Finite Group Databasebeta

← #10694858

Subgroups — #10694858

Order 768.

434 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 117 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×#5078C211yesmaximal, characteristic
2C3×#1966C211yesmaximal, characteristic
3#71773C211yesmaximal, characteristic
4#70506C211yesmaximal, characteristic
5#70490C211yesmaximal, characteristic
6#69877C211yesmaximal, characteristic
7#69847C211yesmaximal, characteristic
8#9808—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×#101—22no—
13C3×C4×M16C2211yescharacteristic
14C2×#824—22no—
15#5078—21no—
16#5078—21no—
17#5078—22no—
18C2×#807C414yes—
19#4925C2211yescharacteristic
20#4924C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22#4547—22no—
23#4482—22no—
24#4480—22no—
25#4454—21no—
26#4454—21no—
27C2×#500—31no—
28#1966—31no—
29#1966D311yescharacteristic
30#1566—31no—
31#1546—31no—
32#1546—31no—
33#1451—31no—
34C22×C3×Q8—21no—
35C22×Dic6—21no—
36C2×C3×M16—21no—
37C2×C3×M16D412yes—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C412yes—
44C2×C3×C42C2311yescharacteristic
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3C2×C412yes—
47C2×#208—21no—
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×#101—44no—
56C3×#97—21no—
57C3×C4×C8—21no—
58#830—21no—
59#824—44no—
60#807—24no—
61#807—24no—
62C4⋊(C3⋊C8)—21no—
63C4⋊(C3⋊C8)—21no—
64C4⋊(C3⋊C8)D412yes—
65C4×C3⋊C8—21no—
66C4×C3⋊C8D411yescharacteristic
67C4×C3⋊C8D411yescharacteristic
68C4×C3⋊C8D412yes—
69C2×C4⋊Q8—31no—
70C2×C4⋊Q8C2×D311yescharacteristic
71#500—61no—
72#500—61no—
73#500—62no—
74#431—31no—
75C2×#107—34no—
76#423—31no—
77#423C2×D311yescharacteristic
78C2×#101—22no—
79C2×#101—62no—
80C4×M16C2×D311yescharacteristic
81C2×C4×C8—31no—
82#334—62no—
83#332—62no—
84#332—62no—
85#322—61no—
86#322—61no—
87C2×C3×Q8—41no—
88C2×C3×Q8—41no—
89C2×C3×Q8—42no—
90C2×C3×Q8—42no—
91C22×C3×C4—22no—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4111yescharacteristic
95C22×Dic3—22no—
96C2×Dic6—41no—
97C2×Dic6—41no—
98C2×Dic6—42no—
99C2×Dic6—42no—
100C3×M16—22no—
101C3×M16—22no—
102C3×M16—42no—
103C2×C3×C8—22no—
104C2×C3×C8—41no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C42C2×D411yescharacteristic
110C3×C42C2×D411yescharacteristic
111C3×C42C2×D411yescharacteristic
112C3×C42#4111yescharacteristic
113C4⋊Dic3—22no—
114C4⋊Dic3—22no—
115C4⋊Dic3—41no—
116C4⋊Dic3—41no—
117#208—42no—
118C2×C3⋊C8—24no—
119C2×C3⋊C8—22no—
120C2×C3⋊C8—41no—
121C2×C3⋊C8C22⋊C412yes—
122C2×C3⋊C8C22⋊C412yes—
123C22×Q8—21no—
124C22×Q8—61no—
125C2×M16—21no—
126C2×M16—61no—
127C2×M16#6712yes—
128C22×C8—32no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—61no—
134C4⋊Q8—61no—
135C4⋊Q8—61no—
136C4⋊Q8—61no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—32no—
139C2×C4⋊C4—61no—
140C2×C4⋊C4C4×D312yes—
141C2×C42C22×D311yescharacteristic
142#107—64no—
143#107—64no—
144C4⋊C8—21no—
145C4⋊C8—21no—
146C4⋊C8—32no—
147C4⋊C8—61no—
148C4⋊C8—61no—
149#101—44no—
150#101—124no—
151C22⋊C8—61no—
152#97—21no—
153C4×C8—21no—
154C4×C8—31no—
155C4×C8—31no—
156C4×C8—32no—
157C4×C8—61no—
158C23×C3C4×D411yescharacteristic
