Finite Group Databasebeta

← #10709019

Subgroups — #10709019

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#79607C211yesmaximal, characteristic
2#70887C211yesmaximal, characteristic, complemented
3#69065C211yesmaximal, characteristic, complemented
4#68579C211yesmaximal, characteristic, complemented
5C3×#1331C211yesmaximal, characteristic, complemented
6#67503C211yesmaximal, characteristic, complemented
7#67474C211yesmaximal, characteristic, complemented
8#12375—31nomaximal, Sylow
9C2×#912—21no—
10#5555—21no—
11C3×C4×D8C2211yescharacteristic
12#5019—21no—
13#4887—21no—
14#4842—21no—
15#4783—22no—
16#4699C2211yescharacteristic
17C8×D12C2211yescharacteristic
18#4682C2211yescharacteristic
19C3×C4×C16C2211yescharacteristic
20C3×#335C2211yescharacteristic
21#4562—21no—
22#4503—22no—
23#4490—21no—
24#4481—21no—
25C4⋊(C3⋊C16)C2211yescharacteristic
26#3283—31no—
27#2241—31no—
28#2226—31no—
29#1675—31no—
30#1362—31no—
31#1331D311yescharacteristic, complemented
32D4⋊C16—31no—
33C2×#237—22no—
34C2×C3×D8D411yescharacteristic
35C3×C4×D4—21no—
36C2×#214—22no—
37#930—41no—
38C2×C4⋊Dic3—41no—
39#912—42no—
40#912—42no—
41C2×Dic12—21no—
42C2×Dic12D412yes—
43C2×D3×C8—21no—
44#897—22no—
45#886—42no—
46C4×D12D411yescharacteristic
47#869—21no—
48#869—21no—
49C2×C3×C16—21no—
50C3×#107D411yescharacteristic
51C3×C4⋊C8—21no—
52C3×#102—21no—
53C3×C4×C8C2311yescharacteristic
54#821—41no—
55(C2×D3)⋊C8—21no—
56#818—21no—
57#818—21no—
58#817—41no—
59C2×C3⋊C16—21no—
60#810—22no—
61C4⋊(C3⋊C8)D411yescharacteristic
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×D8C2×D311yescharacteristic
71C8×D4—31no—
72#367—61no—
73C4⋊C16—31no—
74#358—62no—
75#357—61no—
76C4×C16C2×D311yescharacteristic
77#335C2×D311yescharacteristic
78#333—61no—
79C2×C3×D4—21no—
80C22×C3×C4—41no—
81C2×#67—42no—
82C22×Dic3—42no—
83#237—44no—
84#237—82no—
85C2×D12D811yescharacteristic
86C2×C4×D3—21no—
87C2×Dic6—22no—
88C2×Dic6—41no—
89C3×D8—21no—
90C3×D8SD1612yes—
91C2×C3×C8—41no—
92C2×C3×C8C2×D411yescharacteristic
93C2×C3×C8C2×D411yescharacteristic
94C3×C4⋊C4—21no—
95C3×C22⋊C4—41no—
96C3×C42C2×D411yescharacteristic
97#214—44no—
98(C2×D3)⋊C4—21no—
99(C2×D3)⋊C4—42no—
100C4⋊Dic3—42no—
101C4⋊Dic3—41no—
102C4⋊Dic3—41no—
103C4⋊Dic3—41no—
104C4⋊Dic3—41no—
105C4⋊Dic3D811yescharacteristic
106C2×C3⋊C8—21no—
107Dic12—42no—
108Dic12—41no—
109Dic12—41no—
110D3×C8—41no—
111D3×C8—41no—
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×D8D1211yescharacteristic
121C22×C8—61no—
122C4⋊Q8—61no—
123C4⋊Q8—61no—
124#123—122no—
125C22⋊Q8—62no—
126#121—121no—
127C4×D4—21no—
128C4×D4—31no—
129C2×C4⋊C4—121no—
130C2×C16—21no—
131C2×C16—61no—
132#107—61no—
133#107—61no—
134#107D1211yescharacteristic
135#106—121no—
136C4⋊C8—21no—
137C4⋊C8—31no—
138#103—62no—
139#102—21no—
140#102—121no—
141C22⋊C8—61no—
142C4×C8C22×D311yescharacteristic
143C23×C3—41no—
144C22×D3—21no—
145C3×D4—22no—
146C3×D4—41no—
147C2×C3×C4—41no—
148C2×C3×C4—81no—
149C2×C3×C4—81no—
