Finite Group Databasebeta

← #10694865

Subgroups — #10694865

Order 768.

418 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×#5093C211yesmaximal, characteristic
2C3×#1546C211yesmaximal, characteristic
3#71789C211yesmaximal, characteristic
4#70779C211yesmaximal, characteristic
5#70741C211yesmaximal, characteristic
6#70524C211yesmaximal, characteristic
7#70491C211yesmaximal, characteristic
8#9809—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×#836—22no—
14#5093—21no—
15#5093—21no—
16#5093—22no—
17C2×#810C414yes—
18#4924C2211yescharacteristic
19C4×#208C2211yescharacteristic
20C2×C4×C3⋊C8C2211yescharacteristic
21C3×#332—22no—
22#4554—22no—
23#4481—22no—
24#4479—22no—
25#4455—21no—
26#4454—21no—
27C2×#487—31no—
28#1997—31no—
29#1966—31no—
30#1567—31no—
31#1548—31no—
32#1546D311yescharacteristic
33#1451—31no—
34C22×C3×Q8—21no—
35C22×Dic6C2×C412yes—
36C2×C3×M16—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C412yes—
43C2×C3×C42C2311yescharacteristic
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×#208—21no—
47C2×#208D412yes—
48C22×C3⋊C8C2×C412yes—
49#869—21no—
50#869—21no—
51#869—21no—
52#869—21no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8—21no—
55C3×C4⋊C8D412yes—
56C3×C22⋊C8—21no—
57C3×C4×C8—21no—
58#836—44no—
59#810—24no—
60#810—24no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)—21no—
63#803—21no—
64C4×C3⋊C8—21no—
65C4×C3⋊C8D411yescharacteristic
66C4×C3⋊C8D411yescharacteristic
67C4×C3⋊C8D412yes—
68C2×C4⋊Q8—31no—
69C2×C4⋊Q8C2×D311yescharacteristic
70#487—61no—
71#487—61no—
72#487—62no—
73#431C2×D311yescharacteristic
74#423—31no—
75#423C2×D311yescharacteristic
76C2×#103—34no—
77C2×#101—62no—
78C4×M16—31no—
79C2×C4×C8—31no—
80#343—62no—
81#335—62no—
82#333—62no—
83#332—22no—
84#322—61no—
85#322—61no—
86C2×C3×Q8—41no—
87C2×C3×Q8—41no—
88C2×C3×Q8—42no—
89C2×C3×Q8—42no—
90C22×C3×C4—22no—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4#4111yescharacteristic
94C22×Dic3—22no—
95C2×Dic6—22no—
96C2×Dic6—22no—
97C2×Dic6—44no—
98C3×M16—42no—
99C2×C3×C8—22no—
100C2×C3×C8—41no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C42C2×D411yescharacteristic
106C3×C42C2×D411yescharacteristic
107C3×C42C2×D411yescharacteristic
108C3×C42#4111yescharacteristic
109C4⋊Dic3—41no—
110C4⋊Dic3—41no—
111C4⋊Dic3—41no—
112C4⋊Dic3—41no—
113#208—22no—
114#208—22no—
115#208—42no—
116C2×C3⋊C8—22no—
117C2×C3⋊C8—42no—
118C2×C3⋊C8—41no—
119C2×C3⋊C8C22⋊C412yes—
120C2×C3⋊C8C22⋊C412yes—
121C22×Q8—21no—
122C22×Q8—32no—
123C2×M16—21no—
124C2×M16—32no—
125C2×M16—61no—
126C22×C8—32no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—61no—
132C4⋊Q8—61no—
133C4⋊Q8—61no—
134C4⋊Q8—61no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—61no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4C4×D312yes—
139C2×C42C22×D311yescharacteristic
140C4⋊C8—21no—
141C4⋊C8—21no—
142C4⋊C8—61no—
143C4⋊C8—61no—
144C4⋊C8#6712yes—
145#103—64no—
146#103—64no—
147#101—124no—
148C22⋊C8—21no—
