Finite Group Databasebeta

← #10711995

Subgroups — #10711995

Order 768.

516 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 95 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
1#82957C211yesmaximal, characteristic
2C3×#1885C211yesmaximal, characteristic
3C2×#4541C211yesmaximal, characteristic
4#71657C212yesmaximal
5#67698C212yesmaximal
6#12967—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8C3×#518C2211yescharacteristic
9C3×#419—21no—
10C2×C3×#100C2211yescharacteristic
11#5229—21no—
12#5220C2212yes—
13C2×#834C2212yes—
14#4951—22no—
15#4950—22no—
16#4946—22no—
17#4936—21no—
18C3×#339—21no—
19#4541C414yescomplemented
20#4526—22no—
21#2379—31no—
22C2×#354—31no—
23#1885D311yescharacteristic
24#1867—32no—
25#1390—32no—
26C22×C3×Q8D411yescharacteristic
27C22×C3×D4C2311yescharacteristic
28C23×Dic3D412yes—
29C2×C3×M16—21no—
30C3×#124—22no—
31C3×#124C2×C412yes—
32C3×#124Q812yes—
33C3×C4×D4—41no—
34C3×C4×D4—41no—
35C3×#117—21no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C42D411yescharacteristic
39C2×#217—22no—
40C2×#217—22no—
41C2×#217D412yes—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×Dic3⋊C4—42no—
46C2×C4×Dic3—21no—
47C3×#103—41no—
48C3×#103—41no—
49C3×#100—22no—
50#834—22no—
51#834C2×C414yes—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95—22no—
55C3⋊#95—42no—
56C2×#124C2×D311yescharacteristic
57#518C2×D311yescharacteristic
58#419—21no—
59C2×#100C2×D311yescharacteristic
60C2×#99—32no—
61#391—61no—
62#388—62no—
63#386—32no—
64#385—62no—
65#385—62no—
66#382—61no—
67#354—34no—
68#342—62no—
69#339—21no—
70C24×C3C2×D412yes—
71C2×C3×Q8—21no—
72C2×C3×Q8—41no—
73C2×C3×Q8—41no—
74C2×C3×Q8C4⋊C412yescomplemented
75C2×C3×D4—41no—
76C2×C3×D4—41no—
77C2×C3×D4C22×C411yescharacteristic
78C2×C3×D4C2×Q811yescharacteristic
79C2×C3×D4#4112yes—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4C2×D411yescharacteristic
85C22×Dic3—22no—
86C22×Dic3—22no—
87C22×Dic3—42no—
88C22×Dic3—42no—
89C22×Dic3—82no—
90C3×M16—41no—
91C3×M16—41no—
92C2×C3×C8—41no—
93C3×C4⋊C4—21no—
94C3×C4⋊C4—21no—
95C3×C22⋊C4—24no—
96C3×C22⋊C4—24no—
97C3×C22⋊C4—42no—
98C3×C22⋊C4—41no—
99C3×C42—21no—
100C3×C42—41no—
101C3×C42C4⋊C412yescomplemented
102#217—24no—
103#217—42no—
104#217—42no—
105#217—82no—
106C4⋊Dic3—81no—
107C4⋊Dic3—81no—
108C4⋊Dic3—81no—
109Dic3⋊C4—84no—
110C4×Dic3—81no—
111C22×Q8#6711yescharacteristic
112C22×D4C22×D311yescharacteristic
113C23×C4—32no—
114C2×M16—21no—
115#124—22no—
116#124Dic612yes—
117#124C4×D312yes—
118C4×D4—41no—
119C4×D4—41no—
120#117—21no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4—61no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—122no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—32no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4—62no—
130C2×C42—61no—
131C2×C42#6711yescharacteristic
132#103—41no—
133#103—41no—
134#100—22no—
135#99—34no—
136#99—62no—
137#95—62no—
138#95—62no—
139#95—62no—
140#95—122no—
141C23×C3—22no—
142C23×C3—22no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3C4×D412yes—
146C23×C3#12011yescharacteristic
147C23×C3C22⋊Q812yes—
148C3×Q8—42no—
149C3×Q8—81no—
150C3×D4—41no—
151C3×D4—41no—
152C3×D4—42no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—41no—
158C2×C3×C4—41no—
159C2×C3×C4—41no—
160C2×C3×C4—41no—
161C2×C3×C4—41no—
162C2×C3×C4—41no—
