Finite Group Databasebeta

← #10693624

Subgroups — #10693624

Order 768.

338 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 101 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#70768C211yesmaximal, characteristic
2#70514C211yesmaximal, characteristic
3#70493C211yesmaximal, characteristic
4Dic3×C4⋊C8C211yesmaximal, characteristic
5#70437C211yesmaximal, characteristic
6C3×#1301C211yesmaximal, characteristic
7#67736C211yesmaximal, characteristic
8#9579—31nomaximal, Sylow
9#5470C2211yescharacteristic
10C3×#431C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C8×Dic3—21no—
14C42×Dic3C2211yescharacteristic
15#4925C2211yescharacteristic
16C2×C4⋊(C3⋊C8)C2211yescharacteristic
17#4555—21no—
18#4555—21no—
19#4550—21no—
20#4486C412yescomplemented
21#4481C412yescomplemented
22#4480—22no—
23#4456—22no—
24#1994—31no—
25C4×C4⋊C8—31no—
26#1570—31no—
27#1542—31no—
28#1481—31no—
29#1301D311yescharacteristic
30#1276—31no—
31C2×C3×M16—21no—
32C22×C3×C8—21no—
33C2×C3×C42C2311yescharacteristic
34#926—21no—
35#924—21no—
36#924—21no—
37C2×C4×Dic3—21no—
38C2×C4×Dic3—21no—
39C2×C4×Dic3D411yescharacteristic
40C2×C4×Dic3D411yescharacteristic
41C22×C3⋊C8—21no—
42#870—21no—
43#869C2×C411yescharacteristic
44#869C2×C411yescharacteristic
45C4×Dic6—21no—
46C4×Dic6—21no—
47C3×C4⋊C8—21no—
48C3×C4⋊C8—21no—
49C3×C4⋊C8—21no—
50C3×C4⋊C8—21no—
51C3×C4⋊C8C2×C412yes—
52C3×C22⋊C8—21no—
53C3×C4×C8—21no—
54#830—21no—
55C8×Dic3—42no—
56C4⋊(C3⋊C8)—21no—
57C4⋊(C3⋊C8)C2×C412yes—
58C4⋊(C3⋊C8)D412yes—
59C4×C3⋊C8—21no—
60#533—31no—
61#431—31no—
62#431C2×D311yescharacteristic
63#423C2×D311yescharacteristic
64C2×C4⋊C8—31no—
65C2×C4⋊C8C2×D311yescharacteristic
66C2×C4×C8—61no—
67C43—31no—
68#339—61no—
69#339—61no—
70#336—61no—
71#334—62no—
72#333—32no—
73#332—32no—
74Q8⋊C8—62no—
75C22×C3×C4C2×D411yescharacteristic
76C22×C3×C4#4111yescharacteristic
77C22×C3×C4#4111yescharacteristic
78C22×Dic3—21no—
79C22×Dic3—21no—
80C2×Dic6—21no—
81C2×Dic6—21no—
82C3×M16—41no—
83C3×M16—41no—
84C2×C3×C8—22no—
85C2×C3×C8—41no—
86C2×C3×C8—41no—
87C2×C3×C8—41no—
88C2×C3×C8—41no—
89C3×C42C22×C411yescharacteristic
90C3×C42C2×D411yescharacteristic
91C3×C42#4111yescharacteristic
92C3×C42#4111yescharacteristic
93#217—41no—
94C4⋊Dic3—21no—
95C4⋊Dic3—21no—
96C4⋊Dic3—21no—
97C4⋊Dic3—21no—
98Dic3⋊C4—41no—
99Dic3⋊C4—41no—
100C4×Dic3—24no—
101C4×Dic3—21no—
102C4×Dic3—21no—
103C4×Dic3—21no—
104C4×Dic3—42no—
105C4×Dic3—41no—
106C4×Dic3SD1612yes—
107C2×C3⋊C8—22no—
108C2×C3⋊C8—22no—
109C2×C3⋊C8—41no—
110C2×C3⋊C8—41no—
111C2×M16—21no—
112C22×C8—21no—
113C22×C8—61no—
114C4⋊Q8—31no—
115C4⋊Q8—31no—
116#125—61no—
117C22⋊Q8—61no—
118C22⋊Q8—61no—
119C4×Q8—61no—
120C4×Q8—61no—
121#117—61no—
122C2×C42—31no—
123C2×C42—31no—
124C2×C42—61no—
125C2×C42—61no—
126C2×C42C22×D311yescharacteristic
127C4⋊C8—21no—
128C4⋊C8—21no—
129C4⋊C8—21no—
130C4⋊C8—21no—
131C4⋊C8—32no—
132C4⋊C8—32no—
133C4⋊C8—61no—
134C4⋊C8C4×D312yes—
135C22⋊C8—21no—
136C22⋊C8—61no—
137C4×C8—21no—
138C4×C8—61no—
139C4×C8—122no—
140C23×C3#12311yescharacteristic
141C2×C3×C4—22no—
142C2×C3×C4—41no—
143C2×C3×C4#11711yescharacteristic
144C2×C3×C4C4×D411yescharacteristic
145C2×C3×C4C4×D411yescharacteristic
146C2×C3×C4#12111yescharacteristic
