Finite Group Databasebeta

← #10709074

Subgroups — #10709074

Order 768.

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

normal subgroups only

This group has 294 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#79912C211yesmaximal, characteristic
2#70944C211yesmaximal, characteristic, complemented
3#69289C211yesmaximal, characteristic
4#69065C211yesmaximal, characteristic, complemented
5C3×Q8⋊C16C211yesmaximal, characteristic, complemented
6#67561C211yesmaximal, characteristic, complemented
7#67504C211yesmaximal, characteristic
8#12383—31nomaximal, Sylow
9#5584—21no—
10C3×C8×Q8C2211yescharacteristic
11#5031—21no—
12#4845—22no—
13#4841—21no—
14#4699C2211yescharacteristic
15C4×Dic12—21no—
16#4696—21no—
17C4×D24C2211yescharacteristic
18#4682C2211yescharacteristic
19C3×C4⋊C16C2211yescharacteristic
20C3×C4×C16C2211yescharacteristic
21#4509—21no—
22#4504—21no—
23#4503—22no—
24#4486—21no—
25#4458C2211yescharacteristic
26#3393—31no—
27#2241—31no—
28#2224—31no—
29#1674—31no—
30#1362—31no—
31Q8⋊C16D311yescharacteristic, complemented
32#1331—31no—
33C3×C4×Q8D411yescharacteristic
34#946—41no—
35C2×#216—41no—
36C2×Dic12—21no—
37C2×Dic12D412yes—
38C2×D24D411yescharacteristic
39C2×#204—22no—
40#898—42no—
41#892—41no—
42#875—22no—
43C4×D12—21no—
44#869—21no—
45#869—21no—
46C4×Dic6—22no—
47C2×C3×C16—21no—
48C2×C3×C16—21no—
49C3×C4⋊C8—21no—
50C3×C4⋊C8D411yescharacteristic
51C3×C4×C8—21no—
52C3×C4×C8C2311yescharacteristic
53#821—21no—
54#818—21no—
55#818—21no—
56#818D411yescharacteristic
57#816—22no—
58#810—41no—
59C4⋊(C3⋊C8)—21no—
60#556—61no—
61#500—31no—
62#495—61no—
63#494—31no—
64#464—62no—
65#462—61no—
66#461—61no—
67C8×Q8C2×D311yescharacteristic
68C4×Q16—61no—
69C4×D8—31no—
70#366—61no—
71C4⋊C16C2×D311yescharacteristic
72#358—62no—
73#357—61no—
74C4×C16C2×D311yescharacteristic
75#335—31no—
76#332—61no—
77C2×C3×Q8D811yescharacteristic
78C2×D12—21no—
79C2×C4×D3—41no—
80C2×Dic6—22no—
81C2×Dic6—41no—
82C2×C3×C8—21no—
83C2×C3×C8C2×D411yescharacteristic
84C2×C3×C8C2×D411yescharacteristic
85C3×C4⋊C4—21no—
86C3×C4⋊C4D811yescharacteristic
87C3×C42—21no—
88C3×C42C2×D411yescharacteristic
89#216—82no—
90(C2×D3)⋊C4—42no—
91(C2×D3)⋊C4—42no—
92(C2×D3)⋊C4—41no—
93C4⋊Dic3—21no—
94C4⋊Dic3—41no—
95C4⋊Dic3—41no—
96C4⋊Dic3—41no—
97C4⋊Dic3—41no—
98Dic3⋊C4—42no—
99C4×Dic3—42no—
100C2×C3⋊C8—41no—
101Dic12—42no—
102Dic12—41no—
103Dic12—41no—
104D24—21no—
105D24—21no—
106#204—82no—
107C3×C16—41no—
108C3×C16—41no—
109C2×Q16—32no—
110C2×Q16—61no—
111C2×Q16—121no—
112C2×SD16—62no—
113C2×D8—31no—
114C4⋊Q8—61no—
115C4⋊Q8—61no—
116#126—122no—
117#124—62no—
118C22⋊Q8—121no—
119C4×Q8—62no—
120C4×Q8D1211yescharacteristic
121C4×D4—61no—
122#117—121no—
123C2×C16—21no—
124C2×C16—21no—
125#107—31no—
126#107—61no—
127#107—61no—
128C4⋊C8—21no—
129C4⋊C8—61no—
130C4⋊C8D1211yescharacteristic
131#103—62no—
132#103—121no—
133#102—61no—
