Finite Group Databasebeta

← #10709018

Subgroups — #10709018

Order 768.

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

normal subgroups only

This group has 350 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#79436C211yesmaximal, characteristic
2#70853C211yesmaximal, characteristic, complemented
3#69290C211yesmaximal, characteristic, complemented
4#69065C211yesmaximal, characteristic, complemented
5C3×D4⋊C16C211yesmaximal, characteristic, complemented
6#67561C211yesmaximal, characteristic, complemented
7#67504C211yesmaximal, characteristic, complemented
8#12375—31nomaximal, Sylow
9C2×#904—21no—
10#5551—21no—
11C3×C8×D4C2211yescharacteristic
12#5116—21no—
13#5008—21no—
14#4841—21no—
15#4733—22no—
16#4699C2211yescharacteristic
17C4×D24C2211yescharacteristic
18#4682C2211yescharacteristic
19C3×C4⋊C16C2211yescharacteristic
20C3×C4×C16C2211yescharacteristic
21#4509—21no—
22#4505—21no—
23#4503—22no—
24#4481—21no—
25#4458C2211yescharacteristic
26#3283—31no—
27#2241—31no—
28#2226—31no—
29#1675—31no—
30#1362—31no—
31#1331—31no—
32D4⋊C16D311yescharacteristic, complemented
33C2×#235—22no—
34C22×C3×C8—21no—
35C3×C4×D4D411yescharacteristic
36#938—41no—
37#924—22no—
38C2×Dic12—21no—
39C2×Dic12D412yes—
40#904—42no—
41#904—42no—
42C2×D24D411yescharacteristic
43C2×#204—22no—
44D3×C4⋊C4—41no—
45#883—42no—
46C4×D12—21no—
47#869—21no—
48#869—21no—
49C2×C3×C16—21no—
50C2×C3×C16—21no—
51C3×C4⋊C8D411yescharacteristic
52C3×C22⋊C8—21no—
53C3×C4×C8C2311yescharacteristic
54#832—41no—
55#821—21no—
56#818—21no—
57#818—21no—
58#818D411yescharacteristic
59#816—22no—
60#808—41no—
61C4⋊(C3⋊C8)—21no—
62C2×#135—61no—
63#547—61no—
64#500—31no—
65#494—31no—
66#476—61no—
67#467—61no—
68#461—61no—
69#452—62no—
70C4×D8—31no—
71C8×D4C2×D311yescharacteristic
72#367—61no—
73C4⋊C16C2×D311yescharacteristic
74#358—62no—
75#357—61no—
76C4×C16C2×D311yescharacteristic
77#335—31no—
78#333—61no—
79C2×C3×D4D811yescharacteristic
80C22×C3×C4—21no—
81C2×#67—42no—
82#235—44no—
83#235—82no—
84C2×D12—21no—
85C2×C4×D3—42no—
86C2×C4×D3—41no—
87C2×Dic6—22no—
88C2×Dic6—41no—
89C2×C3×C8—21no—
90C2×C3×C8—41no—
91C2×C3×C8—41no—
92C2×C3×C8C2×D411yescharacteristic
93C2×C3×C8C2×D411yescharacteristic
94C3×C4⋊C4D811yescharacteristic
95C3×C22⋊C4—21no—
96C3×C42C2×D411yescharacteristic
97#217—42no—
98(C2×D3)⋊C4—41no—
99C4⋊Dic3—21no—
100C4⋊Dic3—41no—
101C4⋊Dic3—41no—
102C4⋊Dic3—41no—
103C4⋊Dic3—41no—
104Dic3⋊C4—42no—
105C2×C3⋊C8—41no—
106Dic12—42no—
107Dic12—41no—
108Dic12—41no—
109D24—21no—
110D24SD1612yes—
111#204—44no—
112C3×C16—41no—
113C3×C16—41no—
114C2×#41—62no—
115#135—122no—
116#135—122no—
117C2×Q16—32no—
118C2×Q16—61no—
119C2×SD16—62no—
120C2×D8—31no—
121C22×C8—21no—
122C4⋊Q8—61no—
123C4⋊Q8—61no—
124#123—122no—
125C22⋊Q8—62no—
126#121—121no—
127C4×D4—61no—
128C4×D4D1211yescharacteristic
129C2×C4⋊C4—121no—
130C2×C16—21no—
131C2×C16—21no—
132#107—31no—
133#107—61no—
134#107—61no—
135#106—121no—
136C4⋊C8—61no—
137C4⋊C8D1211yescharacteristic
138#103—62no—
139#102—61no—
140#102—121no—
141C22⋊C8—21no—
142C4×C8C22×D311yescharacteristic
143C23×C3—21no—
144C22×D3—41no—
145C3×D4—21no—
146C3×D4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—41no—
