—81no—
104C2×C263—21no—
105C2×C263—21no—
106C2×C263—21no—
107C2×C263—81no—
108C2×C263—82no—
109C2×C263#39411yescharacteristic
110C263#204811yescharacteristic, complemented, Sylow
111#2048—2631nomaximal, Sylow
112C2×AGL(1,ℤ/8)D26311yescharacteristic
113#430—2631no—
114#421—2631no—
115#418—2631no—
116#413—2631no—
117C2×#100—2631no—
118#344—2631no—
119C2×#41C2×D26311yescharacteristic
120C22×D4C2×D26311yescharacteristic
121AGL(1,ℤ/8)—22no—
122AGL(1,ℤ/8)—22no—
123C2×SD16—21no—
124C2×D8—21no—
125C2×M16—2631no—
126C2×M16—2631no—
127C2×M16—2631no—
128C2×M16C2×D26311yescharacteristic
129#117—2631no—
130#108—5262no—
131#104—5262no—
132#102—5262no—
133#101—5261no—
134#100—5262no—
135#100—5261no—
136C24—21no—
137#41—21no—
138#41—21no—
139#41—21no—
140#41—21no—
141C2×Q8C22⋊D26311yescharacteristic
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4C22⋊D26311yescharacteristic
149C2×D4C22⋊D26312yes—
150C22×C4—21no—
151C22×C4C22×D26311yescharacteristic
152SD16—24no—
153D8—24no—
154M16—21no—
155M16—2632no—
156M16—5261no—
157M16—5261no—
158M16—5261no—
159M16—5261no—
160M16—5261no—
161M16C22⋊D26312yes—
162C2×C8—21no—
163C2×C8—5261no—
164C2×C8—5261no—
165C2×C8—5261no—
166C4⋊C4—5261no—
167C22⋊C4—5261no—
168C42—5261no—
169C23—22no—
170C23—21no—
171C23—41no—
172C23—41no—
173C23—82no—
174C23#1686237511yescharacteristic
175Q8—21no—
176Q8—21no—
177D4—21no—
178D4—22no—
179D4—21no—
180D4—22no—
181D4—41no—
182D4—41no—
183D4—41no—
184D4—41no—
185C2×C4—21no—
186C2×C4—41no—
187C2×C4—41no—
188C2×C4—10521no—
189C2×C4D4×D26311yescharacteristic
190C2×C4C2×C22⋊D26311yescharacteristic
191C2×C4C2×C22⋊D26311yescharacteristic
192C2×C4C2×C22⋊D26311yescharacteristic
193C2×C4#1686237511yescharacteristic
194C2×C4#1686237511yescharacteristic
195C8—22no—
196C8—5262no—
197C8—10521no—
198C8—10521no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—82no—
203C22—81no—
204C22—82no—
205C22—82no—
206C22#2255444211yescharacteristic
207C22#2255445111yescharacteristic
208C22#2255445711yescharacteristic
209C4—41no—
210C4—21041no—
211C4#2255443911yescharacteristic
212C4#2255444111yescharacteristic
213C4#2255444211yescharacteristic
214C4#2255445111yescharacteristic
215C2—21no—
216C2—21no—
217C2—21no—
218C2—82no—
219C2—81no—
220C2#3068185211yescharacteristic
subgroupG/H
centerC2#30681852
commutator, cosocleC22×C4×C263C23
FrattiniC22×C4C22×D263
Fitting, prime radicalC2×AGL(1,ℤ/8)×C263C2
socleC2×C263#394
hypercenterC2×AGL(1,ℤ/8)D263
hypocenterC263#2048
LCSG ▷ C22×C4×C263 ▷ C2×C263 ▷ C263
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #30681986C2semidirect
2. #30681955C2semidirect
3. #30681943C2semidirect
4. #30681922C2semidirect
5. C2×#22554387C2semidirect
6. #30681710C2semidirect
110. C263#2048semidirect

Maximal subgroups

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

subgroupindex
#306819862
#306819552
#306819432
#306819222
C2×#225543872
#306817102
C2×AGL(1,ℤ/8)×C2632
#2048263