Finite Group Databasebeta

← #10694667

Subgroups — #10694667

Order 768.

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

normal subgroups only

This group has 406 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×#1549C211yesmaximal, characteristic
3#71774C211yesmaximal, characteristic
4#70778C211yesmaximal, characteristic
5#70742C211yesmaximal, characteristic
6#70518C211yesmaximal, characteristic
7#70506C211yesmaximal, characteristic
8#9775—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#431C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×#822—22no—
14#5093—21no—
15#5093—21no—
16#5093C414yes—
17C2×#810—22no—
18#4925C2211yescharacteristic
19#4924C2211yescharacteristic
20C2×C4×C3⋊C8C2211yescharacteristic
21C3×#332—22no—
22#4546—22no—
23#4486—22no—
24#4482—22no—
25#4455—21no—
26#4454—21no—
27C2×#487—31no—
28#1848—31no—
29#1549D311yescharacteristic
30#1548—31no—
31#1546—31no—
32#1546—31no—
33#1452—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×C4⋊Dic3—21no—
45C2×C4⋊Dic3—21no—
46C2×#208—21no—
47C22×C3⋊C8D412yes—
48#869—21no—
49#869—21no—
50#869—21no—
51#869C2×C412yes—
52C3×C4⋊C8—21no—
53C3×C4⋊C8—21no—
54C3×C4⋊C8D412yes—
55C3×C22⋊C8—21no—
56C3×C4×C8—21no—
57#830—21no—
58#822—44no—
59#810—24no—
60#810—24no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)D412yes—
64C4×C3⋊C8—21no—
65C4×C3⋊C8C2×C412yes—
66C4×C3⋊C8D411yescharacteristic
67C4×C3⋊C8D411yescharacteristic
68C2×C4⋊Q8—31no—
69C2×C4⋊Q8C2×D311yescharacteristic
70#487—34no—
71#487—61no—
72#487—61no—
73#431—31no—
74#431C2×D311yescharacteristic
75C2×#108—62no—
76#423—31no—
77#423C2×D311yescharacteristic
78C2×#103—62no—
79C2×C4×C8—31no—
80#333—62no—
81#332—22no—
82#332—62no—
83#332—62no—
84#322—61no—
85#322—61no—
86C2×C3×Q8—22no—
87C2×C3×Q8—41no—
88C2×C3×Q8—82no—
89C22×C3×C4—41no—
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4#4111yescharacteristic
92C22×C3×C4#4111yescharacteristic
93C22×Dic3—41no—
94C2×Dic6—22no—
95C2×Dic6—42no—
96C2×Dic6—42no—
97C2×Dic6—41no—
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×C42C22×C411yescharacteristic
106C3×C42C2×D411yescharacteristic
107C3×C42C2×D411yescharacteristic
108C3×C42C2×D411yescharacteristic
109C4⋊Dic3—22no—
110C4⋊Dic3—22no—
111C4⋊Dic3—41no—
112C4⋊Dic3—41no—
113#208—42no—
114C2×C3⋊C8—22no—
115C2×C3⋊C8—22no—
116C2×C3⋊C8—22no—
117C2×C3⋊C8—41no—
118C2×C3⋊C8C22⋊C412yes—
119C2×C3⋊C8C22⋊C412yes—
120C22×Q8—21no—
121C22×Q8—61no—
122C2×M16—21no—
123C2×M16—61no—
124C22×C8—32no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—21no—
128C4⋊Q8—32no—
129C4⋊Q8—61no—
130C4⋊Q8—61no—
131C4⋊Q8—61no—
132C4⋊Q8C4×D312yes—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—61no—
136C2×C4⋊C4—61no—
137C2×C42C22×D311yescharacteristic
138#108—124no—
139C4⋊C8—21no—
140C4⋊C8—21no—
141C4⋊C8—32no—
142C4⋊C8—61no—
143C4⋊C8—61no—
144C4⋊C8#6712yes—
145#103—64no—
146#103—64no—
147C22⋊C8—21no—
148C22⋊C8—61no—
149C4×C8—21no—
150C4×C8—32no—
151C4×C8—31no—
152C4×C8—31no—
153C4×C8—61no—
154C23×C3#12411yescharacteristic
