Finite Group Databasebeta

← #10712003

Subgroups — #10712003

Order 768.

424 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 99 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#83278C211yesmaximal, characteristic
2C3×#2126C211yesmaximal, characteristic
3#71817C211yesmaximal, characteristic
4#70044C212yesmaximal, complemented
5#67696C212yesmaximal, complemented
6#12968—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8#5819C2211yescharacteristic
9C3×#410C2212yes—
10#5240C2211yescharacteristic
11#5236—21no—
12#5234—21no—
13#5234C2212yes—
14#5012—22no—
15D4×C3⋊C8—22no—
16#4999—22no—
17#4925—21no—
18C3×#354C412yescomplemented
19C3×#353C412yescomplemented
20#4533—22no—
21#4523—21no—
22#2979—31no—
23#2126D311yescharacteristic
24#1884—31no—
25#1868—32no—
26#1391—32no—
27C2×C3×#41D411yescharacteristic
28C2×C3×#41C2311yescharacteristic
29C3×C4⋊Q8Q811yescharacteristic
30C3×#127C2×C411yescharacteristic
31C3×#124C2×C411yescharacteristic
32C3×#124Q811yescharacteristic
33C3×#121—22no—
34C3×C4×D4—22no—
35C3×#117—22no—
36C2×C3×C42D411yescharacteristic
37#948—42no—
38#926—21no—
39C2×#208—22no—
40C2×#208—21no—
41C2×#208—21no—
42C2×#208D412yes—
43C22×C3⋊C8—21no—
44C22×C3⋊C8D412yes—
45C3×#99—22no—
46C3×#99C2×C414yes—
47#837—42no—
48#834—22no—
49#830—22no—
50#830—22no—
51#830—22no—
52C4⋊(C3⋊C8)—21no—
53C4⋊(C3⋊C8)—21no—
54C4⋊(C3⋊C8)—44no—
55C4×C3⋊C8—22no—
56#568—31no—
57#532C2×D311yescharacteristic
58#436—62no—
59C8×D4—62no—
60#431—61no—
61#424—62no—
62#421—61no—
63#410—31no—
64#410C2×D312yes—
65#408—32no—
66#408—61no—
67#354Dic312yescomplemented
68#353Dic312yescomplemented
69#344—62no—
70#339—61no—
71C3×#41—41no—
72C3×#41—41no—
73C3×#41—41no—
74C3×#41—41no—
75C2×C3×Q8C4⋊C411yescharacteristic
76C2×C3×Q8C2×Q811yescharacteristic
77C2×C3×D4—22no—
78C2×C3×D4—21no—
79C2×C3×D4C4⋊C411yescharacteristic
80C2×C3×D4C22×C411yescharacteristic
81C2×C3×D4#4112yes—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D412yes—
86C3×C4⋊C4—22no—
87C3×C4⋊C4—42no—
88C3×C22⋊C4—22no—
89C3×C22⋊C4—22no—
90C3×C22⋊C4—24no—
91C3×C42—21no—
92C3×C42—42no—
93C3×C42C4⋊C411yescharacteristic
94C3×C42C4⋊C411yescharacteristic
95#217—41no—
96C4⋊Dic3—41no—
97C4×Dic3—41no—
98#208—22no—
99#208—42no—
100#208—41no—
101#208—42no—
102#208—81no—
103C2×C3⋊C8—24no—
104C2×C3⋊C8—22no—
105C2×C3⋊C8—22no—
106C2×C3⋊C8—41no—
107C2×C3⋊C8—42no—
108C2×C3⋊C8—81no—
109C2×#41D1211yescharacteristic
110C2×#41C22×D311yescharacteristic
111#131—122no—
112C2×M16—32no—
113C2×M16—62no—
114C2×M16—61no—
115C2×M16—61no—
116C22×C8—32no—
117C22×C8—61no—
118C4⋊Q8Dic611yescharacteristic
119#127C2×Dic311yescharacteristic
120#124Dic611yescharacteristic
121#124C2×Dic311yescharacteristic
122#121—22no—
123C4×D4—22no—
124#117—22no—
125#117—61no—
126C2×C42D1211yescharacteristic
127C4⋊C8—61no—
128C4⋊C8—61no—
129C4⋊C8—124no—
130#104—122no—
131#99—22no—
132#99—62no—
133#99C2×Dic314yes—
134C22⋊C8—62no—
135C22⋊C8—62no—
136C22⋊C8—62no—
137C4×C8—62no—
138C23×C3—21no—
139C23×C3C4×D412yes—
