Finite Group Databasebeta

← #10707622

Subgroups — #10707622

Order 768.

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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#74217C211yesmaximal, characteristic, complemented
2C3×#2040C211yesmaximal, characteristic
3#71874C212yesmaximal
4#67632C211yesmaximal, characteristic, complemented
5#67495C212yesmaximal, complemented
6#12157—31nomaximal, Sylow
7C3×#537C2211yescharacteristic
8C2×#870—21no—
9C2×C3×#104—21no—
10C3×#417—21no—
11C3×#382C2211yescharacteristic
12#5033—22no—
13#4981—21no—
14#4981C2212yes—
15#4661—22no—
16#4659—22no—
17C3×#343C2211yescharacteristic
18#4537—22no—
19#4528C2212yes—
20#2822—31no—
21#2040D311yescharacteristic
22#1610—32no—
23#1346—32no—
24#1324—31no—
25C2×C3×#41—21no—
26C2×C3×M16—21no—
27C3×C22⋊Q8—41no—
28C3×#121—22no—
29C3×#121—41no—
30C2×C3×C4⋊C4D411yescharacteristic
31C2×C3×C4⋊C4C2311yescharacteristic
32C2×C3×C42D411yescharacteristic
33C2×C4⋊Dic3—21no—
34C2×C4⋊Dic3—21no—
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3D412yes—
37C2×Dic3⋊C4—22no—
38C2×Dic3⋊C4—22no—
39#870—42no—
40#870—42no—
41C3×#104—42no—
42C3×#103—41no—
43C3×#102—41no—
44C3×#95—22no—
45#832—42no—
46C3⋊#95—21no—
47C3⋊#95—22no—
48C3⋊#95—22no—
49C3⋊#95D412yes—
50#830—22no—
51#807—42no—
52#537C2×D311yescharacteristic
53C2×#125—61no—
54#480—62no—
55C2×#104—21no—
56#417—21no—
57#400—62no—
58#398—32no—
59#398—62no—
60#398—61no—
61#382C2×D311yescharacteristic
62#343C2×D311yescharacteristic
63#327—62no—
64#324—32no—
65C3×#41—41no—
66C3×#41—41no—
67C3×#41—41no—
68C3×#41—41no—
69C2×C3×Q8—21no—
70C2×C3×D4—21no—
71C2×C3×D4—22no—
72C2×C3×D4—41no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—41no—
76C22×C3×C4C2×D412yes—
77C22×C3×C4C2×D411yescharacteristic
78C22×Dic3—22no—
79C22×Dic3—22no—
80C3×M16—41no—
81C3×M16—41no—
82C2×C3×C8—41no—
83C3×C4⋊C4—22no—
84C3×C4⋊C4—41no—
85C3×C4⋊C4—41no—
86C3×C22⋊C4—42no—
87C3×C42—21no—
88C3×C42D812yes—
89C4⋊Dic3—42no—
90C4⋊Dic3—42no—
91C4⋊Dic3—42no—
92C4⋊Dic3—81no—
93Dic3⋊C4—44no—
94Dic3⋊C4—42no—
95Dic3⋊C4—42no—
96C2×C3⋊C8—42no—
97C2×#41—21no—
98C2×M16—21no—
99#125—122no—
100#125—122no—
101C22⋊Q8—41no—
102#121—22no—
103#121—41no—
104C2×C4⋊C4—32no—
105C2×C4⋊C4—61no—
106C2×C4⋊C4—62no—
107C2×C4⋊C4—61no—
108C2×C4⋊C4—62no—
109C2×C4⋊C4—61no—
110C2×C4⋊C4#6711yescharacteristic
111C2×C4⋊C4C22×D311yescharacteristic
112C2×C42#6711yescharacteristic
113#107—122no—
114#104—42no—
115#103—41no—
116#102—41no—
117#102—122no—
118C22⋊C8—62no—
119#95—22no—
120#95—32no—
121#95—62no—
122#95—61no—
123#95—62no—
124C23×C3—41no—
125C23×C3#12011yescharacteristic
126C3×Q8—41no—
127C3×Q8—41no—
128C3×D4—41no—
129C3×D4—41no—
130C3×D4—82no—
131C3×D4—81no—
132C3×D4—81no—
133C2×C3×C4—21no—
134C2×C3×C4—21no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—41no—
