Finite Group Databasebeta

← #10708657

Subgroups — #10708657

Order 768.

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

normal subgroups only

This group has 374 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×#2247C211yesmaximal, characteristic, complemented
3#71315C211yesmaximal, characteristic, complemented
4#71305C211yesmaximal, characteristic, complemented
5#67793C211yesmaximal, characteristic, complemented
6#67504C211yesmaximal, characteristic, complemented
7#67503C211yesmaximal, characteristic, complemented
8#12323—31nomaximal, Sylow
9C2×#919—21no—
10#5726—21no—
11C2×C3×Q32—22no—
12C3×#494C2211yescharacteristic
13C3×#494C2211yescharacteristic
14#5188—21no—
15#5187—22no—
16C2×#828—21no—
17#5087—21no—
18#5083—21no—
19C4×D24C2211yescharacteristic
20C8×D12C2211yescharacteristic
21C3×C4×C16C2211yescharacteristic
22#4564—21no—
23#4547—21no—
24C4⋊(C3⋊C16)C2211yescharacteristic
25#4458C2211yescharacteristic
26#3283—31no—
27#2247D311yescharacteristic, complemented
28#2208—31no—
29#1669—31no—
30#1362—31no—
31#1331—31no—
32D4⋊C16—31no—
33C2×#239—22no—
34C2×C3×Q16—21no—
35C2×C3×Q16—21no—
36C2×C3×Q16—21no—
37C2×C3×Q16D412yes—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40#946—22no—
41C2×#215—22no—
42#919—42no—
43#919—42no—
44C2×D24D411yescharacteristic
45C2×D3×C8—21no—
46#895—41no—
47#894—42no—
48D3×C4⋊C4—41no—
49C4×D12—21no—
50C4×D12D411yescharacteristic
51C3×Q32—44no—
52C2×C3×C16—21no—
53C3×C4×C8C2311yescharacteristic
54#835—22no—
55#828—42no—
56#821—21no—
57(C2×D3)⋊C8—21no—
58#818D411yescharacteristic
59C2×C3⋊C16—21no—
60#809—41no—
61#807—41no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)D411yescharacteristic
64C2×#135—61no—
65#547—61no—
66C2×Q32—22no—
67C2×SD32—61no—
68#494C2×D311yescharacteristic
69#494C2×D311yescharacteristic
70#474—61no—
71#467—61no—
72#461—61no—
73#452—62no—
74C4×D8—31no—
75C8×D4—31no—
76C4⋊C16—31no—
77#358—61no—
78C4×C16C2×D311yescharacteristic
79#335—31no—
80#332—61no—
81C2×C3×Q8—22no—
82C2×C3×Q8—41no—
83#239—44no—
84#239—82no—
85C2×D12—21no—
86C2×D12—42no—
87C2×D12D811yescharacteristic
88C2×C4×D3—21no—
89C2×C4×D3—42no—
90C2×C4×D3—41no—
91C3×Q16—42no—
92C3×Q16—42no—
93C3×Q16—41no—
94C3×Q16—41no—
95C3×Q16—41no—
96C3×Q16—41no—
97C2×C3×C8C2×D411yescharacteristic
98C2×C3×C8C2×D411yescharacteristic
99C3×C4⋊C4—41no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C42C2×D411yescharacteristic
104#215—44no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—42no—
107(C2×D3)⋊C4—41no—
108C4⋊Dic3—21no—
109C4⋊Dic3D811yescharacteristic
110Dic3⋊C4—42no—
111C2×C3⋊C8—21no—
112C2×C3⋊C8—41no—
113D24—21no—
114D24D812yes—
115D3×C8—41no—
116D3×C8—41no—
117C3×C16—22no—
118C3⋊C16—41no—
119C2×#41—62no—
120#135—122no—
121#135—122no—
122C2×Q16—21no—
123C2×Q16—21no—
124C2×Q16—21no—
125C2×Q16#6712yes—
126C2×SD16—62no—
127C2×D8—31no—
128C22×C8—61no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131#123—122no—
132C22⋊Q8—62no—
133#121—121no—
134C4×D4—31no—
135C4×D4—61no—
136C2×C4⋊C4—121no—
137Q32—44no—
138SD32—122no—
139C2×C16—21no—
140C2×C16—61no—
141#107—31no—
142#107—121no—
143C4⋊C8—31no—
144C4⋊C8—61no—
145#103—62no—
146#102—61no—
147#102—121no—
148C22⋊C8—61no—
149C4×C8C22×D311yescharacteristic