159C3×Q8—82no—
160C3×Q8—81no—
161C3×Q8—81no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—42no—
165C2×C3×C4—44no—
166C2×C3×C4C2×C22⋊C411yescharacteristic
167C2×C3×C4C4×D411yescharacteristic
168C2×C3×C4#12111yescharacteristic
169C2×C3×C4#12111yescharacteristic
170C2×C3×C4#12411yescharacteristic
171C2×C3×C4#12711yescharacteristic
172C2×C3×C4AGL(1,ℤ/8)12yes—
173C2×C3×C4AGL(1,ℤ/8)12yes—
174C2×Dic3—42no—
175C2×Dic3—44no—
176Dic6—82no—
177Dic6—81no—
178Dic6—81no—
179C3×C8—42no—
180C3×C8—81no—
181C3⋊C8—22no—
182C3⋊C8—42no—
183C3⋊C8—81no—
184C3⋊C8#10114yescomplemented
185C2×Q8—42no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—42no—
189C2×Q8—122no—
190C2×Q8—121no—
191C2×Q8—121no—
192C2×Q8—122no—
193C22×C4—22no—
194C22×C4—62no—
195C22×C4C2×C4×D311yescharacteristic
196C22×C4#23511yescharacteristic
197C22×C4C2×#6711yescharacteristic
198M16—22no—
199M16—22no—
200M16—42no—
201M16—122no—
202C2×C8—22no—
203C2×C8—32no—
204C2×C8—32no—
205C2×C8—41no—
206C2×C8—64no—
207C2×C8—62no—
208C2×C8—121no—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—41no—
212C4⋊C4—41no—
213C4⋊C4—62no—
214C4⋊C4—62no—
215C4⋊C4—121no—
216C4⋊C4—121no—
217C42D3×D411yescharacteristic
218C42D3×D411yescharacteristic
219C42D3×D411yescharacteristic
220C42#23711yescharacteristic
221C22×C3—21no—
222C22×C3—21no—
223C22×C3#39011yescharacteristic
224C22×C3#44212yes—
225C3×C4—21no—
226C3×C4—21no—
227C3×C4—82no—
228C3×C4C2×#10112yes—
229C3×C4#46912yes—
230C3×C4#47312yes—
231C3×C4#49611yescharacteristic
232C3×C4#49611yescharacteristic
233Dic3—82no—
234C23C4×#6711yescharacteristic
235Q8—82no—
236Q8—81no—
237Q8—81no—
238Q8—242no—
239Q8—241no—
240Q8—241no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—44no—
245C2×C4—122no—
246C2×C4—124no—
247C2×C4D3×C22⋊C411yescharacteristic
248C2×C4#67⋊C411yescharacteristic
249C2×C4#88411yescharacteristic
250C2×C4#88511yescharacteristic
251C2×C4#93212yes—
252C2×C4#93911yescharacteristic
253C2×C4#94011yescharacteristic
254C2×C4#94912yes—
255C8—34no—
256C8—42no—
257C8—62no—
258C8—81no—
259C8—122no—
260C8—241no—
261C2×C3—22no—
262C2×C3#196611yescharacteristic
263C2×C3#197211yescharacteristic
264C2×C3#197911yescharacteristic
265C22—21no—
266C22—21no—
267C22#495711yescharacteristic
268C22#501412yes—
269C4—21no—
270C4—21no—
271C4—82no—
272C4—242no—
273C4D3×#10112yes—
274C4#475512yes—
275C4#503912yes—
276C4#505811yescharacteristic
277C4#507411yescharacteristic
278C3#980811yescharacteristic, complemented, Sylow
279C2—22no—
280C2#6964811yescharacteristic
281C2#7009811yescharacteristic
282C2#7010811yescharacteristic
subgroupG/H
centerC22#4957
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1966C2
socleC22×C3#390
prime radicalC23×C3C4×D4
cosocleC2×C3×C42C23
hypercenter#1966D3
hypocenterC3#9808
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1966

Complements

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

NH ≅ G/Ntype
184. C3⋊C8#101semidirect
278. C3#9808semidirect

Maximal subgroups

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

subgroupindex
C2×#50782
C3×#19662
#717732
#705062
#704902
#698772
#698472
#98083