Finite Group Databasebeta

← #10694768

Subgroups — #10694768

Order 768.

426 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 121 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×#5080C211yesmaximal, characteristic
2C3×#1546C211yesmaximal, characteristic
3#71789C211yesmaximal, characteristic
4#70780C211yesmaximal, characteristic
5#70739C211yesmaximal, characteristic
6#70524C211yesmaximal, characteristic
7#70490C211yesmaximal, characteristic
8#9794—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×#836—22no—
14#5080—21no—
15#5080—21no—
16#5080—22no—
17C2×#808C414yes—
18#4924C2211yescharacteristic
19C4×#208C2211yescharacteristic
20C2×C4×C3⋊C8C2211yescharacteristic
21C3×#332—22no—
22#4554—22no—
23#4482—22no—
24#4480—22no—
25#4454—21no—
26C8×C3⋊C8—21no—
27C2×#498—31no—
28#1997—31no—
29#1966—31no—
30#1566—31no—
31#1549—31no—
32#1546D311yescharacteristic
33C8×M16—31no—
34C22×C3×Q8—21no—
35C22×Dic6—21no—
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⋊Dic3C2×C412yes—
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#808—24no—
60#808—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#498—61no—
71#498—61no—
72#498—62no—
73#431C2×D311yescharacteristic
74C2×#106—34no—
75#423—31no—
76#423C2×D311yescharacteristic
77C2×#101—62no—
78C4×M16—31no—
79C2×C4×C8—31no—
80#343—62no—
81#334—62no—
82#332—22no—
83#332—62no—
84#322—61no—
85C82—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—41no—
96C2×Dic6—41no—
97C2×Dic6—82no—
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—22no—
110C4⋊Dic3—22no—
111C4⋊Dic3—41no—
112C4⋊Dic3—41no—
113#208—22no—
114#208—42no—
115#208SD1614yes—
116C2×C3⋊C8—22no—
117C2×C3⋊C8—24no—
118C2×C3⋊C8—41no—
119C2×C3⋊C8C22⋊C412yes—
120C2×C3⋊C8C22⋊C412yes—
121C22×Q8—21no—
122C22×Q8—61no—
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—32no—
137C2×C4⋊C4—61no—
138C2×C4⋊C4C4×D312yes—
139C2×C42C22×D311yescharacteristic
140#106—64no—
141#106—64no—
142C4⋊C8—21no—
143C4⋊C8—21no—
144C4⋊C8—61no—
145C4⋊C8—61no—
146C4⋊C8#6712yes—
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×C4C2×SD1612yes—
170C2×C3×C4#13512yes—
171C2×Dic3—42no—
172C2×Dic3—82no—
173Dic6—82no—
174Dic6—82no—
175C3×C8—42no—
176C3×C8—81no—
177C3⋊C8—22no—
178C3⋊C8—24no—
179C3⋊C8—81no—
180C3⋊C8#10114yescomplemented
181C2×Q8—41no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×Q8—121no—
186C2×Q8—121no—
187C2×Q8—242no—
188C22×C4—22no—
189C22×C4—62no—
190C22×C4C2×C4×D311yescharacteristic
191C22×C4D3×D411yescharacteristic
192C22×C4#23711yescharacteristic
193M16—34no—
194M16—42no—
195M16—62no—
196M16—122no—
197C2×C8—22no—
198C2×C8—32no—
199C2×C8—32no—
200C2×C8—41no—
201C2×C8—64no—
202C2×C8—62no—
203C2×C8—121no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C4⋊C4—121no—
211C4⋊C4—121no—
212C42#23511yescharacteristic
213C42D3×D411yescharacteristic
214C42D3×D411yescharacteristic
215C42C2×#6711yescharacteristic
216C22×C3—21no—
217C22×C3—21no—
218C22×C3#39011yescharacteristic
219C22×C3C4×SD1612yes—
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—82no—
223C3×C4C2×#10112yes—
224C3×C4#46512yes—
225C3×C4#47112yes—
226C3×C4#49211yescharacteristic
227C3×C4#49511yescharacteristic
228Dic3—82no—
229C23#67⋊C411yescharacteristic
230Q8—81no—
231Q8—81no—
232Q8—82no—
233Q8—242no—
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#91212yes—
245C2×C4D3×SD1612yes—
246C2×C4C4×#6711yescharacteristic
247C2×C4#93911yescharacteristic
248C2×C4#94011yescharacteristic
249C8—34no—
250C8—42no—
251C8—62no—
252C8—64no—
253C8—81no—
254C8—241no—
255C2×C3—22no—
256C2×C3#196611yescharacteristic
257C2×C3#196811yescharacteristic
258C2×C3#197711yescharacteristic
259C22—21no—
260C22—21no—
261C22#475812yes—
262C22#495711yescharacteristic
263C4—21no—
264C4—21no—
265C4—82no—
266C4—242no—
267C4D3×#10112yes—
268C4#477812yes—
269C4#515511yescharacteristic
270C4#517511yescharacteristic
271C4#520312yes—
272C3#979411yescharacteristic, complemented, Sylow
273C2—22no—
274C2#7009811yescharacteristic
275C2#7018911yescharacteristic
276C2#7033511yescharacteristic
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#9794
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
180. C3⋊C8#101semidirect
272. C3#9794semidirect

Maximal subgroups

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

subgroupindex
C2×#50802
C3×#15462
#717892
#707802
#707392
#705242
#704902
#97943