Finite Group Databasebeta

← #10690910

Subgroups — #10690910

Order 768.

694 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 135 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#82075C211yesmaximal, characteristic
2C2×#4932C211yesmaximal, characteristic
3C3×#1817C211yesmaximal, characteristic
4C2×#4467C212yesmaximal
5#67675C212yesmaximal
6#9145—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#924C2211yescharacteristic
9C22×C4×Dic3—21no—
10C2×C3×C22⋊C8C2212yes—
11#5212C2211yescharacteristic
12#5211—21no—
13#5210—21no—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2212yes—
16#4933—22no—
17#4932—24no—
18#4929C412yescomplemented
19#4928C412yescomplemented
20C3×#336—22no—
21#4467—24no—
22#2855—31no—
23C2×#376—31no—
24#1817D311yescharacteristic
25C2×#328—32no—
26#1294—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C23×Dic3—22no—
31C22×Dic6C2×C411yescharacteristic
32C22×C3×C8—22no—
33C2×C3×C42—22no—
34C2×C3×C42—22no—
35C2×C3×C42C2×C412yes—
36C2×C3×C42D412yes—
37C2×#217—21no—
38C2×#217—21no—
39C2×C4⋊Dic3C2×C411yescharacteristic
40C2×C4⋊Dic3C2×C412yes—
41#924—22no—
42#924—22no—
43C2×Dic3⋊C4—22no—
44C2×Dic3⋊C4—21no—
45C2×C4×Dic3—24no—
46C2×C4×Dic3—28no—
47C3×C22⋊C8—22no—
48C3×C22⋊C8—22no—
49C3⋊#95—24no—
50C3⋊#95—22no—
51C3⋊#95—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95—21no—
55C3⋊#95D414yes—
56C2×C22⋊Q8—31no—
57C22×C42—61no—
58C22×C42C2×D311yescharacteristic
59C2×C22⋊C8C2×D312yes—
60#391—32no—
61#389—32no—
62#387—62no—
63#385—61no—
64#381—61no—
65#380—31no—
66#376—64no—
67C2×#95—32no—
68C2×#95—61no—
69#336—22no—
70#328—64no—
71C24×C3C2×D411yescharacteristic
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—24no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—42no—
80C22×C3×C4C22⋊C412yes—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D412yes—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4#4112yes—
86C22×Dic3—28no—
87C22×Dic3—24no—
88C22×Dic3—22no—
89C22×Dic3—21no—
90C22×Dic3—21no—
91C22×Dic3—44no—
92C22×Dic3—42no—
93C22×Dic3—42no—
94C2×Dic6—22no—
95C2×Dic6—42no—
96C2×C3×C8—44no—
97C2×C3×C8—42no—
98C3×C42—24no—
99C3×C42—24no—
100#217—41no—
101#217—41no—
102#217—42no—
103C4⋊Dic3—22no—
104C4⋊Dic3—42no—
105Dic3⋊C4—44no—
106Dic3⋊C4—42no—
107C4×Dic3—216no—
108C22×Q8—31no—
109C23×C4—62no—
110C23×C4D1211yescharacteristic
111C23×C4D1211yescharacteristic
112C23×C4C22×D311yescharacteristic
113C22×C8—22no—
114C22⋊Q8—62no—
115C22⋊Q8—62no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—32no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—61no—
120C2×C22⋊C4—61no—
121C2×C22⋊C4—61no—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—64no—
125C2×C42—68no—
126C2×C42C4×D312yes—
127C2×C42#6712yes—
128C22⋊C8—22no—
129C22⋊C8—22no—
130#95—34no—
131#95—64no—
132#95—62no—
133#95—62no—
134#95—62no—
135#95—62no—
136#95—61no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3C2×C22⋊C411yescharacteristic
140C23×C3#12011yescharacteristic
141C23×C3#12311yescharacteristic
142C23×C3#13512yes—
143C23×C3#13712yes—
144C2×C3×C4—24no—
