Finite Group Databasebeta

← #10690682

Subgroups — #10690682

Order 768.

566 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 139 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×#2972C211yesmaximal, characteristic
2#82328C211yesmaximal, characteristic
3#71601C211yesmaximal, characteristic
4#70767C212yesmaximal, complemented
5#67737C212yesmaximal, complemented
6#9099—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#936C411yescharacteristic
9C2×C4×Dic6C2211yescharacteristic
10C3×C8×D4—24no—
11C3×#431C2212yes—
12C2×C3×C22⋊C8—22no—
13#5221C411yescharacteristic
14#5210—22no—
15#4975C411yescharacteristic
16#4973C411yescharacteristic
17#4944—22no—
18#4932C2211yescharacteristic
19#4925C2212yes—
20#4525—22no—
21#4456—24no—
22#2972D311yescharacteristic
23#2326—31no—
24#1479—32no—
25#1460—31no—
26#1274—32no—
27C22×C3×D4C2×C411yescharacteristic
28C23×C3×C4D412yes—
29C22×Dic6C2×C411yescharacteristic
30C22×C3×C8—24no—
31C3×C4×D4—22no—
32C3×C4×D4—22no—
33C2×C3×C4⋊C4C2×C411yescharacteristic
34C2×C3×C22⋊C4D412yes—
35C2×C3×C42C2311yescharacteristic
36C2×#217—22no—
37C2×#217—22no—
38#936—22no—
39#936C814yescomplemented
40C2×C4⋊Dic3C2×C411yescharacteristic
41C2×Dic3⋊C4—21no—
42C2×Dic3⋊C4—22no—
43C2×C4×Dic3—21no—
44C2×C4×Dic3C2×C411yescharacteristic
45C2×C4×Dic3C2×C411yescharacteristic
46C4×Dic6—22no—
47C4×Dic6—22no—
48C3×C4⋊C8—22no—
49C3×C22⋊C8—24no—
50C3×C22⋊C8—22no—
51C3×C4×C8D414yes—
52C3⋊#95—22no—
53C3⋊#95—21no—
54C3⋊#95—21no—
55#830—22no—
56C4⋊(C3⋊C8)—22no—
57C4×C3⋊C8—22no—
58C2×#124—31no—
59C2×C4×Q8—31no—
60C2×C4×D4C2×D311yescharacteristic
61C8×D4—24no—
62#431—32no—
63#431C2×D312yes—
64C2×C22⋊C8—22no—
65#391—31no—
66#390—31no—
67#388—62no—
68#385—62no—
69#384—31no—
70#376—31no—
71Q8⋊C8—64no—
72#323—62no—
73C24×C3C22⋊C412yes—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4C2×C812yes—
77C2×C3×D4M1612yes—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—82no—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×Dic3—21no—
86C22×Dic3—21no—
87C22×Dic3—21no—
88C22×Dic3—21no—
89C2×Dic6—22no—
90C2×Dic6—22no—
91C2×Dic6—21no—
92C2×Dic6C2×C812yescomplemented
93C2×C3×C8—24no—
94C2×C3×C8—84no—
95C3×C4⋊C4—22no—
96C3×C22⋊C4—24no—
97C3×C42C2×D412yes—
98C3×C42#4112yes—
99#217—28no—
100#217—82no—
101C4⋊Dic3—22no—
102Dic3⋊C4—41no—
103Dic3⋊C4—41no—
104Dic3⋊C4—82no—
105C4×Dic3—21no—
106C4×Dic3—22no—
107C4×Dic3—42no—
108C4×Dic3C2×C812yescomplemented
109C2×C3⋊C8—42no—
110C22×Q8—31no—
111C22×D4C4×D311yescharacteristic
112C23×C4#6712yes—
113C22×C8—24no—
114#124—34no—
115#124—62no—
116C4×Q8—62no—
117C4×Q8—62no—
118C4×D4—22no—
119C4×D4—22no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—62no—
123C2×C4⋊C4C4×D311yescharacteristic
124C2×C22⋊C4—62no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4D1212yes—
127C2×C42—31no—
128C2×C42—31no—
129C2×C42—61no—
130C2×C42C22×D311yescharacteristic
131C4⋊C8—22no—
132C4⋊C8—62no—
133C22⋊C8—24no—
134C22⋊C8—22no—
135C22⋊C8—62no—
136C4×C8—62no—
137C4×C8#6714yes—
138#95—62no—
139#95—61no—
140#95—61no—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3C22⋊C814yes—
146C23×C3C2×C22⋊C411yescharacteristic
147C3×D4—24no—
148C3×D4—24no—
149C2×C3×C4—22no—
150C2×C3×C4—41no—
151C2×C3×C4—41no—
152C2×C3×C4—44no—
153C2×C3×C4—42no—
154C2×C3×C4—82no—
155C2×C3×C4C4×D412yes—
156C2×C3×C4#12011yescharacteristic
