Finite Group Databasebeta

← #10696119

Subgroups — #10696119

Order 768.

564 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 83 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#82310C212yesmaximal, complemented
2#79026C211yesmaximal, characteristic, complemented
3C3×#1955C211yesmaximal, characteristic, complemented
4#69798C212yesmaximal, complemented
5#67708C211yesmaximal, characteristic, complemented
6#10026—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8#5477C2212yes—
9C2×C3×C4⋊C8C2211yescharacteristic
10C2×C3×C22⋊C8—22no—
11C2×C3×#97C2211yescharacteristic
12#5217—22no—
13#5115—22no—
14#5015—22no—
15#5014—24no—
16#5008—22no—
17#4968—22no—
18#4926C2212yes—
19#4846—22no—
20#4843—22no—
21#4464—24no—
22#3245—31no—
23#3109—32no—
24#1955D311yescharacteristic, complemented
25#1530—32no—
26#1290—31no—
27C22×C3×D4D411yescharacteristic
28C23×C3×C4—21no—
29C2×#235—22no—
30C22×C3×C8—21no—
31C22×C3×C8D412yes—
32C3×C4×D4—22no—
33C3×C4×D4—22no—
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C22⋊C4—21no—
36C2×C3×C42C2311yescharacteristic
37C2×#213—24no—
38#930—22no—
39#930—22no—
40C2×D24—22no—
41#875—22no—
42#874D412yes—
43C4×D12—22no—
44C4×D12—22no—
45#869D412yes—
46C3×C4⋊C8—22no—
47C3×C22⋊C8—44no—
48C3×#97—22no—
49#832—24no—
50#830—22no—
51#821—24no—
52#818—22no—
53#809—24no—
54#808—24no—
55C4⋊(C3⋊C8)—22no—
56#803D414yes—
57#532—32no—
58C2×C4×D4C2×D311yescharacteristic
59#481—62no—
60#476—62no—
61#474—62no—
62#466—62no—
63#459—62no—
64#459—62no—
65#447—62no—
66#442—64no—
67#432—32no—
68C2×C4⋊C8C2×D311yescharacteristic
69C2×C22⋊C8—22no—
70C2×#97C2×D311yescharacteristic
71#329—64no—
72C24×C3—21no—
73C2×C3×D4—22no—
74C2×C3×D4—42no—
75C22×C3×C4—21no—
76C22×C3×C4—42no—
77C22×C3×C4—42no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4#4111yescharacteristic
83C2×#67—42no—
84#235—84no—
85C2×D12—22no—
86C2×D12—22no—
87C2×D12—22no—
88C2×C4×D3—42no—
89C2×Dic6—22no—
90C2×C3×C8—22no—
91C2×C3×C8—22no—
92C2×C3×C8—41no—
93C2×C3×C8—41no—
94C2×C3×C8—42no—
95C3×C4⋊C4—22no—
96C3×C22⋊C4—42no—
97C3×C42C2×D412yes—
98C3×C42C2×D412yes—
99#213—48no—
100(C2×D3)⋊C4—42no—
101(C2×D3)⋊C4—42no—
102C4⋊Dic3—22no—
103C4⋊Dic3—22no—
104C2×C3⋊C8—24no—
105D24—82no—
106C2×#41—62no—
107C22×D4D1211yescharacteristic
108C23×C4—21no—
109C2×D8—62no—
110C2×D8—64no—
111C22×C8—21no—
112C22×C8#6712yes—
113C4⋊Q8—32no—
114#127—32no—
115#124—62no—
116#121—62no—
117#121—62no—
118C4×D4—22no—
119C4×D4—22no—
120C4×D4—62no—
121C4×D4—62no—
122C2×C4⋊C4D1211yescharacteristic
123C2×C22⋊C4—21no—
124C2×C42C22×D311yescharacteristic
125#107—62no—
126#106—64no—
127C4⋊C8—22no—
128C4⋊C8—62no—
129#102—64no—
130#102—64no—
131#102—64no—
132C22⋊C8—44no—
133C22⋊C8—62no—
134#97—22no—
135#97—34no—
136C23×C3—22no—
137C23×C3—42no—
138C23×C3—41no—
139C23×C3—41no—
140C23×C3—42no—
141C23×C3#12111yescharacteristic
142C22×D3—42no—
143C3×D4—24no—
144C3×D4—42no—
145C2×C3×C4—22no—
146C2×C3×C4—21no—
