Finite Group Databasebeta

← #10708985

Subgroups — #10708985

Order 768.

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

normal subgroups only

This group has 356 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#81806C211yesmaximal, characteristic
2C3×#2241C211yesmaximal, characteristic, complemented
3#71387C211yesmaximal, characteristic, complemented
4#71371C211yesmaximal, characteristic, complemented
5#67793C211yesmaximal, characteristic, complemented
6#67504C211yesmaximal, characteristic, complemented
7#67503C211yesmaximal, characteristic, complemented
8#12371—31nomaximal, Sylow
9C2×#919—21no—
10#5726—21no—
11C3×#500C2211yescharacteristic
12C3×#494C2211yescharacteristic
13#5188—21no—
14#5187—22no—
15#5083—21no—
16#5083—21no—
17C4×D24C2211yescharacteristic
18C8×D12C2211yescharacteristic
19C3×#358—22no—
20C3×C4×C16C2211yescharacteristic
21#4547—21no—
22#4491—21no—
23#4489—21no—
24C4⋊(C3⋊C16)C2211yescharacteristic
25#4458C2211yescharacteristic
26#3283—31no—
27#2241D311yescharacteristic, complemented
28#2224—31no—
29#1673—31no—
30#1362—31no—
31#1331—31no—
32D4⋊C16—31no—
33C2×#239—22no—
34C2×C3×Q16—21no—
35C2×C3×Q16D412yes—
36C3×C4⋊Q8—21no—
37C3×C4⋊Q8—21no—
38#946—22no—
39C2×#215—22no—
40C2×#215—41no—
41#919—42no—
42#919—42no—
43C2×D24D411yescharacteristic
44C2×D3×C8—21no—
45#895—41no—
46#894—42no—
47D3×C4⋊C4—41no—
48C4×D12—21no—
49C4×D12D411yescharacteristic
50C2×C3×C16—21no—
51C3×#107—21no—
52C3×#107—21no—
53C3×C4×C8C2311yescharacteristic
54#835—22no—
55#835—41no—
56#821—21no—
57(C2×D3)⋊C8—21no—
58#818D411yescharacteristic
59C2×C3⋊C16—21no—
60C4⋊(C3⋊C8)—21no—
61C4⋊(C3⋊C8)D411yescharacteristic
62C2×#135—61no—
63#547—61no—
64#500C2×D311yescharacteristic
65#494C2×D311yescharacteristic
66#467—61no—
67#461—61no—
68#461—61no—
69#452—62no—
70C4×D8—31no—
71C8×D4—31no—
72#366—61no—
73C4⋊C16—31no—
74#358—22no—
75#357—61no—
76C4×C16C2×D311yescharacteristic
77#335—31no—
78#332—61no—
79C2×C3×Q8—22no—
80C2×C3×Q8—41no—
81#239—44no—
82#239—82no—
83C2×D12—21no—
84C2×D12—42no—
85C2×D12D811yescharacteristic
86C2×C4×D3—21no—
87C2×C4×D3—42no—
88C2×C4×D3—41no—
89C3×Q16—42no—
90C3×Q16—41no—
91C3×Q16—41no—
92C2×C3×C8C2×D411yescharacteristic
93C2×C3×C8C2×D411yescharacteristic
94C3×C4⋊C4—41no—
95C3×C4⋊C4—41no—
96C3×C4⋊C4—41no—
97C3×C4⋊C4—41no—
98C3×C42C2×D411yescharacteristic
99#215—44no—
100#215—82no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—42no—
103(C2×D3)⋊C4—41no—
104C4⋊Dic3—21no—
105C4⋊Dic3D811yescharacteristic
106Dic3⋊C4—42no—
107C2×C3⋊C8—21no—
108C2×C3⋊C8—41no—
109D24—21no—
110D24SD1612yescomplemented
111D3×C8—41no—
112D3×C8—41no—
113C3×C16—41no—
114C3⋊C16—41no—
115C2×#41—62no—
116#135—122no—
117#135—122no—
118C2×Q16—21no—
119C2×Q16#6712yes—
120C2×SD16—62no—
121C2×SD16—121no—
122C2×D8—31no—
123C22×C8—61no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126#123—122no—
127C22⋊Q8—62no—
128#121—121no—
129C4×D4—31no—
130C4×D4—61no—
131C2×C4⋊C4—121no—
132C2×C16—21no—
133C2×C16—61no—
134#107—21no—
135#107—21no—
136#107—31no—
137C4⋊C8—31no—
138C4⋊C8—61no—
139#103—62no—
140#103—121no—
141#102—61no—
142C22⋊C8—61no—
143C4×C8C22×D311yescharacteristic
144C22×D3—21no—
145C22×D3—41no—
