Finite Group Databasebeta

← #10698657

Subgroups — #10698657

Order 768.

534 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 91 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#74254C211yesmaximal, characteristic
2C3×#1847C211yesmaximal, characteristic, complemented
3C2×#4464C212yesmaximal, complemented
4#69581C212yesmaximal, complemented
5#67630C211yesmaximal, characteristic, complemented
6#10424—31nomaximal, Sylow
7C2×#875C2212yes—
8C2×C3×#107—21no—
9C2×C3×#106—21no—
10C2×C3×#97C2211yescharacteristic
11C3×#384C2211yescharacteristic
12C3×#382C2211yescharacteristic
13#4991—21no—
14#4991—21no—
15C2×#810—22no—
16C2×#809—22no—
17C2×#803C2212yes—
18#4673—22no—
19#4671—22no—
20#4464—24no—
21#2819—31no—
22#1959—32no—
23#1847D311yescharacteristic, complemented
24C2×#329—32no—
25#1295—31no—
26C22×D12—21no—
27C22×D12—21no—
28C22×D12D412yes—
29C22×Dic6D412yes—
30C22×C3×C8—21no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C42C2311yescharacteristic
36C2×(C2×D3)⋊C4—22no—
37C2×(C2×D3)⋊C4—22no—
38C2×(C2×D3)⋊C4—22no—
39C2×(C2×D3)⋊C4—22no—
40C22×C3⋊C8—22no—
41#875—22no—
42#875—22no—
43#875—22no—
44#875—22no—
45C3×#107—42no—
46C3×#106—42no—
47C3×#97—21no—
48C3×#97D412yescomplemented
49C3×#95—22no—
50C3⋊#95—21no—
51C3⋊#95—21no—
52#810—44no—
53#809—44no—
54#803—22no—
55#803—22no—
56C2×#124—32no—
57C2×#107—21no—
58C2×#106—21no—
59C2×#103—62no—
60C2×#102—62no—
61C2×#97—32no—
62C2×#97C2×D311yescharacteristic
63#394—61no—
64#394—61no—
65#393—62no—
66#392—62no—
67#384C2×D311yescharacteristic
68#382C2×D311yescharacteristic
69#329—64no—
70C23×D3—22no—
71C22×C3×C4—21no—
72C22×C3×C4—21no—
73C22×C3×C4—41no—
74C22×C3×C4C2×D412yes—
75C22×C3×C4C2×D411yescharacteristic
76C22×Dic3—22no—
77C2×D12—22no—
78C2×D12—22no—
79C2×D12—42no—
80C2×D12—42no—
81C2×D12—82no—
82C2×D12—81no—
83C2×D12—81no—
84C2×D12—82no—
85C2×Dic6—22no—
86C2×Dic6—22no—
87C2×Dic6—42no—
88C2×Dic6—42no—
89C2×C3×C8—22no—
90C2×C3×C8—22no—
91C2×C3×C8—22no—
92C3×C4⋊C4—41no—
93C3×C4⋊C4—41no—
94C3×C4⋊C4—42no—
95C3×C4⋊C4—42no—
96C3×C4⋊C4—41no—
97C3×C4⋊C4—41no—
98C3×C42C2×D411yescharacteristic
99C3×C42#4111yescharacteristic
100C3×C42#4112yes—
101(C2×D3)⋊C4—42no—
102(C2×D3)⋊C4—42no—
103(C2×D3)⋊C4—42no—
104(C2×D3)⋊C4—42no—
105(C2×D3)⋊C4—82no—
106(C2×D3)⋊C4—82no—
107C2×C3⋊C8—24no—
108C2×C3⋊C8—42no—
109C2×C3⋊C8—42no—
110C22×Q8—32no—
111C22×D4—32no—
112C22×D4—61no—
113C22×D4—61no—
114C22×C8—21no—
115C22×C8—62no—
116#124—62no—
117#124—62no—
118#124—62no—
119#124—62no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4#6711yescharacteristic
123C2×C4⋊C4#6711yescharacteristic
124C2×C22⋊C4—62no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4—62no—
127C2×C22⋊C4—62no—
128C2×C42C22×D311yescharacteristic
129#107—42no—
130#106—42no—
131#103—124no—
132#102—124no—
133#97—21no—
134#97—62no—
135#97—62no—
136#97D1212yescomplemented
137#95—22no—
138#95—61no—
139#95—61no—
140C23×C3#12011yescharacteristic
141C22×D3—42no—
142C22×D3—42no—
143C22×D3—42no—
144C22×D3—42no—
145C22×D3—42no—
146C22×D3—42no—
147C22×D3—42no—
148C2×C3×C4—22no—
149C2×C3×C4—21no—
150C2×C3×C4—22no—
