Finite Group Databasebeta

← #10694648

Subgroups — #10694648

Order 768.

478 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 129 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×#5092C211yesmaximal, characteristic
2C3×#1546C211yesmaximal, characteristic
3#71773C211yesmaximal, characteristic
4#70778C211yesmaximal, characteristic
5#70740C211yesmaximal, characteristic
6#70518C211yesmaximal, characteristic
7#70506C211yesmaximal, characteristic
8#9772—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C22×#216—22no—
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×#822—22no—
15#5092—21no—
16#5092—21no—
17#5092C414yes—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20C2×C4×C3⋊C8C2211yescharacteristic
21C3×#332—22no—
22#4547—22no—
23#4486—22no—
24#4482—22no—
25#4454—21no—
26C8×C3⋊C8—21no—
27C2×#494—31no—
28#1848—31no—
29#1546—31no—
30#1546—31no—
31#1546—31no—
32#1546D311yescharacteristic
33#1450—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⋊Q8C2×C412yes—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C42C2311yescharacteristic
44C2×#216—24no—
45C2×#216—24no—
46C2×#216—44no—
47C2×#216—44no—
48C2×C4⋊Dic3—21no—
49C2×C4⋊Dic3—21no—
50C2×#208—21no—
51C22×C3⋊C8D412yes—
52#869—21no—
53#869—21no—
54#869—21no—
55#869C2×C412yes—
56C3×C4⋊C8—21no—
57C3×C4⋊C8—21no—
58C3×C4⋊C8D412yes—
59C3×C22⋊C8—21no—
60C3×C4×C8—21no—
61#830—21no—
62#822—44no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)—21no—
65C4⋊(C3⋊C8)D412yes—
66C4×C3⋊C8—21no—
67C4×C3⋊C8C2×C412yes—
68C4×C3⋊C8D411yescharacteristic
69C4×C3⋊C8D411yescharacteristic
70C22×Q16—62no—
71C2×C4⋊Q8—31no—
72C2×C4⋊Q8C2×D311yescharacteristic
73#494—34no—
74#494—61no—
75#494—61no—
76#431—31no—
77#431C2×D311yescharacteristic
78C2×#108—62no—
79#423—31no—
80#423C2×D311yescharacteristic
81C2×C4×C8—31no—
82#332—22no—
83#332—62no—
84#332—62no—
85#332—62no—
86#322—61no—
87C82—61no—
88C2×C3×Q8—22no—
89C2×C3×Q8—41no—
90C2×C3×Q8—42no—
91C2×C3×Q8—42no—
92C22×C3×C4—41no—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4111yescharacteristic
95C22×C3×C4#4111yescharacteristic
96C22×Dic3—41no—
97C2×Dic6—22no—
98C2×Dic6—42no—
99C2×Dic6—42no—
100C2×Dic6—41no—
101C3×M16—42no—
102C2×C3×C8—22no—
103C2×C3×C8—41no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C42C22×C411yescharacteristic
109C3×C42C2×D411yescharacteristic
110C3×C42C2×D411yescharacteristic
111C3×C42C2×D411yescharacteristic
112#216—48no—
113#216—48no—
114C4⋊Dic3—22no—
115C4⋊Dic3—22no—
116C4⋊Dic3—41no—
117C4⋊Dic3—41no—
118#208—42no—
119C2×C3⋊C8—22no—
120C2×C3⋊C8—41no—
121C2×C3⋊C8C22⋊C412yes—
122C2×C3⋊C8C22⋊C412yes—
123C2×C3⋊C8D814yes—
124C2×C3⋊C8SD1614yes—
125C22×Q8—21no—
126C22×Q8—61no—
127C2×Q16—64no—
128C2×Q16—64no—
129C2×Q16—124no—
130C2×Q16—124no—
131C2×M16—21no—
132C2×M16—61no—
133C22×C8—32no—
134C4⋊Q8—21no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—32no—
138C4⋊Q8—61no—
139C4⋊Q8—61no—
140C4⋊Q8—61no—
141C4⋊Q8C4×D312yes—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—61no—
145C2×C4⋊C4—61no—
146C2×C42C22×D311yescharacteristic
147#108—124no—
148C4⋊C8—21no—
149C4⋊C8—21no—
150C4⋊C8—32no—
151C4⋊C8—61no—
152C4⋊C8—61no—
153C4⋊C8#6712yes—
154C22⋊C8—21no—
155C22⋊C8—61no—