150C22×D3—21no—
151C22×D3—41no—
152C3×Q8—42no—
153C3×Q8—42no—
154C3×Q8—81no—
155C3×Q8—81no—
156C2×C3×C4—42no—
157C2×C3×C4—81no—
158C2×C3×C4#12011yescharacteristic
159C2×C3×C4C2×D811yescharacteristic
160C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
161C2×Dic3—21no—
162C2×Dic3—41no—
163D12—22no—
164D12—21no—
165D12—21no—
166D12—41no—
167D12—84no—
168C4×D3—41no—
169C4×D3—41no—
170C4×D3—84no—
171C4×D3—81no—
172C4×D3—81no—
173C3×C8—21no—
174C3×C8C2×D811yescharacteristic
175C3×C8AGL(1,ℤ/8)11yescharacteristic
176C3⋊C8—41no—
177C3⋊C8—81no—
178#41—124no—
179#41—242no—
180C2×Q8—22no—
181C2×Q8—41no—
182C2×D4—31no—
183C2×D4—61no—
184C2×D4—122no—
185C22×C4—61no—
186C22×C4—122no—
187C22×C4—121no—
188Q16—42no—
189Q16—42no—
190Q16—41no—
191Q16—41no—
192Q16—41no—
193Q16—41no—
194SD16—124no—
195D8—32no—
196D8—61no—
197C2×C8—61no—
198C2×C8—121no—
199C2×C8—121no—
200C2×C8—121no—
201C2×C8D3×D411yescharacteristic
202C2×C8C2×#6711yescharacteristic
203C4⋊C4—31no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—61no—
209C4⋊C4—122no—
210C22⋊C4—61no—
211C22⋊C4—122no—
212C22⋊C4—121no—
213C42D3×D411yescharacteristic
214C16—22no—
215C16—121no—
216C22×C3#44711yescharacteristic
217C2×D3—41no—
218C2×D3—42no—
219C2×D3—41no—
220C2×D3—41no—
221C2×D3—81no—
222C2×D3—81no—
223C3×C4—21no—
224C3×C4—82no—
225C3×C4—161no—
226C3×C4#44711yescharacteristic
227C3×C4#45311yescharacteristic
228C3×C4#50811yescharacteristic
229C3×C4#51011yescharacteristic
230Dic3—41no—
231Dic3—81no—
232C23—61no—
233C23—121no—
234Q8—42no—
235Q8—42no—
236Q8—81no—
237Q8—81no—
238D4—62no—
239D4—61no—
240D4—61no—
241D4—121no—
242D4—244no—
243C2×C4—42no—
244C2×C4—61no—
245C2×C4—81no—
246C2×C4—121no—
247C2×C4—121no—
248C2×C4—121no—
249C2×C4—244no—
250C2×C4—241no—
251C2×C4—241no—
252C2×C4D3×D811yescharacteristic
253C2×C4#91111yescharacteristic
254C2×C4#93711yescharacteristic
255C8—21no—
256C8—121no—
257C8—241no—
258C8D3×D811yescharacteristic
259C8#91111yescharacteristic
260C2×C3#161911yescharacteristic
261C2×C3#218911yescharacteristic
262C2×C3#219511yescharacteristic
263D3—81no—
264D3—82no—
265C22—121no—
266C22—121no—
267C22—121no—
268C22—122no—
269C22—241no—
270C22—241no—
271C22#515611yescharacteristic
272C4—21no—
273C4—82no—
274C4—121no—
275C4—161no—
276C4—241no—
277C4#491811yescharacteristic
278C4#491911yescharacteristic
279C4#507711yescharacteristic
280C4#515611yescharacteristic
281C3#1232311yescharacteristic, complemented, Sylow
282C2—241no—
283C2—242no—
284C2#7128111yescharacteristic
285C2#7150311yescharacteristic
286C2#7150611yescharacteristic
subgroupG/H
centerC22#5156
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#2247C2
socleC22×C3#447
prime radical#81806C2
hypercenter#2247D3
hypocenterC3#12323
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2247

Complements

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

NH ≅ G/Ntype
2. C3×#2247C2semidirect
3. #71315C2semidirect
4. #71305C2semidirect
5. #67793C2semidirect
6. #67504C2semidirect
7. #67503C2semidirect
27. #2247D3semidirect
281. C3#12323semidirect

Maximal subgroups

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

subgroupindex
#818062
C3×#22472
#713152
#713052
#677932
#675042
#675032
#123233