151C2×C3×C4—21no—
152C2×C3×C4—42no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—82no—
157C2×C3×C4—81no—
158C2×C3×C4#12112yes—
159C2×C3×C4#12111yescharacteristic
160C2×C3×C4#12312yes—
161C2×C3×C4#12411yescharacteristic
162C2×Dic3—42no—
163C2×Dic3—42no—
164C2×Dic3—42no—
165D12—44no—
166D12—82no—
167D12—84no—
168D12—84no—
169Dic6—44no—
170Dic6—82no—
171C3×C8—24no—
172C3⋊C8—44no—
173C24—62no—
174C2×Q8—62no—
175C2×Q8—62no—
176C2×Q8—122no—
177C2×Q8—122no—
178C2×D4—62no—
179C2×D4—62no—
180C2×D4—122no—
181C2×D4—122no—
182C2×D4—242no—
183C2×D4—242no—
184C2×D4—241no—
185C2×D4—241no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4—62no—
190C22×C4D3×D412yes—
191C22×C4C2×#6711yescharacteristic
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—64no—
196C2×C8—122no—
197C2×C8—122no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C4⋊C4—41no—
204C22⋊C4—122no—
205C22⋊C4—122no—
206C22⋊C4—122no—
207C22⋊C4—122no—
208C22⋊C4—242no—
209C22⋊C4—242no—
210C42C2×D1211yescharacteristic
211C42#23511yescharacteristic
212C42#23912yes—
213C22×C3#39411yescharacteristic
214C22×C3#45312yes—
215C22×C3#45311yescharacteristic
216C22×C3#45512yes—
217C22×C3#45611yescharacteristic
218C2×D3—82no—
219C2×D3—82no—
220C2×D3—82no—
221C2×D3—82no—
222C2×D3—82no—
223C2×D3—82no—
224C2×D3—82no—
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—21no—
228C3×C4—21no—
229C3×C4—81no—
230C3×C4—81no—
231C3×C4—161no—
232Dic3—82no—
233C23—122no—
234C23—122no—
235C23—122no—
236C23—122no—
237C23—122no—
238C23—122no—
239C23—122no—
240C23#93711yescharacteristic
241Q8—124no—
242Q8—242no—
243D4—124no—
244D4—242no—
245D4—244no—
246D4—244no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—82no—
256C2×C4—81no—
257C2×C4—122no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4#89412yes—
261C2×C4#89512yes—
262C2×C4#93011yescharacteristic
263C2×C4#94711yescharacteristic
264C8—24no—
265C8—124no—
266C2×C3#165511yescharacteristic
267C2×C3#165911yescharacteristic
268C2×C3#166112yes—
269C2×C3#203712yes—
270C2×C3#203911yescharacteristic
271D3—162no—
272C22—242no—
273C22—242no—
274C22—242no—
275C22—242no—
276C22—242no—
277C22—242no—
278C22—242no—
279C22#499111yescharacteristic
280C22#507711yescharacteristic
281C22#509511yescharacteristic
282C22#519812yes—
283C22#520212yes—
284C4—22no—
285C4—22no—
286C4—21no—
287C4—21no—
288C4—81no—
289C4—81no—
290C4—161no—
291C4—242no—
292C3#1042411yescharacteristic, complemented, Sylow
293C2—482no—
294C2#6959311yescharacteristic
295C2#7067512yes—
296C2#7132611yescharacteristic
297C2#7132911yescharacteristic
298C2#7139212yes—
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1847C2
socleC23×C3#120
prime radical#74254C2
hypercenter#1847D3
hypocenterC3#10424
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
2. C3×#1847C2semidirect
3. C2×#4464C2semidirect
4. #69581C2semidirect
5. #67630C2semidirect
23. #1847D3semidirect
48. C3×#97D4semidirect
136. #97D12semidirect
292. C3#10424semidirect

Maximal subgroups

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

subgroupindex
#742542
C3×#18472
C2×#44642
#695812
#676302
#104243