Finite Group Databasebeta

← #10709242

Subgroups — #10709242

Order 768.

290 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 69 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#79930C211yesmaximal, characteristic
2#70944C211yesmaximal, characteristic, complemented
3#69289C211yesmaximal, characteristic
4#69046C211yesmaximal, characteristic, complemented
5C3×Q8⋊C16C211yesmaximal, characteristic, complemented
6#67565C211yesmaximal, characteristic, complemented
7#67504C211yesmaximal, characteristic
8#12407—31nomaximal, Sylow
9#5579—21no—
10C3×C8×Q8C2211yescharacteristic
11#5031—21no—
12#4847—22no—
13#4842—21no—
14C4×D24—21no—
15C4×D24C2211yescharacteristic
16#4683—21no—
17#4682C2211yescharacteristic
18#4682C2211yescharacteristic
19C3×C4⋊C16C2211yescharacteristic
20C3×C4×C16C2211yescharacteristic
21#4509—21no—
22#4505—22no—
23#4504—21no—
24#4486—21no—
25#4458C2211yescharacteristic
26#3380—31no—
27#2254—31no—
28#2224—31no—
29#1674—31no—
30#1365—31no—
31Q8⋊C16D311yescharacteristic, complemented
32#1331—31no—
33C3×C4×Q8D411yescharacteristic
34#946—41no—
35C2×#216—41no—
36C2×D24D411yescharacteristic
37#896—42no—
38D3×C4⋊C4—41no—
39C4×D12—22no—
40C4×D12—21no—
41#870—22no—
42#869—21no—
43#869—21no—
44C2×C3×C16—21no—
45C2×C3×C16—21no—
46C3×C4⋊C8—21no—
47C3×C4⋊C8D411yescharacteristic
48C3×C4×C8—21no—
49C3×C4×C8C2311yescharacteristic
50#821—22no—
51#821—21no—
52#818—21no—
53#818—21no—
54#818—21no—
55#818D412yes—
56#818D411yescharacteristic
57#817—22no—
58#810—41no—
59C4⋊(C3⋊C8)—21no—
60#554—61no—
61#500—31no—
62#500—31no—
63#499—61no—
64#478—62no—
65#476—61no—
66#462—61no—
67C8×Q8C2×D311yescharacteristic
68C4×D8—31no—
69C4×D8—61no—
70#367—62no—
71#366—61no—
72C4⋊C16C2×D311yescharacteristic
73#357—61no—
74C4×C16C2×D311yescharacteristic
75#335—31no—
76#332—61no—
77C2×C3×Q8D811yescharacteristic
78C2×D12—21no—
79C2×C4×D3—41no—
80C2×C4×D3—42no—
81C2×Dic6—41no—
82C2×Dic6—41no—
83C2×C3×C8—21no—
84C2×C3×C8C2×D411yescharacteristic
85C2×C3×C8C2×D411yescharacteristic
86C3×C4⋊C4—21no—
87C3×C4⋊C4D811yescharacteristic
88C3×C42—21no—
89C3×C42C2×D411yescharacteristic
90#216—82no—
91(C2×D3)⋊C4—42no—
92(C2×D3)⋊C4—41no—
93C4⋊Dic3—21no—
94C4⋊Dic3—22no—
95C4⋊Dic3—41no—
96C4⋊Dic3—41no—
97C4⋊Dic3—41no—
98Dic3⋊C4—42no—
99Dic3⋊C4—42no—
100C2×C3⋊C8—41no—
101D24—21no—
102D24Q1612yescomplemented
103C3×C16—22no—
104C3×C16—41no—
105C2×Q16—121no—
106C2×D8—31no—
107C4⋊Q8—61no—
108C4⋊Q8—61no—
109#125—62no—
110C22⋊Q8—122no—
111C22⋊Q8—121no—
112C4×Q8D1211yescharacteristic
113C4×D4—62no—
114C4×D4—61no—
115C2×C4⋊C4—121no—
116C2×C16—21no—
117C2×C16—21no—
118#107—31no—
119#107—32no—
120#107—61no—
121#107—61no—
122#107—61no—
123#106—62no—
124C4⋊C8—21no—
125C4⋊C8—61no—
126C4⋊C8D1211yescharacteristic
127#103—121no—
128#102—62no—
129#102—61no—
130C4×C8—21no—
131C4×C8C22×D311yescharacteristic
132C22×D3—41no—
133C3×Q8—21no—
134C3×Q8—21no—