145C2×C3×C4—24no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4#10414yes—
158C2×C3×C4C4×D412yes—
159C2×C3×C4#12112yes—
160C2×Dic3—28no—
161C2×Dic3—44no—
162C2×Dic3—48no—
163C2×Dic3—48no—
164C2×Dic3—44no—
165C2×Dic3—41no—
166C2×Dic3—41no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—42no—
170C2×Dic3—42no—
171C2×Dic3—42no—
172Dic6—82no—
173C3×C8—82no—
174C24C2×D1211yescharacteristic
175C2×Q8—62no—
176C2×Q8—122no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—24no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—42no—
185C22×C4—68no—
186C22×C4—64no—
187C22×C4—62no—
188C22×C4—61no—
189C22×C4—61no—
190C22×C4—124no—
191C22×C4—122no—
192C22×C4—122no—
193C22×C4(C2×D3)⋊C412yes—
194C22×C4(C2×D3)⋊C412yes—
195C22×C4C2×C4×D311yescharacteristic
196C22×C4D3×D412yes—
197C22×C4#23712yes—
198C22×C4C2×#6711yescharacteristic
199C2×C8—44no—
200C2×C8—42no—
201C4⋊C4—62no—
202C4⋊C4—122no—
203C4⋊C4—124no—
204C4⋊C4—122no—
205C22⋊C4—121no—
206C22⋊C4—121no—
207C22⋊C4—122no—
208C42—24no—
209C42—24no—
210C42—616no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3—24no—
216C22×C3—24no—
217C22×C3#38611yescharacteristic
218C22×C3#41012yes—
219C22×C3#41611yescharacteristic
220C22×C3#41911yescharacteristic
221C22×C3C2×#10412yes—
222C22×C3#45212yes—
223C22×C3#48512yes—
224C3×C4—24no—
225C3×C4—22no—
226C3×C4—22no—
227C3×C4—42no—
228Dic3—48no—
229Dic3—81no—
230Dic3—81no—
231Dic3—82no—
232C23—22no—
233C23—22no—
234C23#88211yescharacteristic
235C23#88611yescharacteristic
236C23#90412yes—
237C23#90912yes—
238C23C2×(C2×D3)⋊C411yescharacteristic
239Q8—242no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—68no—
254C2×C4—128no—
255C2×C4—124no—
256C2×C4—128no—
257C2×C4—124no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4—121no—
261C2×C4—121no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4#82514yes—
266C2×C4#67⋊C412yes—
267C2×C4#93912yes—
268C8—82no—
269C2×C3—22no—
270C2×C3—22no—
271C2×C3#150012yes—
272C2×C3#150612yes—
273C2×C3#187311yescharacteristic
274C2×C3#189611yescharacteristic
275C2×C3#189911yescharacteristic
276C22—21no—
277C22—21no—
278C22—24no—
279C22—21no—
280C22—21no—
281C22—24no—
282C22#472312yes—
283C22#473312yes—
284C22#495811yescharacteristic
285C22#500612yes—
286C22#511311yescharacteristic
287C22#512011yescharacteristic
288C22C2×#82512yes—
289C4—24no—
290C4—22no—
291C4—22no—
292C4—42no—
293C4—128no—
294C4—241no—
295C4—242no—
296C4—241no—
297C3#914511yescharacteristic, complemented, Sylow
298C2—22no—
299C2—22no—
300C2#6953511yescharacteristic
301C2#6993111yescharacteristic
302C2#6996511yescharacteristic
303C2#7078812yes—
304C2#7083612yes—
subgroupG/H
centerC23C2×(C2×D3)⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1817C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1817D3
hypocenterC3#9145
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
18. #4929C4semidirect
19. #4928C4semidirect
297. C3#9145semidirect

Maximal subgroups

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

subgroupindex
#820752
C2×#49322
C3×#18172
C2×#44672
#676752
#91453