147C2×C3×C4#12411yescharacteristic
148C2×C3×C4#12611yescharacteristic
149C2×C3×C4C2×SD1611yescharacteristic
150C2×C3×C4#13511yescharacteristic
151C2×C3×C4#13711yescharacteristic
152C2×C3×C4#13711yescharacteristic
153C2×Dic3—22no—
154C2×Dic3—22no—
155C2×Dic3—22no—
156C2×Dic3—22no—
157C2×Dic3—41no—
158C2×Dic3—41no—
159C2×Dic3—41no—
160C2×Dic3—41no—
161Dic6—42no—
162Dic6—81no—
163C3×C8—42no—
164C3×C8—81no—
165C3×C8—81no—
166C3⋊C8—42no—
167C3⋊C8—42no—
168C2×Q8—61no—
169C2×Q8—61no—
170C22×C4—61no—
171C22×C4—61no—
172C22×C4#23511yescharacteristic
173C22×C4D3×D411yescharacteristic
174C22×C4#23911yescharacteristic
175M16—41no—
176M16—41no—
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—62no—
183C2×C8—62no—
184C2×C8—121no—
185C2×C8—121no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4—61no—
189C4⋊C4—61no—
190C4⋊C4—121no—
191C4⋊C4—121no—
192C22⋊C4—121no—
193C42—32no—
194C42—64no—
195C42—61no—
196C42—61no—
197C42—61no—
198C42—122no—
199C42—121no—
200C42C2×C4×D311yescharacteristic
201C42#23511yescharacteristic
202C42D3×D411yescharacteristic
203C42#23711yescharacteristic
204C22×C3—21no—
205C22×C3—21no—
206C22×C3#38811yescharacteristic
207C22×C3#48311yescharacteristic
208C22×C3#48511yescharacteristic
209C3×C4—21no—
210C3×C4—21no—
211C3×C4—21no—
212C3×C4C4×SD1611yescharacteristic
213C3×C4#44111yescharacteristic
214C3×C4#44312yes—
215C3×C4#46111yescharacteristic
216C3×C4#46411yescharacteristic
217Dic3—24no—
218Dic3—42no—
219Dic3—81no—
220Dic3—81no—
221C23#89411yescharacteristic
222Q8—122no—
223Q8—241no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—62no—
227C2×C4—62no—
228C2×C4—62no—
229C2×C4—62no—
230C2×C4—121no—
231C2×C4—121no—
232C2×C4—121no—
233C2×C4—121no—
234C2×C4#87211yescharacteristic
235C2×C4#67⋊C411yescharacteristic
236C2×C4#88411yescharacteristic
237C2×C4#88511yescharacteristic
238C2×C4#89311yescharacteristic
239C2×C4#89811yescharacteristic
240C2×C4#91211yescharacteristic
241C2×C4D3×SD1611yescharacteristic
242C2×C4#91511yescharacteristic
243C2×C4#91811yescharacteristic
244C8—42no—
245C8—81no—
246C8—81no—
247C8—122no—
248C8—122no—
249C2×C3—21no—
250C2×C3—21no—
251C2×C3#192711yescharacteristic
252C2×C3#193211yescharacteristic
253C2×C3#193711yescharacteristic
254C22—21no—
255C22—21no—
256C22#466911yescharacteristic
257C22#486211yescharacteristic
258C22#488811yescharacteristic
259C4—21no—
260C4—21no—
261C4—21no—
262C4—64no—
263C4—122no—
264C4—241no—
265C4—241no—
266C4#475811yescharacteristic
267C4#476711yescharacteristic
268C4#478911yescharacteristic
269C4#481811yescharacteristic
270C4#489512yes—
271C3#957911yescharacteristic, complemented, Sylow
272C2—21no—
273C2—21no—
274C2#6879311yescharacteristic
275C2#6891811yescharacteristic
276C2#6901211yescharacteristic
subgroupG/H
centerC2×C4#884
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#388
prime radicalC23×C3#123
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#9579
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
20. #4486C4semidirect
21. #4481C4semidirect
271. C3#9579semidirect

Maximal subgroups

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

subgroupindex
#707682
#705142
#704932
Dic3×C4⋊C82
#704372
C3×#13012
#677362
#95793