135C2×C3×C4—21no—
136C2×C3×C4—21no—
137C2×C3×C4#12011yescharacteristic
138C2×C3×C4C2×D811yescharacteristic
139C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
140C2×Dic3—41no—
141C2×Dic3—42no—
142C2×Dic3—81no—
143D12—22no—
144D12—41no—
145C4×D3—81no—
146C4×D3—81no—
147C4×D3—162no—
148Dic6—81no—
149Dic6—81no—
150Dic6—161no—
151C3×C8—21no—
152C3×C8—41no—
153C3×C8C2×Q1611yescharacteristic
154C3×C8#13711yescharacteristic
155C3⋊C8—81no—
156C2×Q8—121no—
157C2×Q8—121no—
158C2×Q8D2411yescharacteristic
159C2×D4—61no—
160C22×C4—122no—
161C22×C4—121no—
162Q16—242no—
163D8—32no—
164D8—61no—
165C2×C8—21no—
166C2×C8—121no—
167C2×C8D3×D411yescharacteristic
168C2×C8D3×D411yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—61no—
171C4⋊C4—62no—
172C4⋊C4—122no—
173C4⋊C4—122no—
174C4⋊C4—121no—
175C4⋊C4—121no—
176C4⋊C4—121no—
177C4⋊C4D2411yescharacteristic
178C22⋊C4—122no—
179C22⋊C4—121no—
180C42—21no—
181C42C2×D1211yescharacteristic
182C16—22no—
183C16—41no—
184C22×C3#44711yescharacteristic
185C2×D3—42no—
186C2×D3—81no—
187C2×D3—81no—
188C3×C4—21no—
189C3×C4—41no—
190C3×C4—41no—
191C3×C4#45111yescharacteristic
192C3×C4#45611yescharacteristic
193C3×C4#50811yescharacteristic
194C3×C4#50911yescharacteristic
195Dic3—81no—
196Dic3—82no—
197Dic3—161no—
198C23—121no—
199Q8—21no—
200Q8—21no—
201Q8—241no—
202Q8—241no—
203Q8—481no—
204D4—62no—
205D4—121no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—121no—
209C2×C4—122no—
210C2×C4—241no—
211C2×C4—241no—
212C2×C4—241no—
213C2×C4—482no—
214C2×C4#88211yescharacteristic
215C2×C4C2×D2411yescharacteristic
216C2×C4#90811yescharacteristic
217C8—21no—
218C8—41no—
219C8—241no—
220C8#91511yescharacteristic
221C8D3×Q1611yescharacteristic
222C2×C3#164911yescharacteristic
223C2×C3#218611yescharacteristic
224C2×C3#219411yescharacteristic
225D3—82no—
226C22—122no—
227C22—241no—
228C22—241no—
229C22#473211yescharacteristic
230C4—21no—
231C4—41no—
232C4—41no—
233C4—242no—
234C4—241no—
235C4—481no—
236C4#480911yescharacteristic
237C4#482611yescharacteristic
238C4#490411yescharacteristic
239C4#490811yescharacteristic
240C3#1240711yescharacteristic, complemented, Sylow
241C2—242no—
242C2#6827311yescharacteristic
243C2#6910911yescharacteristic
244C2#6912811yescharacteristic
subgroupG/H
centerC22#4732
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×Q8⋊C16C2
socleC22×C3#447
prime radicalC2×D24D4
hypercenterQ8⋊C16D3
hypocenterC3#12407
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ Q8⋊C16

Complements

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

NH ≅ G/Ntype
2. #70944C2semidirect
4. #69046C2semidirect
5. C3×Q8⋊C16C2semidirect
6. #67565C2semidirect
31. Q8⋊C16D3semidirect
102. D24Q16semidirect
240. C3#12407semidirect

Maximal subgroups

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

subgroupindex
#799302
#709442
#692892
#690462
C3×Q8⋊C162
#675652
#675042
#124073