Finite Group Databasebeta

← #10707623

Subgroups — #10707623

Order 768.

372 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C3×#2822C211yesmaximal, characteristic, complemented
2#69712C212yesmaximal
3#69563C211yesmaximal, characteristic
4#67650C212yesmaximal, complemented
5#67632C211yesmaximal, characteristic, complemented
6#12157—31nomaximal, Sylow
7C2×C3×#125—21no—
8#5510C2211yescharacteristic
9C3×#400—22no—
10C3×#398—21no—
11C3×#398—22no—
12C3×#398C2212yes—
13C3×#382C2211yescharacteristic
14#4967—22no—
15#4966—21no—
16C2×#805—21no—
17#4530C2211yescharacteristic
18#4528C2212yes—
19#4451—22no—
20#2822D311yescharacteristic, complemented
21#2040—31no—
22#1610—32no—
23#1346—32no—
24#1324—31no—
25C2×#235—21no—
26C3×#125—42no—
27C3×#125—42no—
28C2×C3×C4⋊C4—21no—
29C2×C3×C4⋊C4—22no—
30C2×C3×C4⋊C4—21no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C4⋊C4D412yes—
35C2×C3×C4⋊C4C2311yescharacteristic
36C2×C3×C42D411yescharacteristic
37#930—22no—
38C2×#208—21no—
39#897—41no—
40#895—41no—
41C3×#95—22no—
42C3×#95—21no—
43C3×#95—22no—
44C3×#95—22no—
45C3×#95D412yes—
46#830—22no—
47#810—41no—
48#809—41no—
49#809—42no—
50#807—42no—
51#805—42no—
52#537—31no—
53C2×#125—21no—
54#480—62no—
55C2×#104—61no—
56#417—61no—
57#400—22no—
58#398—21no—
59#398—22no—
60#398C2×D312yes—
61#382C2×D311yescharacteristic
62#343—31no—
63#327—62no—
64#324—32no—
65C22×C3×C4—21no—
66C22×C3×C4—21no—
67C22×C3×C4—22no—
68C22×C3×C4—22no—
69C22×C3×C4C2×D411yescharacteristic
70C22×C3×C4C2×D412yes—
71C2×#67—41no—
72#235—41no—
73#235—41no—
74#235—41no—
75#235—41no—
76C2×D12—21no—
77C2×D12—22no—
78C2×C4×D3—41no—
79C2×Dic6—21no—
80C3×C4⋊C4—41no—
81C3×C4⋊C4—42no—
82C3×C4⋊C4—41no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4—44no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—42no—
87C3×C4⋊C4—42no—
88C3×C4⋊C4—81no—
89C3×C42—21no—
90C3×C42D812yes—
91(C2×D3)⋊C4—42no—
92C4⋊Dic3—22no—
93#208—41no—
94#208—41no—
95C2×C3⋊C8—42no—
96C2×C3⋊C8—41no—
97C2×#41—61no—
98C2×M16—61no—
99#125—42no—
100#125—42no—
101C22⋊Q8—121no—
102#121—62no—
103#121—121no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4#6711yescharacteristic
110C2×C4⋊C4#6712yes—
111C2×C4⋊C4C22×D311yescharacteristic
112C2×C42#6711yescharacteristic
113#107—122no—
114#104—122no—
115#103—121no—
116#102—121no—
117#102—122no—
118C22⋊C8—62no—
119#95—22no—
120#95—21no—
121#95—22no—
122#95—22no—
123#95#6712yes—
124C23×C3#12011yescharacteristic
125C22×D3—41no—
126C2×C3×C4—21no—
127C2×C3×C4—21no—
128C2×C3×C4—22no—
129C2×C3×C4—22no—
130C2×C3×C4—41no—
131C2×C3×C4—41no—
132C2×C3×C4—42no—
133C2×C3×C4—42no—
134C2×C3×C4—42no—
135C2×C3×C4—42no—
136C2×C3×C4—42no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4—42no—
140C2×C3×C4—41no—
141C2×C3×C4C2×D811yescharacteristic
142C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
143#67—81no—
144#67—81no—
145C2×Dic3—41no—
146D12—41no—
147D12—41no—
148D12—82no—
149C4×D3—81no—
150C4×D3—81no—
151Dic6—41no—
152Dic6—41no—
153C3⋊C8—81no—
154C3⋊C8—82no—
155#41—121no—
156#41—121no—
157#41—121no—
158#41—121no—
159C2×Q8—61no—
160C2×D4—61no—
161C2×D4—62no—
162C2×D4—121no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—22no—
166C22×C4—22no—
167C22×C4—121no—
168C22×C4C2×#6711yescharacteristic
169C22×C4C2×#6712yes—
170M16—121no—
171M16—121no—
172C2×C8—122no—
173C2×C8—121no—
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—42no—
178C4⋊C4—44no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—62no—
183C4⋊C4—81no—
184C22⋊C4—122no—
185C42—21no—
186C42#21312yes—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3#44711yescharacteristic
190C22×C3#45411yescharacteristic
191C22×C3B(4,𝔽2)11yescharacteristic
192C2×D3—81no—
193C2×D3—81no—
194C2×D3—81no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—42no—
198C3×C4—42no—
199C3×C4—82no—
200C3×C4—82no—
201C3×C4—81no—
202Dic3—81no—
203C23—121no—
204C23#95311yescharacteristic
205Q8—121no—
206Q8—121no—
207D4—121no—
208D4—121no—
209D4—242no—
210D4—241no—
211D4—241no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—121no—
228C2×C4—241no—
229C2×C4—241no—
230C2×C4C2×#21311yescharacteristic
231C2×C4#93211yescharacteristic
232C8—241no—
233C8—242no—
234C2×C3—21no—
235C2×C3—21no—
236C2×C3#159811yescharacteristic
237C2×C3#220111yescharacteristic
238C2×C3#220311yescharacteristic
239D3—161no—
240C22—21no—
241C22—21no—
242C22—241no—
243C22—241no—
244C22—241no—
245C22#521711yescharacteristic
246C22#524411yescharacteristic
247C22#524511yescharacteristic
248C4—21no—
249C4—21no—
250C4—42no—
251C4—42no—
252C4—82no—
253C4—81no—
254C4—82no—
255C4—241no—
256C3#1215711yescharacteristic, complemented, Sylow
257C2—21no—
258C2—21no—
259C2—481no—
260C2#7187811yescharacteristic
261C2#7195811yescharacteristic
262C2#7195911yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2822C2
socleC22×C3B(4,𝔽2)
prime radical#5510C22
hypercenter#2822D3
hypocenterC3#12157
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2822

Complements

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

NH ≅ G/Ntype
1. C3×#2822C2semidirect
4. #67650C2semidirect
5. #67632C2semidirect
20. #2822D3semidirect
256. C3#12157semidirect

Maximal subgroups

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

subgroupindex
C3×#28222
#697122
#695632
#676502
#676322
#121573