140C23×C3#12311yescharacteristic
141C3×Q8—41no—
142C3×Q8—41no—
143C3×D4—41no—
144C3×D4—42no—
145C3×D4—41no—
146C3×D4—44no—
147C3×D4—81no—
148C2×C3×C4—21no—
149C2×C3×C4—22no—
150C2×C3×C4—21no—
151C2×C3×C4—22no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—42no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—44no—
159C2×C3×C4C2×C4⋊C411yescharacteristic
160C2×C3×C4#12011yescharacteristic
161C2×C3×C4C22⋊Q812yes—
162C2×Dic3—81no—
163C3⋊C8—42no—
164C3⋊C8—42no—
165C3⋊C8—42no—
166#41—41no—
167#41—41no—
168#41—41no—
169#41—41no—
170C2×Q8C4⋊Dic311yescharacteristic
171C2×Q8C2×Dic611yescharacteristic
172C2×D4—22no—
173C2×D4—21no—
174C2×D4C4⋊Dic311yescharacteristic
175C2×D4#23712yes—
176C2×D4C22×Dic311yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4C2×D1211yescharacteristic
180C22×C4D3×D412yes—
181M16—62no—
182M16—122no—
183M16—121no—
184M16—122no—
185M16—241no—
186C2×C8—64no—
187C2×C8—62no—
188C2×C8—62no—
189C2×C8—121no—
190C2×C8—122no—
191C2×C8—241no—
192C4⋊C4—22no—
193C4⋊C4—42no—
194C4⋊C4—121no—
195C22⋊C4—22no—
196C22⋊C4—22no—
197C22⋊C4—24no—
198C22⋊C4—121no—
199C42—21no—
200C42—42no—
201C42—121no—
202C42C4⋊Dic311yescharacteristic
203C42C4⋊Dic311yescharacteristic
204C22×C3—21no—
205C22×C3—22no—
206C22×C3—41no—
207C22×C3—42no—
208C22×C3—42no—
209C22×C3—41no—
210C22×C3#38511yescharacteristic
211C3×C4—22no—
212C3×C4—21no—
213C3×C4—22no—
214C3×C4—41no—
215C3×C4—41no—
216C3×C4—84no—
217C3×C4B(4,𝔽2)11yescharacteristic
218C3×C4#45811yescharacteristic
219Dic3—161no—
220C23—21no—
221C23#88611yescharacteristic
222C23D4×Dic312yes—
223Q8—41no—
224Q8—41no—
225D4—42no—
226D4—41no—
227D4—41no—
228D4—44no—
229D4—81no—
230C2×C4—21no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—44no—
241C2×C4—241no—
242C2×C4#87812yes—
243C2×C4#88211yescharacteristic
244C2×C4C2×C4⋊Dic311yescharacteristic
245C8—122no—
246C8—122no—
247C8—122no—
248C2×C3—21no—
249C2×C3—41no—
250C2×C3—42no—
251C2×C3—81no—
252C2×C3#186711yescharacteristic
253C22—22no—
254C22—21no—
255C22—41no—
256C22—42no—
257C22—42no—
258C22—41no—
259C22#495011yescharacteristic
260C4—22no—
261C4—22no—
262C4—21no—
263C4—41no—
264C4—41no—
265C4—84no—
266C4—481no—
267C4#474111yescharacteristic
268C4#474211yescharacteristic
269C3#1296811yescharacteristic, complemented, Sylow
270C2—21no—
271C2—42no—
272C2—41no—
273C2—81no—
274C2#7002111yescharacteristic
subgroupG/H
centerC4#4741
commutatorC2×C3×D4C22×C4
FrattiniC2×#41C22×D3
FittingC3×#2126C2
socleC2×C3#1867
prime radicalC3×#532C22
cosocleC2×C3×#41C23
hypercenter#2126D3
hypocenterC3#12968
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2126

Complements

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

NH ≅ G/Ntype
4. #70044C2semidirect
5. #67696C2semidirect
18. C3×#354C4semidirect
19. C3×#353C4semidirect
67. #354Dic3semidirect
68. #353Dic3semidirect
269. C3#12968semidirect

Maximal subgroups

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

subgroupindex
#832782
C3×#21262
#718172
#700442
#676962
#129683