150C2×C3×C4#12011yescharacteristic
151C2×C3×C4C2×D811yescharacteristic
152C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
153#67—84no—
154C2×Dic3—21no—
155C2×Dic3—42no—
156C2×Dic3—84no—
157C2×Dic3—81no—
158D12—21no—
159D12—21no—
160C4×D3—41no—
161C4×D3—41no—
162Dic6—42no—
163Dic6—42no—
164Dic6—161no—
165C3×C8—21no—
166C3×C8—81no—
167C3×C8C2×SD1611yescharacteristic
168C3×C8AGL(1,ℤ/8)11yescharacteristic
169C3⋊C8—41no—
170#41—124no—
171#41—242no—
172C2×Q8—62no—
173C2×Q8—121no—
174C2×D4—21no—
175C2×D4—31no—
176C2×D4—122no—
177C22×C4—41no—
178C22×C4—61no—
179C22×C4—122no—
180Q16—122no—
181Q16—121no—
182Q16—121no—
183SD16—124no—
184D8—21no—
185D8#20412yes—
186C2×C8—41no—
187C2×C8—61no—
188C2×C8—121no—
189C2×C8—121no—
190C2×C8C2×D1211yescharacteristic
191C2×C8D3×D411yescharacteristic
192C4⋊C4—21no—
193C4⋊C4—31no—
194C4⋊C4—122no—
195C4⋊C4—121no—
196C4⋊C4—121no—
197C4⋊C4—121no—
198C4⋊C4—121no—
199C22⋊C4—41no—
200C22⋊C4—61no—
201C22⋊C4—122no—
202C42D3×D411yescharacteristic
203C16—41no—
204C16—121no—
205C22×C3—42no—
206C22×C3—81no—
207C22×C3—81no—
208C22×C3#44711yescharacteristic
209C2×D3—41no—
210C2×D3—41no—
211C2×D3—41no—
212C3×C4—21no—
213C3×C4—81no—
214C3×C4#45011yescharacteristic
215C3×C4#45511yescharacteristic
216C3×C4#50811yescharacteristic
217C3×C4#51011yescharacteristic
218Dic3—41no—
219Dic3—82no—
220Dic3—161no—
221C23—41no—
222C23—61no—
223Q8—122no—
224Q8—122no—
225Q8—481no—
226D4—22no—
227D4—41no—
228D4—61no—
229D4—61no—
230D4—244no—
231C2×C4—41no—
232C2×C4—61no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4—121no—
236C2×C4—121no—
237C2×C4—122no—
238C2×C4—244no—
239C2×C4—241no—
240C2×C4#88211yescharacteristic
241C2×C4D3×D811yescharacteristic
242C2×C4#91411yescharacteristic
243C8—21no—
244C8—81no—
245C8—121no—
246C8C2×#20411yescharacteristic
247C8#90811yescharacteristic
248C2×C3—82no—
249C2×C3#163911yescharacteristic
250C2×C3#218911yescharacteristic
251C2×C3#219311yescharacteristic
252D3—81no—
253C22—42no—
254C22—81no—
255C22—81no—
256C22—121no—
257C22—121no—
258C22—121no—
259C22#477311yescharacteristic
260C4—21no—
261C4—81no—
262C4—121no—
263C4—242no—
264C4—481no—
265C4#473111yescharacteristic
266C4#482511yescharacteristic
267C4#491211yescharacteristic
268C4#491511yescharacteristic
269C3#1237511yescharacteristic, complemented, Sylow
270C2—82no—
271C2—241no—
272C2#6818211yescharacteristic
273C2#6919411yescharacteristic
274C2#6924311yescharacteristic
subgroupG/H
centerC22#4773
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#1331C2
socleC22×C3#447
prime radical#79607C2
hypercenter#1331D3
hypocenterC3#12375
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
2. #70887C2semidirect
3. #69065C2semidirect
4. #68579C2semidirect
5. C3×#1331C2semidirect
6. #67503C2semidirect
7. #67474C2semidirect
31. #1331D3semidirect
269. C3#12375semidirect

Maximal subgroups

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

subgroupindex
#796072
#708872
#690652
#685792
C3×#13312
#675032
#674742
#123753