138C2×C3×C4—41no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—41no—
142C2×C3×C4—41no—
143C2×C3×C4—81no—
144C2×C3×C4—81no—
145C2×C3×C4C2×D811yescharacteristic
146C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
147C2×Dic3—42no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—42no—
151C2×Dic3—42no—
152C2×Dic3—42no—
153C3×C8—81no—
154C3⋊C8—82no—
155#41—41no—
156#41—41no—
157#41—41no—
158#41—41no—
159C2×Q8—21no—
160C2×D4—21no—
161C2×D4—22no—
162C2×D4—41no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—62no—
167C22×C4—62no—
168C22×C4D3×D412yes—
169C22×C4C2×#6711yescharacteristic
170M16—41no—
171M16—41no—
172C2×C8—41no—
173C2×C8—122no—
174C4⋊C4—22no—
175C4⋊C4—41no—
176C4⋊C4—41no—
177C4⋊C4—122no—
178C4⋊C4—124no—
179C4⋊C4—122no—
180C4⋊C4—122no—
181C4⋊C4—122no—
182C4⋊C4—122no—
183C4⋊C4—241no—
184C22⋊C4—42no—
185C42—21no—
186C42#21312yes—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3—81no—
190C22×C3—81no—
191C22×C3—81no—
192C22×C3#44711yescharacteristic
193C22×C3#45411yescharacteristic
194C22×C3B(4,𝔽2)11yescharacteristic
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—42no—
198C3×C4—42no—
199C3×C4—81no—
200C3×C4—81no—
201Dic3—82no—
202Dic3—82no—
203C23—41no—
204C23#93711yescharacteristic
205Q8—41no—
206Q8—41no—
207D4—41no—
208D4—41no—
209D4—82no—
210D4—81no—
211D4—81no—
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—41no—
221C2×C4—41no—
222C2×C4—81no—
223C2×C4—81no—
224C2×C4—122no—
225C2×C4—122no—
226C2×C4—122no—
227C2×C4—122no—
228C2×C4—122no—
229C2×C4—122no—
230C2×C4C2×#21311yescharacteristic
231C2×C4#94911yescharacteristic
232C8—81no—
233C8—242no—
234C2×C3—21no—
235C2×C3—21no—
236C2×C3—161no—
237C2×C3#159811yescharacteristic
238C2×C3#220111yescharacteristic
239C2×C3#220311yescharacteristic
240C22—21no—
241C22—21no—
242C22—81no—
243C22—81no—
244C22—81no—
245C22#503211yescharacteristic
246C22#503611yescharacteristic
247C22#506211yescharacteristic
248C4—21no—
249C4—21no—
250C4—42no—
251C4—42no—
252C4—81no—
253C4—81no—
254C4—242no—
255C4—242no—
256C3#1215711yescharacteristic, complemented, Sylow
257C2—21no—
258C2—21no—
259C2—161no—
260C2#6970311yescharacteristic
261C2#7098911yescharacteristic
262C2#7099211yescharacteristic
subgroupG/H
centerC22#5032
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2040C2
socleC22×C3B(4,𝔽2)
prime radicalC3×#537C22
hypercenter#2040D3
hypocenterC3#12157
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2040

Complements

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

NH ≅ G/Ntype
1. #74217C2semidirect
4. #67632C2semidirect
5. #67495C2semidirect
256. C3#12157semidirect

Maximal subgroups

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

subgroupindex
#742172
C3×#20402
#718742
#676322
#674952
#121573