149C2×C3×C4—41no—
150C2×C3×C4#12011yescharacteristic
151C2×C3×C4C2×D811yescharacteristic
152C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
153#67—84no—
154C2×Dic3—41no—
155C2×Dic3—42no—
156C2×Dic3—81no—
157D12—22no—
158D12—41no—
159C4×D3—84no—
160C4×D3—81no—
161C4×D3—81no—
162Dic6—42no—
163Dic6—42no—
164Dic6—161no—
165C3×C8—21no—
166C3×C8—41no—
167C3×C8C2×SD1611yescharacteristic
168C3×C8AGL(1,ℤ/8)11yescharacteristic
169C3⋊C8—81no—
170#41—124no—
171#41—242no—
172C2×Q8—62no—
173C2×Q8—121no—
174C2×D4—61no—
175C2×D4—122no—
176C2×D4D2411yescharacteristic
177C22×C4—21no—
178C22×C4—122no—
179C22×C4—121no—
180Q16—122no—
181Q16—121no—
182Q16—121no—
183SD16—124no—
184D8—32no—
185D8—61no—
186C2×C8—21no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—121no—
190C2×C8D3×D411yescharacteristic
191C2×C8D3×D411yescharacteristic
192C4⋊C4—61no—
193C4⋊C4—122no—
194C4⋊C4—121no—
195C4⋊C4—121no—
196C4⋊C4—121no—
197C4⋊C4—121no—
198C4⋊C4D2411yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—122no—
201C22⋊C4—121no—
202C42C2×D1211yescharacteristic
203C16—41no—
204C16—41no—
205C22×C3—41no—
206C22×C3—41no—
207C22×C3—41no—
208C22×C3#44711yescharacteristic
209C2×D3—42no—
210C2×D3—81no—
211C2×D3—81no—
212C3×C4—21no—
213C3×C4—41no—
214C3×C4#45011yescharacteristic
215C3×C4#45511yescharacteristic
216C3×C4#50811yescharacteristic
217C3×C4#51011yescharacteristic
218Dic3—82no—
219Dic3—81no—
220Dic3—161no—
221C23—21no—
222C23—121no—
223Q8—122no—
224Q8—122no—
225Q8—481no—
226D4—21no—
227D4—21no—
228D4—62no—
229D4—121no—
230D4—244no—
231C2×C4—21no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—121no—
235C2×C4—122no—
236C2×C4—244no—
237C2×C4—241no—
238C2×C4—241no—
239C2×C4—241no—
240C2×C4#88211yescharacteristic
241C2×C4C2×D2411yescharacteristic
242C2×C4#90811yescharacteristic
243C8—21no—
244C8—41no—
245C8—241no—
246C8#91111yescharacteristic
247C8D3×SD1611yescharacteristic
248C2×C3—81no—
249C2×C3#163911yescharacteristic
250C2×C3#218911yescharacteristic
251C2×C3#219311yescharacteristic
252D3—82no—
253C22—41no—
254C22—41no—
255C22—41no—
256C22—122no—
257C22—241no—
258C22—241no—
259C22#473211yescharacteristic
260C4—21no—
261C4—41no—
262C4—242no—
263C4—241no—
264C4—481no—
265C4#477611yescharacteristic
266C4#482511yescharacteristic
267C4#490411yescharacteristic
268C4#490911yescharacteristic
269C3#1237511yescharacteristic, complemented, Sylow
270C2—81no—
271C2—242no—
272C2#6819611yescharacteristic
273C2#6911111yescharacteristic
274C2#6912711yescharacteristic
subgroupG/H
centerC22#4732
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×D4⋊C16C2
socleC22×C3#447
prime radical#79436C2
hypercenterD4⋊C16D3
hypocenterC3#12375
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ D4⋊C16

Complements

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

NH ≅ G/Ntype
2. #70853C2semidirect
3. #69290C2semidirect
4. #69065C2semidirect
5. C3×D4⋊C16C2semidirect
6. #67561C2semidirect
7. #67504C2semidirect
32. D4⋊C16D3semidirect
269. C3#12375semidirect

Maximal subgroups

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

subgroupindex
#794362
#708532
#692902
#690652
C3×D4⋊C162
#675612
#675042
#123753