Finite Group Databasebeta

← #10694641

Subgroups — #10694641

Order 768.

398 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 127 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×C3×#487C211yesmaximal, characteristic
2#71775C211yesmaximal, characteristic
3#71774C211yesmaximal, characteristic
4#71773C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic
6#69791C211yesmaximal, characteristic
7#69724C211yesmaximal, characteristic
8#9770—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C3×#487—21no—
12C3×#487—21no—
13C3×#487C414yes—
14C2×C3×#103—22no—
15C2×C3×C4×C8C2211yescharacteristic
16#4925C2211yescharacteristic
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19#4924C2211yescharacteristic
20#4552—22no—
21#4547—22no—
22#4547—22no—
23#4547—22no—
24#4546—22no—
25#4455—21no—
26C8×C3⋊C8—21no—
27C2×#487D311yescharacteristic
28#1837—31no—
29#1549—31no—
30#1548—31no—
31#1546—31no—
32#1546—31no—
33#1450—31no—
34C22×C3×Q8—21no—
35C22×C3×Q8—21no—
36C22×C3×C8D412yes—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8—21no—
43C3×C4⋊Q8C2×C412yes—
44C3×C4⋊Q8C2×C412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#208—21no—
51C2×#208—21no—
52C3×#103—24no—
53C3×#103—24no—
54C3×C4×C8C2×C412yes—
55C3×C4×C8D411yescharacteristic
56C3×C4×C8D411yescharacteristic
57#830—21no—
58#830—21no—
59C4⋊(C3⋊C8)—21no—
60C4⋊(C3⋊C8)—21no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)D412yes—
64C4⋊(C3⋊C8)D412yes—
65C4×C3⋊C8—21no—
66C4×C3⋊C8—21no—
67C2×C4⋊Q8C2×D311yescharacteristic
68C2×C4⋊Q8C2×D311yescharacteristic
69#487—21no—
70#487—21no—
71#487Dic314yes—
72#431—31no—
73#431—31no—
74#423—31no—
75#423—31no—
76C2×#103—22no—
77C2×C4×C8C2×D311yescharacteristic
78#341—62no—
79#333—62no—
80#332—62no—
81#332—62no—
82#332—62no—
83#322—61no—
84C82—61no—
85C2×C3×Q8—22no—
86C2×C3×Q8—22no—
87C2×C3×Q8—41no—
88C2×C3×Q8—41no—
89C2×C3×Q8—42no—
90C2×C3×Q8—42no—
91C2×C3×Q8—82no—
92C22×C3×C4—41no—
93C22×C3×C4—41no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4#4111yescharacteristic
96C22×C3×C4#4111yescharacteristic
97C2×C3×C8—22no—
98C2×C3×C8—22no—
99C2×C3×C8C22⋊C412yes—
100C2×C3×C8C22⋊C412yes—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—22no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C42C22×C411yescharacteristic
110C3×C42C2×D411yescharacteristic
111C3×C42C2×D411yescharacteristic
112C3×C42C2×D411yescharacteristic
113#208—42no—
114#208—42no—
115C2×C3⋊C8—22no—
116C2×C3⋊C8—22no—
117C2×C3⋊C8—41no—
118C2×C3⋊C8—41no—
119C22×Q8—21no—
120C22×Q8—21no—
121C2×M16—61no—
122C2×M16—61no—
123C22×C8#6712yes—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8C2×Dic312yes—
131C4⋊Q8C2×Dic312yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C42C22×D311yescharacteristic
137C4⋊C8—32no—
138C4⋊C8—32no—
139C4⋊C8—61no—
140C4⋊C8—61no—
141C4⋊C8—61no—
142C4⋊C8—61no—
143#103—24no—
144#103—24no—
145C22⋊C8—61no—
146C22⋊C8—61no—
147C4×C8—61no—
148C4×C8—61no—
149C4×C8C2×Dic312yes—
150C4×C8#6711yescharacteristic
151C4×C8#6711yescharacteristic
152C23×C3#12411yescharacteristic
153C3×Q8—44no—
154C3×Q8—82no—
155C3×Q8—82no—
156C3×Q8—81no—
157C3×Q8—81no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—82no—
163C2×C3×C4—82no—
164C2×C3×C4C2×C22⋊C411yescharacteristic
165C2×C3×C4C4×D411yescharacteristic
166C2×C3×C4C4×D411yescharacteristic
167C2×C3×C4#12111yescharacteristic
168C2×C3×C4#12111yescharacteristic
169C2×C3×C4#12711yescharacteristic
170C2×C3×C4#13512yes—
171C2×C3×C4#13512yes—
172C3×C8—22no—
173C3×C8—22no—
174C3⋊C8—42no—
175C3⋊C8—42no—
176C3⋊C8—81no—
177C3⋊C8—81no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—82no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4#23711yescharacteristic
188C22×C4#23711yescharacteristic
189C22×C4C2×#6711yescharacteristic
190M16—122no—
191M16—122no—
192C2×C8—22no—
193C2×C8—22no—
194C2×C8—62no—
195C2×C8—62no—
196C2×C8—121no—
197C2×C8—121no—
198C2×C8#21712yes—
199C2×C8#21712yes—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C42D3×D411yescharacteristic
209C42D3×D411yescharacteristic
210C42C22×Dic311yescharacteristic
211C42C2×#6711yescharacteristic
212C22×C3—21no—
213C22×C3—21no—
214C22×C3#39011yescharacteristic
215C22×C3#48812yes—
216C3×C4—21no—
217C3×C4—21no—
218C3×C4—82no—
219C3×C4—82no—
220C3×C4#41612yes—
221C3×C4#47112yes—
222C3×C4#47312yes—
223C3×C4#49511yescharacteristic
224C3×C4#49511yescharacteristic
225C23#93611yescharacteristic
226Q8—44no—
227Q8—82no—
228Q8—82no—
229Q8—81no—
230Q8—81no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—82no—
236C2×C4—82no—
237C2×C4D4×Dic311yescharacteristic
238C2×C4D4×Dic311yescharacteristic
239C2×C4#93811yescharacteristic
240C2×C4#93811yescharacteristic
241C2×C4#94011yescharacteristic
242C2×C4#95012yes—
243C2×C4#95012yes—
244C2×C4C2×#21711yescharacteristic
245C8—22no—
246C8—22no—
247C8—122no—
248C8—122no—
249C8—241no—
250C8—241no—
251C2×C3—22no—
252C2×C3#195811yescharacteristic
253C2×C3#197711yescharacteristic
254C2×C3#197911yescharacteristic
255C22—21no—
256C22—21no—
257C22#505612yes—
258C22#522111yescharacteristic
259C4—21no—
260C4—21no—
261C4—82no—
262C4—82no—
263C4#505911yescharacteristic
264C4#505911yescharacteristic
265C4#517712yes—
266C4#519212yes—
267C4#523312yes—
268C3#977011yescharacteristic, complemented, Sylow
269C2—22no—
270C2#7172511yescharacteristic
271C2#7205211yescharacteristic
272C2#7206911yescharacteristic
subgroupG/H
centerC22#5221
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#487C2
socleC22×C3#390
prime radicalC23×C3#124
cosocleC2×C3×C42C23
hypercenterC2×#487D3
hypocenterC3#9770
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
268. C3#9770semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4872
#717752
#717742
#717732
#697912
#697242
#97703