Finite Group Databasebeta

← #10697938

Subgroups — #10697938

Order 768.

592 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 89 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×#3253C211yesmaximal, characteristic, complemented
2#83313C212yesmaximal, complemented
3#71815C212yesmaximal, complemented
4#70512C211yesmaximal, characteristic, complemented
5C3⋊#1309C211yesmaximal, characteristic, complemented
6#10300—31nomaximal, Sylow
7C3×#532C2212yes—
8#5477C2211yescharacteristic
9C3×#476—22no—
10C3×#469—22no—
11C3×#459—22no—
12C3×#423C2211yescharacteristic
13C2×#837—24no—
14C2×#825—22no—
15#5084—22no—
16#5072—22no—
17#5039—24no—
18#5015—22no—
19#5008—22no—
20C42×Dic3C2211yescharacteristic
21#4924C2212yes—
22#3253—32no—
23#3253D311yescharacteristic, complemented
24#1962—31no—
25#1962—32no—
26#1309—31no—
27C2×C3×#41—22no—
28C2×#235—21no—
29C2×C3×D8—22no—
30C2×C3×M16D412yes—
31C3×C4⋊Q8D412yes—
32C3×#127D412yes—
33C3×#124—22no—
34C3×#121—22no—
35C3×#121—22no—
36C3×C4×D4—22no—
37C3×C4×D4—22no—
38C2×C3×C42C2311yescharacteristic
39C2×#213—24no—
40#930—22no—
41C2×C4×Dic3—21no—
42C2×C4×Dic3—22no—
43C2×#208D414yes—
44#875—21no—
45#874D411yescharacteristic
46C4×D12—22no—
47#869D411yescharacteristic
48C3×#106—22no—
49C3×C4⋊C8—22no—
50C3×#102—24no—
51#837—48no—
52#832—24no—
53#825—44no—
54#809—24no—
55#808—24no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)—22no—
58#532—31no—
59#532C2×D312yes—
60#476—22no—
61#476—62no—
62#476—62no—
63#469—22no—
64#469—64no—
65#459—22no—
66#459—62no—
67#459—62no—
68#423—32no—
69#423C2×D311yescharacteristic
70C2×#104—64no—
71C2×#104—62no—
72C43—31no—
73C3×#41—44no—
74C3×#41—44no—
75C2×C3×Q8—22no—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—42no—
80C22×C3×C4—42no—
81C22×C3×C4C2×D412yes—
82C22×C3×C4C2×D411yescharacteristic
83C2×#67—41no—
84C22×Dic3—41no—
85#235—42no—
86#235—42no—
87C2×D12—22no—
88C2×D12—21no—
89C2×C4×D3—41no—
90C2×Dic6—21no—
91C3×D8—82no—
92C3×M16—42no—
93C2×C3×C8—22no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—22no—
96C3×C22⋊C4—42no—
97C3×C22⋊C4—42no—
98C3×C42C2×D411yescharacteristic
99C3×C42C2×D412yes—
100C3×C42#4111yescharacteristic
101#213—84no—
102(C2×D3)⋊C4—42no—
103C4⋊Dic3—22no—
104C4×Dic3—22no—
105C4×Dic3—24no—
106C4×Dic3—41no—
107C4×Dic3—42no—
108#208—44no—
109C2×C3⋊C8—24no—
110C2×#41—22no—
111C2×#41—61no—
112C2×D8—22no—
113C2×D8—64no—
114C2×M16—34no—
115C2×M16#6712yes—
116C4⋊Q8—31no—
117C4⋊Q8#6712yes—
118#127—31no—
119#127#6712yes—
120#124—22no—
121#124—61no—
122#121—22no—
123#121—22no—
124#121—62no—
125C4×D4—22no—
126C4×D4—22no—
127C4×D4—62no—
128C2×C42—62no—
129C2×C42—61no—
130C2×C42C22×D311yescharacteristic
131#106—22no—
132#106—64no—
133C4⋊C8—22no—
134C4⋊C8—62no—
135C4⋊C8—62no—
136#104—124no—
137#104—128no—
138#102—24no—
139#102—64no—
140#102—64no—
141C23×C3—42no—
142C23×C3#12711yescharacteristic
143C22×D3—41no—
144C3×Q8—44no—
145C3×D4—44no—
146C3×D4—42no—
147C3×D4—42no—
148C3×D4—42no—
149C3×D4—42no—
150C3×D4—84no—
151C2×C3×C4—22no—
