Finite Group Databasebeta

← #10698654

Subgroups — #10698654

Order 768.

534 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 93 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×#2819C211yesmaximal, characteristic
2#71726C212yesmaximal, complemented
3C2×#4540C212yesmaximal, complemented
4#69550C211yesmaximal, characteristic, complemented
5#67630C211yesmaximal, characteristic, complemented
6#10424—31nomaximal, Sylow
7C2×C3×#124C2212yes—
8C3×#394—21no—
9C3×#394—21no—
10C3×#393—22no—
11C3×#392—22no—
12C3×#382C2211yescharacteristic
13C2×#835—22no—
14C2×#832—22no—
15C2×#808—21no—
16C2×#807—21no—
17#4935C2211yescharacteristic
18C2×#803C2212yes—
19C2×#803C2211yescharacteristic
20#4540—24no—
21#2819D311yescharacteristic
22#1959—32no—
23#1847—31no—
24C2×#329—32no—
25#1295—31no—
26C22×C3×Q8D412yes—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4D412yes—
30C3×#124—22no—
31C3×#124—22no—
32C3×#124—22no—
33C3×#124—22no—
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C42C2311yescharacteristic
41C2×C4⋊Dic3—22no—
42C2×C4⋊Dic3—21no—
43C22×C3⋊C8—21no—
44C22×C3⋊C8—22no—
45C3×#95—21no—
46C3×#95—21no—
47C3×#95—22no—
48#835—44no—
49#832—44no—
50#808—42no—
51#807—42no—
52#803—21no—
53#803—22no—
54#803—22no—
55#803D412yescomplemented
56C2×#124C2×D312yes—
57C2×#107—61no—
58C2×#106—61no—
59C2×#103—62no—
60C2×#102—62no—
61C2×#97—31no—
62C2×#97—32no—
63#394—21no—
64#394—21no—
65#393—22no—
66#392—22no—
67#384—31no—
68#382C2×D311yescharacteristic
69#329—64no—
70C24×C3—22no—
71C2×C3×Q8—22no—
72C2×C3×Q8—22no—
73C2×C3×Q8—42no—
74C2×C3×Q8—42no—
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—42no—
78C2×C3×D4—42no—
79C2×C3×D4—82no—
80C2×C3×D4—81no—
81C2×C3×D4—81no—
82C2×C3×D4—82no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—22no—
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D411yescharacteristic
88C22×Dic3—41no—
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91C3×C22⋊C4—42no—
92C3×C22⋊C4—42no—
93C3×C22⋊C4—42no—
94C3×C22⋊C4—42no—
95C3×C22⋊C4—82no—
96C3×C22⋊C4—82no—
97C3×C42C2×D411yescharacteristic
98C3×C42#4111yescharacteristic
99C3×C42#4112yes—
100C4⋊Dic3—41no—
101C4⋊Dic3—41no—
102C4⋊Dic3—42no—
103C4⋊Dic3—42no—
104C2×C3⋊C8—22no—
105C2×C3⋊C8—22no—
106C2×C3⋊C8—22no—
107C2×C3⋊C8—24no—
108C2×C3⋊C8—42no—
109C2×C3⋊C8—42no—
110C22×Q8#6712yes—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4#6712yes—
114C22×C8—61no—
115C22×C8—62no—
116#124—22no—
117#124—22no—
118#124—22no—
119#124—22no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4#6711yescharacteristic
123C2×C4⋊C4#6711yescharacteristic
124C2×C22⋊C4—22no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—22no—
128C2×C42C22×D311yescharacteristic
129#107—122no—
130#106—122no—
131#103—124no—
132#102—124no—
133#97—32no—
134#97—62no—
135#97—61no—
136#97—62no—
137#95—21no—
138#95—21no—
139#95—22no—
140C23×C3—42no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3—42no—
147C23×C3#12011yescharacteristic
148C3×Q8—44no—
149C3×Q8—82no—
150C3×D4—44no—
151C3×D4—82no—
152C3×D4—84no—
153C3×D4—84no—
154C2×C3×C4—21no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—21no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4#12111yescharacteristic
166C2×C3×C4#12112yes—
167C2×C3×C4#12312yes—
168C2×C3×C4#12411yescharacteristic
169C2×Dic3—81no—
170C2×Dic3—82no—
171C3⋊C8—24no—
172C3⋊C8—44no—
173C24—22no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—82no—
183C2×D4—81no—
184C2×D4—81no—
185C2×D4—82no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—22no—
189C22×C4—121no—
190C22×C4C2×#6712yes—
191C22×C4C2×#6711yescharacteristic
192C2×C8—62no—
193C2×C8—62no—
194C2×C8—62no—
195C2×C8—64no—
196C2×C8—122no—
197C2×C8—122no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—122no—
201C4⋊C4—122no—
202C4⋊C4—121no—
203C4⋊C4—121no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—82no—
209C22⋊C4—82no—
210C42D3×D411yescharacteristic
211C42#23711yescharacteristic
212C42#23712yes—
213C22×C3—82no—
214C22×C3—82no—
215C22×C3—82no—
216C22×C3—82no—
217C22×C3—82no—
218C22×C3—82no—
219C22×C3—82no—
220C22×C3#39411yescharacteristic
221C22×C3#45311yescharacteristic
222C22×C3#45312yes—
223C22×C3#45512yes—
224C22×C3#45611yescharacteristic
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—81no—
230C3×C4—81no—
231C3×C4—82no—
232Dic3—161no—
233C23—42no—
234C23—42no—
235C23—42no—
236C23—42no—
237C23—42no—
238C23—42no—
239C23—42no—
240C23#95311yescharacteristic
241Q8—44no—
242Q8—82no—
243D4—44no—
244D4—82no—
245D4—84no—
246D4—84no—
247C2×C4—21no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—241no—
259C2×C4—242no—
260C2×C4#93512yes—
261C2×C4#93611yescharacteristic
262C2×C4#93811yescharacteristic
263C2×C4#93912yes—
264C8—64no—
265C8—124no—
266C2×C3—162no—
267C2×C3#165511yescharacteristic
268C2×C3#165911yescharacteristic
269C2×C3#166112yes—
270C2×C3#203712yes—
271C2×C3#203911yescharacteristic
272C22—82no—
273C22—82no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—82no—
278C22—82no—
279C22#522211yescharacteristic
280C22#524111yescharacteristic
281C22#524112yes—
282C22#524212yes—
283C22#524611yescharacteristic
284C4—22no—
285C4—22no—
286C4—21no—
287C4—21no—
288C4—81no—
289C4—81no—
290C4—82no—
291C4—481no—
292C3#1042411yescharacteristic, complemented, Sylow
293C2—162no—
294C2#7185212yes—
295C2#7187611yescharacteristic
296C2#7194811yescharacteristic
297C2#7194911yescharacteristic
298C2#7195312yes—
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2819C2
socleC23×C3#120
hypercenter#2819D3
hypocenterC3#10424
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2819

Complements

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

NH ≅ G/Ntype
2. #71726C2semidirect
3. C2×#4540C2semidirect
4. #69550C2semidirect
5. #67630C2semidirect
55. #803D4semidirect
292. C3#10424semidirect

Maximal subgroups

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

subgroupindex
C3×#28192
#717262
C2×#45402
#695502
#676302
#104243