155C3×Q8—82no—
156C3×Q8—82no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—42no—
160C2×C3×C4—82no—
161C2×C3×C4C2×C22⋊C411yescharacteristic
162C2×C3×C4C4×D411yescharacteristic
163C2×C3×C4C4×D411yescharacteristic
164C2×C3×C4#12111yescharacteristic
165C2×C3×C4#12111yescharacteristic
166C2×C3×C4#12711yescharacteristic
167C2×C3×C4AGL(1,ℤ/8)12yes—
168C2×C3×C4#13712yes—
169C2×Dic3—42no—
170C2×Dic3—82no—
171Dic6—44no—
172Dic6—81no—
173Dic6—81no—
174C3×C8—42no—
175C3×C8—81no—
176C3⋊C8—22no—
177C3⋊C8—22no—
178C3⋊C8—42no—
179C3⋊C8—81no—
180C2×Q8—22no—
181C2×Q8—41no—
182C2×Q8—62no—
183C2×Q8—82no—
184C2×Q8—121no—
185C2×Q8—122no—
186C2×Q8—122no—
187C22×C4—41no—
188C22×C4—121no—
189C22×C4#23511yescharacteristic
190C22×C4D3×D411yescharacteristic
191C22×C4#23711yescharacteristic
192M16—42no—
193M16—122no—
194C2×C8—22no—
195C2×C8—32no—
196C2×C8—32no—
197C2×C8—41no—
198C2×C8—62no—
199C2×C8—62no—
200C2×C8—62no—
201C2×C8—121no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—41no—
205C4⋊C4—41no—
206C4⋊C4—62no—
207C4⋊C4—62no—
208C4⋊C4—121no—
209C4⋊C4—121no—
210C42C2×C4×D311yescharacteristic
211C42D3×D411yescharacteristic
212C42D3×D411yescharacteristic
213C42C2×#6711yescharacteristic
214C22×C3—21no—
215C22×C3—21no—
216C22×C3#39011yescharacteristic
217C22×C3#48912yes—
218C3×C4—21no—
219C3×C4—21no—
220C3×C4—82no—
221C3×C4#41712yes—
222C3×C4#47112yes—
223C3×C4#47312yes—
224C3×C4#49611yescharacteristic
225C3×C4#49711yescharacteristic
226Dic3—82no—
227C23#88511yescharacteristic
228Q8—82no—
229Q8—82no—
230Q8—124no—
231Q8—241no—
232Q8—241no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—82no—
237C2×C4—122no—
238C2×C4—242no—
239C2×C4D3×C22⋊C411yescharacteristic
240C2×C4#67⋊C411yescharacteristic
241C2×C4#88411yescharacteristic
242C2×C4#91412yes—
243C2×C4#91812yes—
244C2×C4C4×#6711yescharacteristic
245C2×C4#93911yescharacteristic
246C2×C4#94011yescharacteristic
247C8—42no—
248C8—62no—
249C8—62no—
250C8—81no—
251C8—122no—
252C8—241no—
253C2×C3—22no—
254C2×C3#195911yescharacteristic
255C2×C3#197711yescharacteristic
256C2×C3#197911yescharacteristic
257C22—21no—
258C22—21no—
259C22#479412yes—
260C22#495711yescharacteristic
261C4—21no—
262C4—21no—
263C4—82no—
264C4—242no—
265C4#475512yes—
266C4#480312yes—
267C4#517611yescharacteristic
268C4#519011yescharacteristic
269C4#520012yes—
270C3#977511yescharacteristic, complemented, Sylow
271C2—22no—
272C2#7010811yescharacteristic
273C2#7022711yescharacteristic
274C2#7031511yescharacteristic
subgroupG/H
centerC22#4957
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1549C2
socleC22×C3#390
prime radicalC23×C3#124
cosocleC2×C3×C42C23
hypercenter#1549D3
hypocenterC3#9775
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1549

Complements

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

NH ≅ G/Ntype
270. C3#9775semidirect

Maximal subgroups

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

subgroupindex
C2×#50932
C3×#15492
#717742
#707782
#707422
#705182
#705062
#97753