163C2×C3×C4—82no—
164C2×C3×C4—82no—
165C2×C3×C4C2×C4⋊C411yescharacteristic
166C2×C3×C4#12311yescharacteristic
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—44no—
170C2×Dic3—42no—
171C2×Dic3—42no—
172C2×Dic3—44no—
173C2×Dic3—42no—
174C2×Dic3—42no—
175C2×Dic3—82no—
176C2×Dic3—82no—
177C2×Dic3—82no—
178C3×C8—81no—
179C24D3×D412yes—
180C2×Q8—21no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8Dic3⋊C412yescomplemented
184C2×D4—41no—
185C2×D4—41no—
186C2×D4C2×Dic611yescharacteristic
187C2×D4C2×C4×D311yescharacteristic
188C2×D4#23712yes—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—62no—
194C22×C4—62no—
195C22×C4—122no—
196C22×C4—122no—
197C22×C4—242no—
198C22×C4C2×#6711yescharacteristic
199M16—41no—
200M16—41no—
201C2×C8—41no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—241no—
205C4⋊C4—241no—
206C4⋊C4—241no—
207C4⋊C4—244no—
208C22⋊C4—24no—
209C22⋊C4—24no—
210C22⋊C4—42no—
211C22⋊C4—41no—
212C22⋊C4—64no—
213C22⋊C4—122no—
214C22⋊C4—122no—
215C22⋊C4—242no—
216C42—21no—
217C42—41no—
218C42—241no—
219C42Dic3⋊C412yescomplemented
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—22no—
223C22×C3—22no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—42no—
228C22×C3—82no—
229C22×C3#38511yescharacteristic
230C22×C3B(4,𝔽2)11yescharacteristic
231C22×C3#45811yescharacteristic
232C3×C4—21no—
233C3×C4—21no—
234C3×C4—42no—
235C3×C4—42no—
236C3×C4—81no—
237C3×C4—81no—
238Dic3—82no—
239Dic3—84no—
240Dic3—82no—
241C23—22no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23#87812yes—
246C23#67⋊C412yes—
247C23#93711yescharacteristic
248Q8—42no—
249Q8—81no—
250D4—41no—
251D4—41no—
252D4—42no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—82no—
264C2×C4—82no—
265C2×C4—124no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4—122no—
269C2×C4—122no—
270C2×C4—124no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—242no—
274C2×C4—242no—
275C2×C4—242no—
276C2×C4C2×Dic3⋊C411yescharacteristic
277C2×C4#93511yescharacteristic
278C8—81no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—42no—
282C2×C3—42no—
283C2×C3#186711yescharacteristic
284C2×C3#219911yescharacteristic
285C2×C3#220111yescharacteristic
286C22—21no—
287C22—21no—
288C22—22no—
289C22—22no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—42no—
294C22—82no—
295C22#494611yescharacteristic
296C22#503211yescharacteristic
297C22#505311yescharacteristic
298C4—21no—
299C4—21no—
300C4—42no—
301C4—42no—
302C4—81no—
303C4—81no—
304C4—242no—
305C4—244no—
306C4—242no—
307C3#1296711yescharacteristic, complemented, Sylow
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#6959711yescharacteristic
313C2#7099111yescharacteristic
314C2#7099211yescharacteristic
subgroupG/H
centerC22#5032
commutatorC2×C3×D4C22×C4
FrattiniC22×D4C22×D3
FittingC3×#1885C2
socleC22×C3B(4,𝔽2)
prime radical, cosocleC22×C3×D4C23
hypercenter#1885D3
hypocenterC3#12967
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #1885

Complements

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

NH ≅ G/Ntype
19. #4541C4semidirect
74. C2×C3×Q8C4⋊C4semidirect
101. C3×C42C4⋊C4semidirect
183. C2×Q8Dic3⋊C4semidirect
219. C42Dic3⋊C4semidirect
307. C3#12967semidirect

Maximal subgroups

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

subgroupindex
#829572
C3×#18852
C2×#45412
#716572
#676982
#129673