Finite Group Databasebeta

← #10714645

Subgroups — #10714645

Order 768.

512 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 77 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
1C3×#2733C211yesmaximal, characteristic
2C2×#4542C212yesmaximal, complemented
3#71724C212yesmaximal, complemented
4#69672C211yesmaximal, characteristic, complemented
5#67644C211yesmaximal, characteristic, complemented
6#13426—31nomaximal, Sylow
7C2×C3×#124C2212yes—
8#5812C2211yescharacteristic
9C3×#394—21no—
10C3×#392—22no—
11C3×#391C2211yescharacteristic
12C3×#390—21no—
13C3×#388—22no—
14C2×#836C2212yes—
15C2×#836C2211yescharacteristic
16#5215—22no—
17#4964—21no—
18#4542—24no—
19#4523—21no—
20#4523—22no—
21#2733D311yescharacteristic
22C2×#355—32no—
23#1888—32no—
24#1857—31no—
25#1394—31no—
26C22×C3×Q8C2311yescharacteristic
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4D412yes—
30C3×#124—22no—
31C3×#124—22no—
32C3×#124—22no—
33C3×#124—22no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4D411yescharacteristic
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4—22no—
40C2×C3×C42D412yes—
41C2×C3×C42D411yescharacteristic
42Q8×Dic3—41no—
43Q8×Dic3—41no—
44#926—22no—
45#926—21no—
46C2×#208—21no—
47C2×#208—22no—
48C3×#95—21no—
49C3×#95—22no—
50#836—21no—
51#836—22no—
52#836—22no—
53#836D412yescomplemented
54#832—42no—
55#832—42no—
56#808—41no—
57#807—41no—
58C2×#124C2×D312yes—
59#519—31no—
60#428—61no—
61#418—62no—
62C2×#101—31no—
63C2×#101—32no—
64#394—21no—
65#392—22no—
66#391C2×D311yescharacteristic
67#390—21no—
68#388—22no—
69#355—64no—
70#339—61no—
71#339—62no—
72C24×C3—22no—
73C2×C3×Q8—41no—
74C2×C3×Q8—41no—
75C2×C3×Q8C2×D411yescharacteristic
76C2×C3×Q8#4111yescharacteristic
77C2×C3×Q8#4112yes—
78C2×C3×D4—22no—
79C2×C3×D4—22no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4—42no—
83C2×C3×D4—81no—
84C2×C3×D4—81no—
85C2×C3×D4—82no—
86C2×C3×D4—81no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—22no—
90C22×C3×C4C2×D412yes—
91C22×C3×C4C2×D411yescharacteristic
92C3×C4⋊C4—41no—
93C3×C4⋊C4—81no—
94C3×C22⋊C4—44no—
95C3×C22⋊C4—42no—
96C3×C22⋊C4—42no—
97C3×C22⋊C4—42no—
98C3×C22⋊C4—42no—
99C3×C22⋊C4—82no—
100C3×C42—22no—
101C3×C42—22no—
102C3×C42—41no—
103#217—41no—
104C4⋊Dic3—21no—
105C4⋊Dic3—21no—
106C4⋊Dic3—22no—
107C4⋊Dic3—22no—
108C4×Dic3—41no—
109C4×Dic3—42no—
110#208—22no—
111#208—41no—
112#208—42no—
113#208—42no—
114C2×C3⋊C8—41no—
115C2×C3⋊C8—42no—
116C22×Q8C22×D311yescharacteristic
117C22×D4—21no—
118C22×D4—21no—
119C22×D4#6712yes—
120C2×M16—61no—
121C2×M16—62no—
122#124—22no—
123#124—22no—
124#124—22no—
125#124—22no—
126C4×Q8—121no—
127C4×Q8—121no—
128#117—62no—
129#117—61no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4#6711yescharacteristic
132C2×C22⋊C4—22no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C42#6712yes—
137C2×C42#6711yescharacteristic
138#107—121no—
139#106—121no—
140#102—122no—
141#102—122no—
142#101—32no—
143#101—61no—
144#101—62no—
145#101—62no—
146#95—21no—
147#95—22no—
148C23×C3—42no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—42no—
152C23×C3—42no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3#12011yescharacteristic