149#97—61no—
150C4×C8—21no—
151C4×C8—31no—
152C4×C8—31no—
153C4×C8—32no—
154C4×C8—61no—
155C23×C3C4×D411yescharacteristic
156C3×Q8—82no—
157C3×Q8—81no—
158C3×Q8—81no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—42no—
162C2×C3×C4—44no—
163C2×C3×C4C2×C22⋊C411yescharacteristic
164C2×C3×C4C4×D411yescharacteristic
165C2×C3×C4#12111yescharacteristic
166C2×C3×C4#12111yescharacteristic
167C2×C3×C4#12411yescharacteristic
168C2×C3×C4#12711yescharacteristic
169C2×C3×C4AGL(1,ℤ/8)12yes—
170C2×C3×C4#13712yes—
171C2×Dic3—42no—
172C2×Dic3—82no—
173Dic6—44no—
174Dic6—82no—
175C3×C8—42no—
176C3×C8—81no—
177C3⋊C8—22no—
178C3⋊C8—22no—
179C3⋊C8—42no—
180C3⋊C8—81no—
181C2×Q8—41no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×Q8—62no—
186C2×Q8—62no—
187C2×Q8—124no—
188C22×C4—22no—
189C22×C4—62no—
190C22×C4C2×C4×D311yescharacteristic
191C22×C4D3×D411yescharacteristic
192C22×C4#23711yescharacteristic
193M16—42no—
194M16—62no—
195M16—62no—
196M16—122no—
197C2×C8—22no—
198C2×C8—32no—
199C2×C8—32no—
200C2×C8—41no—
201C2×C8—62no—
202C2×C8—122no—
203C2×C8—121no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—121no—
209C4⋊C4—121no—
210C4⋊C4—121no—
211C4⋊C4—121no—
212C42#23511yescharacteristic
213C42D3×D411yescharacteristic
214C42D3×D411yescharacteristic
215C42C2×#6711yescharacteristic
216C22×C3—21no—
217C22×C3—21no—
218C22×C3#39011yescharacteristic
219C22×C3#44012yes—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—82no—
223C3×C4#41312yes—
224C3×C4#46812yes—
225C3×C4#47312yes—
226C3×C4#49611yescharacteristic
227C3×C4#49711yescharacteristic
228Dic3—82no—
229C23#67⋊C411yescharacteristic
230Q8—81no—
231Q8—81no—
232Q8—82no—
233Q8—124no—
234Q8—242no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—42no—
238C2×C4—44no—
239C2×C4—122no—
240C2×C4—242no—
241C2×C4D3×C22⋊C411yescharacteristic
242C2×C4#88411yescharacteristic
243C2×C4#88511yescharacteristic
244C2×C4#91112yes—
245C2×C4#91512yes—
246C2×C4C4×#6711yescharacteristic
247C2×C4#93911yescharacteristic
248C2×C4#94011yescharacteristic
249C8—42no—
250C8—62no—
251C8—62no—
252C8—81no—
253C8—122no—
254C8—241no—
255C2×C3—22no—
256C2×C3#196711yescharacteristic
257C2×C3#197311yescharacteristic
258C2×C3#197911yescharacteristic
259C22—21no—
260C22—21no—
261C22#475712yes—
262C22#495711yescharacteristic
263C4—21no—
264C4—21no—
265C4—82no—
266C4—242no—
267C4#475112yes—
268C4#478012yes—
269C4#515411yescharacteristic
270C4#516711yescharacteristic
271C4#520412yes—
272C3#980911yescharacteristic, complemented, Sylow
273C2—22no—
274C2#7009911yescharacteristic
275C2#7019211yescharacteristic
276C2#7033611yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#390
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#1546D3
hypocenterC3#9809
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
272. C3#9809semidirect

Maximal subgroups

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

subgroupindex
C2×#50932
C3×#15462
#717892
#707792
#707412
#705242
#704912
#98093