134C4×C8—21no—
135C4×C8C22×D311yescharacteristic
136C22×D3—41no—
137C3×Q8—21no—
138C3×Q8—21no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4#12011yescharacteristic
142C2×C3×C4C2×D811yescharacteristic
143C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
144C2×Dic3—41no—
145C2×Dic3—42no—
146C2×Dic3—81no—
147D12—41no—
148D12—41no—
149C4×D3—81no—
150C4×D3—81no—
151Dic6—42no—
152Dic6—42no—
153Dic6—81no—
154Dic6—81no—
155C3×C8—21no—
156C3×C8—41no—
157C3×C8#13511yescharacteristic
158C3×C8#13711yescharacteristic
159C3⋊C8—81no—
160C2×Q8—62no—
161C2×Q8—121no—
162C2×Q8D2411yescharacteristic
163C2×D4—61no—
164C22×C4—121no—
165Q16—122no—
166Q16—121no—
167Q16—121no—
168Q16—242no—
169SD16—242no—
170D8—61no—
171D8—61no—
172C2×C8—21no—
173C2×C8—121no—
174C2×C8D3×D411yescharacteristic
175C2×C8D3×D411yescharacteristic
176C4⋊C4—21no—
177C4⋊C4—61no—
178C4⋊C4—122no—
179C4⋊C4—121no—
180C4⋊C4—121no—
181C4⋊C4—121no—
182C4⋊C4—121no—
183C4⋊C4D2411yescharacteristic
184C22⋊C4—122no—
185C22⋊C4—122no—
186C22⋊C4—121no—
187C42—21no—
188C42—122no—
189C42C2×D1211yescharacteristic
190C16—41no—
191C16—41no—
192C22×C3#44711yescharacteristic
193C2×D3—81no—
194C2×D3—81no—
195C2×D3—81no—
196C3×C4—21no—
197C3×C4—41no—
198C3×C4—41no—
199C3×C4#45211yescharacteristic
200C3×C4#45511yescharacteristic
201C3×C4#50811yescharacteristic
202C3×C4#50911yescharacteristic
203Dic3—82no—
204Dic3—81no—
205Dic3—161no—
206C23—121no—
207Q8—21no—
208Q8—21no—
209Q8—122no—
210Q8—122no—
211Q8—241no—
212Q8—241no—
213D4—121no—
214D4—121no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—121no—
218C2×C4—122no—
219C2×C4—241no—
220C2×C4—241no—
221C2×C4—241no—
222C2×C4#88211yescharacteristic
223C2×C4C2×D2411yescharacteristic
224C2×C4#90811yescharacteristic
225C8—21no—
226C8—41no—
227C8—241no—
228C8#91611yescharacteristic
229C8#91811yescharacteristic
230C2×C3#164111yescharacteristic
231C2×C3#218611yescharacteristic
232C2×C3#219511yescharacteristic
233D3—161no—
234C22—241no—
235C22—241no—
236C22—241no—
237C22#473211yescharacteristic
238C4—21no—
239C4—41no—
240C4—41no—
241C4—242no—
242C4—241no—
243C4—481no—
244C4#480811yescharacteristic
245C4#482511yescharacteristic
246C4#490411yescharacteristic
247C4#490811yescharacteristic
248C3#1238311yescharacteristic, complemented, Sylow
249C2—481no—
250C2#6826311yescharacteristic
251C2#6910911yescharacteristic
252C2#6912911yescharacteristic
subgroupG/H
centerC22#4732
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×Q8⋊C16C2
socleC22×C3#447
prime radicalC2×D24D4
hypercenterQ8⋊C16D3
hypocenterC3#12383
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. #69065C2semidirect
5. C3×Q8⋊C16C2semidirect
6. #67561C2semidirect
31. Q8⋊C16D3semidirect
248. C3#12383semidirect

Maximal subgroups

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

subgroupindex
#799122
#709442
#692892
#690652
C3×Q8⋊C162
#675612
#675042
#123833