157C2×C3×C4#12112yes—
158C2×C3×C4#12311yescharacteristic
159C2×C3×C4C22×C811yescharacteristic
160C2×C3×C4C2×M1611yescharacteristic
161C2×C3×C4#13112yes—
162C2×Dic3—22no—
163C2×Dic3—22no—
164C2×Dic3—41no—
165C2×Dic3—41no—
166C2×Dic3—41no—
167C2×Dic3—41no—
168C2×Dic3—42no—
169C2×Dic3—41no—
170C2×Dic3—41no—
171C2×Dic3—41no—
172C2×Dic3—41no—
173Dic6—24no—
174Dic6—42no—
175C3×C8—44no—
176C3⋊C8—82no—
177C24(C2×D3)⋊C412yes—
178C2×Q8—32no—
179C2×Q8—61no—
180C2×Q8—62no—
181C2×Q8—62no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4D3×C812yes—
185C2×D4ΓL(1,𝔽25)12yes—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—61no—
189C22×C4—61no—
190C22×C4—61no—
191C22×C4—61no—
192C22×C4—82no—
193C22×C4(C2×D3)⋊C412yes—
194C22×C4C2×C4×D311yescharacteristic
195C22×C4C2×D1211yescharacteristic
196C22×C4C2×#6711yescharacteristic
197C2×C8—24no—
198C2×C8—84no—
199C2×C8—122no—
200C4⋊C4—22no—
201C4⋊C4—62no—
202C4⋊C4—121no—
203C4⋊C4—121no—
204C4⋊C4—242no—
205C22⋊C4—24no—
206C22⋊C4—68no—
207C22⋊C4—242no—
208C42—32no—
209C42—61no—
210C42—62no—
211C42—122no—
212C42#23512yes—
213C42C2×#6712yes—
214C22×C3—21no—
215C22×C3—21no—
216C22×C3—24no—
217C22×C3—44no—
218C22×C3—42no—
219C22×C3—42no—
220C22×C3—82no—
221C22×C3#38611yescharacteristic
222C22×C3C2×C22⋊C811yescharacteristic
223C22×C3#40811yescharacteristic
224C3×C4—22no—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—42no—
228C3×C4C8×D412yes—
229C3×C4#43512yes—
230Dic3—42no—
231Dic3—42no—
232Dic3—81no—
233Dic3—81no—
234C23—22no—
235C23—22no—
236C23—42no—
237C23—42no—
238C23(C2×D3)⋊C814yes—
239C23C2×(C2×D3)⋊C411yescharacteristic
240Q8—64no—
241Q8—122no—
242D4—24no—
243D4—24no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—62no—
250C2×C4—62no—
251C2×C4—82no—
252C2×C4—121no—
253C2×C4—121no—
254C2×C4—121no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4—122no—
258C2×C4—121no—
259C2×C4—121no—
260C2×C4—121no—
261C2×C4C2×D3×C811yescharacteristic
262C2×C4C2×ΓL(1,𝔽25)11yescharacteristic
263C2×C4#90112yes—
264C2×C4C4×#6712yes—
265C2×C4#92911yescharacteristic
266C2×C4#93012yes—
267C2×C4#95311yescharacteristic
268C8—44no—
269C8—242no—
270C2×C3—21no—
271C2×C3—21no—
272C2×C3—44no—
273C2×C3#186911yescharacteristic
274C2×C3#190911yescharacteristic
275C2×C3#191111yescharacteristic
276C22—21no—
277C22—21no—
278C22—24no—
279C22—44no—
280C22—42no—
281C22—42no—
282C22—82no—
283C22C2×(C2×D3)⋊C811yescharacteristic
284C22#511011yescharacteristic
285C22#521311yescharacteristic
286C4—22no—
287C4—21no—
288C4—21no—
289C4—42no—
290C4—122no—
291C4—122no—
292C4—241no—
293C4—241no—
294C4C8×#6712yes—
295C4#511112yes—
296C3#909911yescharacteristic, complemented, Sylow
297C2—21no—
298C2—21no—
299C2—44no—
300C2#7165211yescharacteristic
301C2#7197511yescharacteristic
302C2#7202411yescharacteristic
subgroupG/H
centerC22#5213
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#2972C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#2972D3
hypocenterC3#9099
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
4. #70767C2semidirect
5. #67737C2semidirect
39. #936C8semidirect
92. C2×Dic6C2×C8semidirect
108. C4×Dic3C2×C8semidirect
296. C3#9099semidirect

Maximal subgroups

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

subgroupindex
C3×#29722
#823282
#716012
#707672
#677372
#90993