156C3×Q8—41no—
157C3×Q8—41no—
158C3×Q8—42no—
159C3×D4—44no—
160C3×D4—82no—
161C3×D4—84no—
162C3×D4—84no—
163C2×C3×C4—21no—
164C2×C3×C4—22no—
165C2×C3×C4—21no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—41no—
175C2×C3×C4#12111yescharacteristic
176C2×C3×C4#12112yes—
177C2×C3×C4#12312yes—
178C2×C3×C4#12411yescharacteristic
179C2×Dic3—41no—
180C2×Dic3—41no—
181C3⋊C8—42no—
182C3⋊C8—82no—
183C24—22no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8D3×D411yescharacteristic
187C2×Q8#23711yescharacteristic
188C2×Q8#23712yes—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4—82no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4C2×#6712yes—
202C22×C4C2×#6711yescharacteristic
203M16—62no—
204M16—121no—
205M16—122no—
206M16—122no—
207C2×C8—121no—
208C2×C8—122no—
209C4⋊C4—41no—
210C4⋊C4—62no—
211C4⋊C4—62no—
212C4⋊C4—61no—
213C4⋊C4—61no—
214C4⋊C4—81no—
215C22⋊C4—44no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—82no—
221C22⋊C4—121no—
222C42—22no—
223C42—22no—
224C42—41no—
225C42—121no—
226C42—122no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—82no—
230C22×C3—82no—
231C22×C3—82no—
232C22×C3—82no—
233C22×C3—82no—
234C22×C3—82no—
235C22×C3—82no—
236C22×C3#39411yescharacteristic
237C22×C3B(4,𝔽2)11yescharacteristic
238C22×C3#45811yescharacteristic
239C3×C4—21no—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—21no—
243C3×C4—42no—
244C3×C4—82no—
245C3×C4—81no—
246Dic3—81no—
247Dic3—81no—
248C23—42no—
249C23—42no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23#95311yescharacteristic
256Q8—41no—
257Q8—41no—
258Q8—42no—
259D4—44no—
260D4—82no—
261D4—84no—
262D4—84no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—41no—
275C2×C4—121no—
276C2×C4—121no—
277C2×C4#93512yes—
278C2×C4#93611yescharacteristic
279C2×C4#93811yescharacteristic
280C2×C4#93912yes—
281C8—122no—
282C8—242no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—162no—
286C2×C3#204411yescharacteristic
287C2×C3#220211yescharacteristic
288C2×C3#220411yescharacteristic
289C22—21no—
290C22—21no—
291C22—82no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—82no—
298C22#522211yescharacteristic
299C22#524311yescharacteristic
300C22#524411yescharacteristic
301C4—21no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—42no—
306C4—82no—
307C4—81no—
308C4—241no—
309C4—241no—
310C3#1342611yescharacteristic, complemented, Sylow
311C2—21no—
312C2—21no—
313C2—162no—
314C2#7185011yescharacteristic
315C2#7193011yescharacteristic
316C2#7195111yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC22×C3×Q8C23
FrattiniC22×Q8C22×D3
Fitting, prime radicalC3×#2733C2
socleC22×C3B(4,𝔽2)
hypercenter#2733D3
hypocenterC3#13426
LCSG ▷ C22×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2733

Complements

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

NH ≅ G/Ntype
2. C2×#4542C2semidirect
3. #71724C2semidirect
4. #69672C2semidirect
5. #67644C2semidirect
53. #836D4semidirect
310. C3#13426semidirect

Maximal subgroups

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

subgroupindex
C3×#27332
C2×#45422
#717242
#696722
#676442
#134263