147C2×C3×C4—21no—
148C2×C3×C4—41no—
149C2×C3×C4—42no—
150C2×C3×C4—41no—
151C2×C3×C4—82no—
152C2×C3×C4—81no—
153C2×C3×C4—81no—
154C2×C3×C4#12012yes—
155C2×C3×C4#12011yescharacteristic
156C2×C3×C4#12112yes—
157C2×C3×C4#12711yescharacteristic
158#67—84no—
159C2×Dic3—42no—
160D12—44no—
161D12—42no—
162D12—42no—
163D12—82no—
164C4×D3—84no—
165Dic6—82no—
166C3×C8—42no—
167C3×C8—42no—
168C3⋊C8—44no—
169C24—21no—
170#41—244no—
171C2×Q8—62no—
172C2×D4—22no—
173C2×D4—42no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—122no—
178C22×C4—21no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—122no—
184C22×C4C2×D1211yescharacteristic
185C22×C4#23511yescharacteristic
186C22×C4C2×#6711yescharacteristic
187D8—128no—
188D8—242no—
189C2×C8—22no—
190C2×C8—22no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—42no—
194C2×C8—64no—
195C4⋊C4—22no—
196C4⋊C4—62no—
197C4⋊C4—62no—
198C22⋊C4—42no—
199C22⋊C4—122no—
200C22⋊C4—122no—
201C42D3×D412yes—
202C42D3×D412yes—
203C22×C3—21no—
204C22×C3—21no—
205C22×C3—44no—
206C22×C3—42no—
207C22×C3—82no—
208C22×C3—81no—
209C22×C3—81no—
210C22×C3#39211yescharacteristic
211C22×C3#47212yes—
212C2×D3—82no—
213C2×D3—82no—
214C2×D3—82no—
215C3×C4—21no—
216C3×C4—22no—
217C3×C4—22no—
218C3×C4—81no—
219C3×C4#45311yescharacteristic
220C3×C4#45411yescharacteristic
221Dic3—82no—
222C23—22no—
223C23—42no—
224C23—41no—
225C23—42no—
226C23—41no—
227C23—122no—
228C23#93011yescharacteristic
229Q8—242no—
230D4—24no—
231D4—42no—
232D4—124no—
233D4—122no—
234D4—122no—
235D4—242no—
236D4—244no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—41no—
243C2×C4—82no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4—122no—
247C2×C4—244no—
248C2×C4#88212yes—
249C2×C4#88412yes—
250C2×C4#93711yescharacteristic
251C2×C4#94011yescharacteristic
252C8—42no—
253C8—42no—
254C8—124no—
255C2×C3—22no—
256C2×C3—82no—
257C2×C3#200611yescharacteristic
258C2×C3#201011yescharacteristic
259C2×C3#201811yescharacteristic
260D3—162no—
261C22—21no—
262C22—21no—
263C22—44no—
264C22—42no—
265C22—82no—
266C22—81no—
267C22—81no—
268C22—242no—
269C22—242no—
270C22—242no—
271C22#496011yescharacteristic
272C22#514212yes—
273C4—21no—
274C4—22no—
275C4—22no—
276C4—81no—
277C4—242no—
278C4#506211yescharacteristic
279C4#507711yescharacteristic
280C3#1002611yescharacteristic, complemented, Sylow
281C2—22no—
282C2—82no—
283C2—482no—
284C2#6966411yescharacteristic
285C2#6999311yescharacteristic
286C2#7000611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1955C2
socleC22×C3#392
hypercenter#1955D3
hypocenterC3#10026
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. #82310C2semidirect
2. #79026C2semidirect
3. C3×#1955C2semidirect
4. #69798C2semidirect
5. #67708C2semidirect
24. #1955D3semidirect
280. C3#10026semidirect

Maximal subgroups

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

subgroupindex
#823102
#790262
C3×#19552
#697982
#677082
#100263