Finite Group Databasebeta

← #10711998

Subgroups — #10711998

Order 768.

516 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 103 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
2C2×C3×#354C211yesmaximal, characteristic
3#71723C211yesmaximal, characteristic
4#70021C212yesmaximal
5#67694C212yesmaximal
6#12967—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8#5798C2211yescharacteristic
9C2×C3×#99C2212yes—
10#5229—21no—
11#5225—21no—
12#5220C2212yes—
13C2×#833C2211yescharacteristic
14#4951—22no—
15#4950—22no—
16#4950—22no—
17#4936—21no—
18C3×#354C414yescomplemented
19#4527—22no—
20#4523—21no—
21#2379—31no—
22C2×#354D311yescharacteristic
23#1885—31no—
24#1867—32no—
25#1390—32no—
26C22×C3×Q8D411yescharacteristic
27C22×C3×D4C2311yescharacteristic
28C23×Dic3D412yes—
29C3×#124—22no—
30C3×#124C2×C412yes—
31C3×#124Q812yes—
32C2×C3×C22⋊C4—22no—
33C2×C3×C22⋊C4—22no—
34C2×C3×C42D411yescharacteristic
35C2×#217—22no—
36C2×#217—22no—
37C2×#217D412yes—
38D4×Dic3—41no—
39D4×Dic3—41no—
40#926—21no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—42no—
45C2×C4×Dic3—21no—
46C2×#208—21no—
47C3×#99—22no—
48C3×#99C2×C414yes—
49#835—41no—
50#835—41no—
51#833—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95—22no—
55C3⋊#95—42no—
56C2×#124C2×D311yescharacteristic
57#518—31no—
58#419—61no—
59C2×#100—31no—
60C2×#99C2×D312yes—
61#391—61no—
62#388—62no—
63#386—32no—
64#385—62no—
65#385—62no—
66#382—61no—
67#354Dic314yescomplemented
68#342—62no—
69#339—61no—
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—42no—
83C22×C3×C4C2×D411yescharacteristic
84C22×Dic3—22no—
85C22×Dic3—22no—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—42no—
89C22×Dic3—82no—
90C3×C22⋊C4—24no—
91C3×C22⋊C4—24no—
92C3×C22⋊C4—24no—
93C3×C22⋊C4—42no—
94C3×C42—21no—
95C3×C42C4⋊C412yescomplemented
96#217—42no—
97#217—41no—
98#217—42no—
99#217—82no—
100C4⋊Dic3—21no—
101C4⋊Dic3—21no—
102C4⋊Dic3—81no—
103C4⋊Dic3—81no—
104C4⋊Dic3—81no—
105C4⋊Dic3—84no—
106C4×Dic3—41no—
107C4×Dic3—81no—
108#208—41no—
109#208—41no—
110C2×C3⋊C8—41no—
111C22×Q8D1211yescharacteristic
112C22×D4C22×D311yescharacteristic
113C23×C4—32no—
114C2×M16—61no—
115#124—22no—
116#124Dic612yes—
117#124C2×Dic312yes—
118C4×D4—121no—
119C4×D4—121no—
120#117—61no—
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×C42D1211yescharacteristic
132#103—121no—
133#103—121no—
134#100—62no—
135#99—22no—
136#99C2×Dic314yes—
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—44no—
162C2×C3×C4—82no—
163C2×C3×C4—82no—
164C2×C3×C4C2×C4⋊C411yescharacteristic
165C2×C3×C4#12311yescharacteristic
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—42no—
170C2×Dic3—44no—
171C2×Dic3—41no—
172C2×Dic3—41no—
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×Q8C4⋊Dic312yescomplemented
184C2×D4—41no—
185C2×D4—41no—
186C2×D4C2×Dic611yescharacteristic
187C2×D4#23712yes—
188C2×D4C22×Dic311yescharacteristic
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—42no—
192C22×C4—62no—
193C22×C4—62no—
194C22×C4—62no—
195C22×C4—62no—
196C22×C4—122no—
197C22×C4—242no—
198C22×C4C2×D1211yescharacteristic
199M16—121no—
200M16—121no—
201C2×C8—121no—
202C4⋊C4—61no—
203C4⋊C4—61no—
204C4⋊C4—241no—
205C4⋊C4—241no—
206C4⋊C4—241no—
207C4⋊C4—244no—
208C22⋊C4—24no—
209C22⋊C4—24no—
210C22⋊C4—24no—
211C22⋊C4—42no—
212C22⋊C4—122no—
213C22⋊C4—121no—
214C22⋊C4—122no—
215C22⋊C4—242no—
216C42—21no—
217C42—121no—
218C42—241no—
219C42C4⋊Dic312yescomplemented
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—84no—
237Dic3—82no—
238Dic3—82no—
239Dic3—81no—
240Dic3—81no—
241C23—22no—
242C23—22no—
243C23—42no—
244C23—42no—
245C23#87812yes—
246C23#88211yescharacteristic
247C23D4×Dic312yes—
248Q8—42no—
249Q8—81no—
250D4—41no—
251D4—42no—
252D4—41no—
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—44no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4—124no—
269C2×C4—121no—
270C2×C4—121no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—242no—
274C2×C4—242no—
275C2×C4—242no—
276C2×C4#88611yescharacteristic
277C2×C4C2×C4⋊Dic311yescharacteristic
278C8—241no—
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#474111yescharacteristic
296C22#474211yescharacteristic
297C22#495011yescharacteristic
298C4—21no—
299C4—21no—
300C4—42no—
301C4—42no—
302C4—84no—
303C4—242no—
304C4—242no—
305C4—241no—
306C4—241no—
307C3#1296711yescharacteristic, complemented, Sylow
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2#6915411yescharacteristic
313C2#6915511yescharacteristic
314C2#7002111yescharacteristic
subgroupG/H
centerC22#4741
commutatorC2×C3×D4C22×C4
FrattiniC22×D4C22×D3
FittingC2×C3×#354C2
socleC22×C3B(4,𝔽2)
prime radical, cosocleC22×C3×D4C23
hypercenterC2×#354D3
hypocenterC3#12967
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#354

Complements

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

NH ≅ G/Ntype
18. C3×#354C4semidirect
67. #354Dic3semidirect
74. C2×C3×Q8C4⋊C4semidirect
95. C3×C42C4⋊C4semidirect
183. C2×Q8C4⋊Dic3semidirect
219. C42C4⋊Dic3semidirect
307. C3#12967semidirect

Maximal subgroups

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

subgroupindex
#829572
C2×C3×#3542
#717232
#700212
#676942
#129673