Finite Group Databasebeta

← #10694866

Subgroups — #10694866

Order 768.

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

normal subgroups only

This group has 418 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#71790C211yesmaximal, characteristic
3#71789C211yesmaximal, characteristic
4#71774C211yesmaximal, characteristic
5#71773C211yesmaximal, characteristic
6#69731C211yesmaximal, characteristic
7#69725C211yesmaximal, characteristic
8#9809—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C3×#487—21no—
12C3×#487—21no—
13C3×#487—22no—
14C2×C3×#103C414yes—
15C2×C3×C4×C8C2211yescharacteristic
16C2×#836—22no—
17#4925C2211yescharacteristic
18#4924C2211yescharacteristic
19#4924C2211yescharacteristic
20C4×#208C2211yescharacteristic
21#4547—22no—
22#4546—22no—
23#4530—22no—
24#4458—22no—
25#4455—21no—
26#4454—21no—
27C2×#487D311yescharacteristic
28#1997—31no—
29#1966—31no—
30#1567—31no—
31#1548—31no—
32#1546—31no—
33#1451—31no—
34C22×C3×Q8—21no—
35C22×C3×Q8C2×C412yes—
36C22×C3×C8C2×C412yes—
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⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4C2×C412yes—
49C2×C3×C42C2311yescharacteristic
50C2×#208—21no—
51C2×#208—21no—
52C2×#208D412yes—
53C3×#103—24no—
54C3×#103—24no—
55C3×C4×C8D411yescharacteristic
56C3×C4×C8D411yescharacteristic
57C3×C4×C8D412yes—
58#836—44no—
59#830—21no—
60C4⋊(C3⋊C8)—21no—
61C4⋊(C3⋊C8)—21no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)D412yes—
65#803—21no—
66C4×C3⋊C8—21no—
67C4×C3⋊C8—21no—
68C2×C4⋊Q8C2×D311yescharacteristic
69C2×C4⋊Q8C2×D311yescharacteristic
70#487—21no—
71#487—21no—
72#487—22no—
73#431—31no—
74#423—31no—
75#423—31no—
76C2×#103Dic314yes—
77C2×#101—62no—
78C4×M16—31no—
79C2×C4×C8C2×D311yescharacteristic
80#343—62no—
81#335—62no—
82#333—62no—
83#332—62no—
84#322—61no—
85#322—61no—
86C2×C3×Q8—22no—
87C2×C3×Q8—22no—
88C2×C3×Q8—44no—
89C2×C3×Q8—41no—
90C2×C3×Q8—41no—
91C2×C3×Q8—42no—
92C2×C3×Q8—42no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4111yescharacteristic
98C2×C3×C8—42no—
99C2×C3×C8C22⋊C412yes—
100C2×C3×C8C22⋊C412yes—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—41no—
106C3×C4⋊C4—41no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C42C2×D411yescharacteristic
110C3×C42C2×D411yescharacteristic
111C3×C42C2×D411yescharacteristic
112C3×C42#4111yescharacteristic
113#208—22no—
114#208—22no—
115#208—42no—
116#208—42no—
117C2×C3⋊C8—22no—
118C2×C3⋊C8—22no—
119C2×C3⋊C8—41no—
120C2×C3⋊C8—41no—
121C22×Q8—21no—
122C22×Q8C2×Dic312yes—
123C2×M16—32no—
124C2×M16—61no—
125C2×M16—61no—
126C22×C8C2×Dic312yes—
127C4⋊Q8—21no—
128C4⋊Q8—21no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4C2×Dic312yes—
139C2×C42C22×D311yescharacteristic
140C4⋊C8—32no—
141C4⋊C8—61no—
142C4⋊C8—61no—
143C4⋊C8—61no—
144C4⋊C8—61no—
145#103—24no—
146#103—24no—
147#101—124no—
148C22⋊C8—61no—
149#97—61no—
150C4×C8—61no—
151C4×C8—61no—
152C4×C8#6711yescharacteristic
153C4×C8#6711yescharacteristic
154C4×C8#6712yes—
155C23×C3C4×D411yescharacteristic
156C3×Q8—44no—
157C3×Q8—82no—
158C3×Q8—82no—
159C3×Q8—81no—
160C3×Q8—81no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—44no—
166C2×C3×C4—82no—
167C2×C3×C4C2×C22⋊C411yescharacteristic
168C2×C3×C4C4×D411yescharacteristic
169C2×C3×C4#12111yescharacteristic
170C2×C3×C4#12111yescharacteristic
171C2×C3×C4#12411yescharacteristic
172C2×C3×C4#12711yescharacteristic
173C2×C3×C4AGL(1,ℤ/8)12yes—
174C2×C3×C4#13712yes—
175C3×C8—22no—
176C3×C8—22no—
177C3⋊C8—42no—
178C3⋊C8—42no—
179C3⋊C8—81no—
180C3⋊C8—81no—
181C2×Q8—22no—
182C2×Q8—22no—
183C2×Q8—44no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—41no—
187C2×Q8—41no—
188C22×C4—22no—
189C22×C4—22no—
190C22×C4D3×D411yescharacteristic
191C22×C4#23711yescharacteristic
192C22×C4C22×Dic311yescharacteristic
193M16—62no—
194M16—62no—
195M16—122no—
196M16—122no—
197C2×C8—42no—
198C2×C8—62no—
199C2×C8—62no—
200C2×C8—121no—
201C2×C8—121no—
202C2×C8#21712yes—
203C2×C8#21712yes—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C42D3×D411yescharacteristic
213C42#23711yescharacteristic
214C42C2×#6711yescharacteristic
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—82no—
224C3×C4#41312yes—
225C3×C4#46812yes—
226C3×C4#47312yes—
227C3×C4#49611yescharacteristic
228C3×C4#49711yescharacteristic
229C23D4×Dic311yescharacteristic
230Q8—44no—
231Q8—82no—
232Q8—81no—
233Q8—82no—
234Q8—81no—
235C2×C4—22no—
236C2×C4—22no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—82no—
241C2×C4#91412yes—
242C2×C4#91512yes—
243C2×C4D4×Dic311yescharacteristic
244C2×C4#93611yescharacteristic
245C2×C4#93811yescharacteristic
246C2×C4#93811yescharacteristic
247C2×C4#94011yescharacteristic
248C2×C4C2×#21711yescharacteristic
249C8—22no—
250C8—22no—
251C8—122no—
252C8—122no—
253C8—241no—
254C8—241no—
255C2×C3—22no—
256C2×C3#196711yescharacteristic
257C2×C3#197311yescharacteristic
258C2×C3#197911yescharacteristic
259C22—21no—
260C22—21no—
261C22#516412yes—
262C22#522111yescharacteristic
263C4—21no—
264C4—21no—
265C4—82no—
266C4—82no—
267C4#516711yescharacteristic
268C4#517412yes—
269C4#517611yescharacteristic
270C4#519212yes—
271C4#523712yes—
272C3#980911yescharacteristic, complemented, Sylow
273C2—22no—
274C2#7172711yescharacteristic
275C2#7205011yescharacteristic
276C2#7206911yescharacteristic
subgroupG/H
centerC22#5221
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#487C2
socleC22×C3#390
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenterC2×#487D3
hypocenterC3#9809
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

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×C3×#4872
#717902
#717892
#717742
#717732
#697312
#697252
#98093