146C3×Q8—42no—
147C3×Q8—42no—
148C3×Q8—81no—
149C3×Q8—81no—
150C2×C3×C4—42no—
151C2×C3×C4—81no—
152C2×C3×C4#12011yescharacteristic
153C2×C3×C4C2×D811yescharacteristic
154C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
155C2×Dic3—21no—
156C2×Dic3—41no—
157D12—22no—
158D12—21no—
159D12—21no—
160D12—41no—
161D12—84no—
162C4×D3—41no—
163C4×D3—41no—
164C4×D3—84no—
165C4×D3—81no—
166C4×D3—81no—
167C3×C8—21no—
168C3×C8C2×SD1611yescharacteristic
169C3×C8AGL(1,ℤ/8)11yescharacteristic
170C3⋊C8—41no—
171C3⋊C8—81no—
172#41—124no—
173#41—242no—
174C2×Q8—22no—
175C2×Q8—41no—
176C2×D4—31no—
177C2×D4—61no—
178C2×D4—122no—
179C22×C4—61no—
180C22×C4—122no—
181C22×C4—121no—
182Q16—42no—
183Q16—41no—
184Q16—41no—
185SD16—124no—
186SD16—242no—
187D8—32no—
188D8—61no—
189C2×C8—61no—
190C2×C8—121no—
191C2×C8—121no—
192C2×C8—121no—
193C2×C8D3×D411yescharacteristic
194C2×C8C2×#6711yescharacteristic
195C4⋊C4—31no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—61no—
201C4⋊C4—122no—
202C22⋊C4—61no—
203C22⋊C4—122no—
204C22⋊C4—121no—
205C42D3×D411yescharacteristic
206C16—41no—
207C16—121no—
208C22×C3#44711yescharacteristic
209C2×D3—41no—
210C2×D3—42no—
211C2×D3—41no—
212C2×D3—41no—
213C2×D3—81no—
214C2×D3—81no—
215C3×C4—21no—
216C3×C4—82no—
217C3×C4—161no—
218C3×C4#45011yescharacteristic
219C3×C4#45511yescharacteristic
220C3×C4#50811yescharacteristic
221C3×C4#50911yescharacteristic
222Dic3—41no—
223Dic3—81no—
224C23—61no—
225C23—121no—
226Q8—42no—
227Q8—42no—
228Q8—81no—
229Q8—81no—
230D4—62no—
231D4—61no—
232D4—61no—
233D4—121no—
234D4—244no—
235C2×C4—42no—
236C2×C4—61no—
237C2×C4—81no—
238C2×C4—121no—
239C2×C4—121no—
240C2×C4—121no—
241C2×C4—244no—
242C2×C4—241no—
243C2×C4—241no—
244C2×C4D3×D811yescharacteristic
245C2×C4#91111yescharacteristic
246C2×C4#93711yescharacteristic
247C8—21no—
248C8—121no—
249C8—241no—
250C8D3×SD1611yescharacteristic
251C8#91411yescharacteristic
252C2×C3#163911yescharacteristic
253C2×C3#218611yescharacteristic
254C2×C3#219511yescharacteristic
255D3—81no—
256D3—82no—
257C22—121no—
258C22—122no—
259C22—121no—
260C22—121no—
261C22—241no—
262C22—241no—
263C22#515611yescharacteristic
264C4—21no—
265C4—82no—
266C4—121no—
267C4—161no—
268C4—241no—
269C4#491411yescharacteristic
270C4#491611yescharacteristic
271C4#506111yescharacteristic
272C4#516911yescharacteristic
273C3#1237111yescharacteristic, complemented, Sylow
274C2—241no—
275C2—242no—
276C2#7119811yescharacteristic
277C2#7148411yescharacteristic
278C2#7150611yescharacteristic
subgroupG/H
centerC22#5156
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#2241C2
socleC22×C3#447
prime radical#81806C2
hypercenter#2241D3
hypocenterC3#12371
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2241

Complements

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

NH ≅ G/Ntype
2. C3×#2241C2semidirect
3. #71387C2semidirect
4. #71371C2semidirect
5. #67793C2semidirect
6. #67504C2semidirect
7. #67503C2semidirect
27. #2241D3semidirect
110. D24SD16semidirect
273. C3#12371semidirect

Maximal subgroups

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

subgroupindex
#818062
C3×#22412
#713872
#713712
#677932
#675042
#675032
#123713