152C2×C3×C4—21no—
153C2×C3×C4—21no—
154C2×C3×C4—22no—
155C2×C3×C4—42no—
156C2×C3×C4—82no—
157C2×C3×C4—82no—
158C2×C3×C4#12012yes—
159C2×C3×C4#12011yescharacteristic
160C2×C3×C4#12111yescharacteristic
161C2×C3×C4#12112yes—
162#67—82no—
163C2×Dic3—42no—
164C2×Dic3—44no—
165C2×Dic3—41no—
166D12—42no—
167D12—42no—
168D12—42no—
169C4×D3—81no—
170C4×D3—81no—
171Dic6—42no—
172C3×C8—42no—
173C3⋊C8—44no—
174#41—44no—
175#41—44no—
176#41—122no—
177#41—122no—
178C2×Q8—22no—
179C2×Q8—61no—
180C2×D4—22no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—42no—
184C2×D4—62no—
185C2×D4—61no—
186C2×D4—121no—
187C22×C4—42no—
188C22×C4—121no—
189C22×C4—121no—
190C22×C4D3×D412yes—
191C22×C4C2×#6711yescharacteristic
192D8—82no—
193D8—244no—
194M16—42no—
195M16—124no—
196C2×C8—22no—
197C2×C8—64no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—62no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C22⋊C4—122no—
204C42—64no—
205C42—62no—
206C42—122no—
207C42—121no—
208C42D3×D411yescharacteristic
209C42#23711yescharacteristic
210C42C2×#6712yes—
211C22×C3—21no—
212C22×C3—22no—
213C22×C3—82no—
214C22×C3—82no—
215C22×C3—82no—
216C22×C3#39211yescharacteristic
217C2×D3—81no—
218C2×D3—82no—
219C3×C4—21no—
220C3×C4—22no—
221C3×C4—82no—
222C3×C4#45312yes—
223C3×C4#45311yescharacteristic
224C3×C4#45412yes—
225C3×C4#45411yescharacteristic
226Dic3—44no—
227Dic3—81no—
228C23—42no—
229C23—121no—
230C23#94011yescharacteristic
231Q8—44no—
232Q8—122no—
233D4—44no—
234D4—42no—
235D4—42no—
236D4—42no—
237D4—42no—
238D4—84no—
239D4—122no—
240D4—122no—
241D4—122no—
242D4—242no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—42no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4—124no—
251C2×C4—122no—
252C2×C4—121no—
253C2×C4—241no—
254C2×C4—241no—
255C2×C4#93712yes—
256C2×C4#93811yescharacteristic
257C2×C4#93912yes—
258C2×C4#95311yescharacteristic
259C8—42no—
260C8—124no—
261C2×C3—41no—
262C2×C3—162no—
263C2×C3#200612yes—
264C2×C3#200611yescharacteristic
265D3—161no—
266C22—22no—
267C22—21no—
268C22—82no—
269C22—82no—
270C22—82no—
271C22—242no—
272C22—241no—
273C22#522311yescharacteristic
274C4—22no—
275C4—21no—
276C4—82no—
277C4—124no—
278C4—241no—
279C4#519812yes—
280C4#520112yes—
281C4#524111yescharacteristic
282C4#524511yescharacteristic
283C3#1030011yescharacteristic, complemented, Sylow
284C2—41no—
285C2—162no—
286C2—481no—
287C2#7187911yescharacteristic
288C2#7209812yes—
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3253C2
socleC22×C3#392
hypercenter#3253D3
hypocenterC3#10300
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3253

Complements

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

NH ≅ G/Ntype
1. C3×#3253C2semidirect
2. #83313C2semidirect
3. #71815C2semidirect
4. #70512C2semidirect
5. C3⋊#1309C2semidirect
23. #3253D3semidirect
283. C3#10300semidirect

Maximal subgroups

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

subgroupindex
C3×#32532
#833132
#718152
#705122
C3⋊#13092
#103003