156C4×C8—21no—
157C4×C8—32no—
158C4×C8—31no—
159C4×C8—31no—
160C4×C8—61no—
161C23×C3#12411yescharacteristic
162C3×Q8—44no—
163C3×Q8—81no—
164C3×Q8—81no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—42no—
168C2×C3×C4—82no—
169C2×C3×C4C2×C22⋊C411yescharacteristic
170C2×C3×C4C4×D411yescharacteristic
171C2×C3×C4C4×D411yescharacteristic
172C2×C3×C4#12111yescharacteristic
173C2×C3×C4#12111yescharacteristic
174C2×C3×C4#12711yescharacteristic
175C2×C3×C4C2×D812yes—
176C2×C3×C4C2×SD1612yes—
177C2×Dic3—42no—
178C2×Dic3—82no—
179Dic6—44no—
180Dic6—81no—
181Dic6—81no—
182C3×C8—42no—
183C3×C8—81no—
184C3⋊C8—42no—
185C3⋊C8—81no—
186C3⋊C8#10214yes—
187C3⋊C8#10214yes—
188C2×Q8—22no—
189C2×Q8—41no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—62no—
193C2×Q8—121no—
194C2×Q8—122no—
195C2×Q8—122no—
196C22×C4—41no—
197C22×C4—121no—
198C22×C4#23511yescharacteristic
199C22×C4D3×D411yescharacteristic
200C22×C4#23711yescharacteristic
201Q16—128no—
202Q16—128no—
203M16—42no—
204M16—122no—
205C2×C8—22no—
206C2×C8—32no—
207C2×C8—32no—
208C2×C8—34no—
209C2×C8—34no—
210C2×C8—41no—
211C2×C8—62no—
212C2×C8—121no—
213C4⋊C4—22no—
214C4⋊C4—22no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4—62no—
218C4⋊C4—62no—
219C4⋊C4—121no—
220C4⋊C4—121no—
221C42C2×C4×D311yescharacteristic
222C42D3×D411yescharacteristic
223C42D3×D411yescharacteristic
224C42C2×#6711yescharacteristic
225C22×C3—21no—
226C22×C3—21no—
227C22×C3#39011yescharacteristic
228C22×C3#48612yes—
229C3×C4—21no—
230C3×C4—21no—
231C3×C4—82no—
232C3×C4C2×#10212yes—
233C3×C4#47312yes—
234C3×C4#47312yes—
235C3×C4#49211yescharacteristic
236C3×C4#49311yescharacteristic
237Dic3—82no—
238C23#88511yescharacteristic
239Q8—44no—
240Q8—81no—
241Q8—81no—
242Q8—124no—
243Q8—241no—
244Q8—241no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—82no—
249C2×C4—122no—
250C2×C4—242no—
251C2×C4D3×C22⋊C411yescharacteristic
252C2×C4#67⋊C411yescharacteristic
253C2×C4#88411yescharacteristic
254C2×C4D3×D812yes—
255C2×C4D3×SD1612yes—
256C2×C4C4×#6711yescharacteristic
257C2×C4#93911yescharacteristic
258C2×C4#94011yescharacteristic
259C8—34no—
260C8—34no—
261C8—42no—
262C8—81no—
263C8—122no—
264C8—241no—
265C2×C3—22no—
266C2×C3#195711yescharacteristic
267C2×C3#197911yescharacteristic
268C2×C3#197911yescharacteristic
269C22—21no—
270C22—21no—
271C22#476012yes—
272C22#495711yescharacteristic
273C4—21no—
274C4—21no—
275C4—82no—
276C4—242no—
277C4#475512yes—
278C4D3×#10212yes—
279C4#515111yescharacteristic
280C4#517511yescharacteristic
281C4#520412yes—
282C3#977211yescharacteristic, complemented, Sylow
283C2—22no—
284C2#7010811yescharacteristic
285C2#7015711yescharacteristic
286C2#7033611yescharacteristic
subgroupG/H
centerC22#4957
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1546C2
socleC22×C3#390
prime radicalC23×C3#124
cosocleC2×C3×C42C23
hypercenter#1546D3
hypocenterC3#9772
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1546

Complements

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

NH ≅ G/Ntype
282. C3#9772semidirect

Maximal subgroups

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

subgroupindex
C2×#50922
C3×#15462
#717732
#707782
#707402
#705182
#705062
#97723