Finite Group Databasebeta

← #10690182

Subgroups — #10690182

Order 768.

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

normal subgroups only

This group has 756 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
1C2×C3×#395C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic
3#72132C212yesmaximal
4C2×#4528C211yesmaximal, characteristic
5#67612C212yesmaximal
6#9014—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C22×C3×C4⋊C4—21no—
9C22×C3×C4⋊C4C2211yescharacteristic
10C3×#395—22no—
11C3×#395C414yescomplemented
12C2×C3×#95—21no—
13C2×C3×#95C2211yescharacteristic
14C2×#835—24no—
15#5212C2212yes—
16#5211—21no—
17C4×#217—21no—
18C2×#830C2212yes—
19#4980—24no—
20#4528—22no—
21C2×#395D311yescharacteristic
22#2313—31no—
23#1787—32no—
24C2×#324—31no—
25#1294—32no—
26C22×C3×Q8—21no—
27C22×C3×Q8—22no—
28C22×C3×Q8—22no—
29C22×C3×Q8—44no—
30C22×C3×Q8C2×C412yes—
31C23×C3×C4—21no—
32C23×C3×C4—21no—
33C23×C3×C4D411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—24no—
39C2×C3×C4⋊C4—44no—
40C2×C3×C4⋊C4C2×C412yes—
41C2×C3×C4⋊C4D412yes—
42C2×#217—22no—
43C2×C4⋊Dic3D414yes—
44C2×C4×Dic3—22no—
45C22×C3⋊C8—22no—
46C3×#95—24no—
47C3×#95—21no—
48C3×#95C2×C412yes—
49#835—48no—
50C3⋊#95—22no—
51C3⋊#95—24no—
52C3⋊#95—22no—
53#830—22no—
54#830—22no—
55C23×Q8C2×D311yescharacteristic
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×D311yescharacteristic
58C2×#103—64no—
59C2×C22⋊C8—32no—
60#395—22no—
61#395Dic314yescomplemented
62#387—64no—
63#381—61no—
64#380—32no—
65C4×C22⋊C4—61no—
66C2×#95—21no—
67C2×#95C2×D311yescharacteristic
68#324—62no—
69C24×C3C2×D411yescharacteristic
70C2×C3×Q8—28no—
71C2×C3×Q8—24no—
72C2×C3×Q8—44no—
73C2×C3×Q8—44no—
74C2×C3×Q8—44no—
75C2×C3×Q8—48no—
76C2×C3×Q8—42no—
77C22×C3×C4—24no—
78C22×C3×C4—24no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—21no—
87C22×C3×C4—42no—
88C22×C3×C4—42no—
89C22×C3×C4—42no—
90C22×C3×C4—41no—
91C22×C3×C4—41no—
92C22×C3×C4—42no—
93C22×C3×C4—42no—
94C22×C3×C4C22⋊C412yes—
95C22×C3×C4C22⋊C412yes—
96C22×C3×C4C22×C411yescharacteristic
97C22×C3×C4C2×D412yes—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4112yes—
100C22×Dic3—24no—
101C3×C4⋊C4—24no—
102C3×C4⋊C4—24no—
103C3×C4⋊C4—48no—
104#217—42no—
105#217—42no—
106C4⋊Dic3—44no—
107C4×Dic3—82no—
108C2×C3⋊C8—44no—
109C2×C3⋊C8—42no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—44no—
114C22×Q8C2×Dic312yes—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4#6711yescharacteristic
118C23×C4#6711yescharacteristic
119C23×C4C22×D311yescharacteristic
120C22×C8—62no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—34no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4C2×Dic312yes—
127C2×C4⋊C4#6712yes—
128C2×C22⋊C4—62no—
129C2×C42—62no—
130#103—128no—
131C22⋊C8—62no—
132C22⋊C8—62no—
133#95—24no—
134#95—21no—
135#95—62no—
136#95—64no—
137#95—62no—
138#95C2×Dic312yes—
139C23×C3—22no—
140C23×C3—22no—
141C23×C3C2×C22⋊C411yescharacteristic
142C23×C3#12011yescharacteristic
143C23×C3#12311yescharacteristic
144C23×C3#13512yes—
145C23×C3#13712yes—
146C3×Q8—28no—
147C3×Q8—48no—
148C3×Q8—44no—
149C2×C3×C4—24no—
150C2×C3×C4—24no—
151C2×C3×C4—24no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—41no—
157C2×C3×C4—41no—
158C2×C3×C4—42no—
159C2×C3×C4—42no—
160C2×C3×C4—48no—
161C2×C3×C4—42no—
162C2×C3×C4—44no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—44no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—81no—
171C2×C3×C4C4×D412yes—
172C2×C3×C4#12112yes—
173C2×Dic3—44no—
174C2×Dic3—44no—
175C2×Dic3—44no—
176C3⋊C8—82no—
177C24C2×#6711yescharacteristic
178C2×Q8—28no—
179C2×Q8—24no—
180C2×Q8—44no—
181C2×Q8—44no—
182C2×Q8—44no—
183C2×Q8—48no—
184C2×Q8—42no—
185C22×C4—24no—
186C22×C4—24no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—21no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—64no—
203C22×C4#21712yes—
204C22×C4#21712yes—
205C22×C4D3×D412yes—
206C22×C4#23712yes—
207C22×C4C22×Dic311yescharacteristic
208C22×C4C2×#6711yescharacteristic
209C2×C8—124no—
210C2×C8—122no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—48no—
214C4⋊C4—124no—
215C22⋊C4—122no—
216C22⋊C4—122no—
217C42—242no—
218C22×C3—24no—
219C22×C3—24no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3#38611yescharacteristic
225C22×C3#41611yescharacteristic
226C22×C3#41911yescharacteristic
227C22×C3#45212yes—
228C22×C3#45612yes—
229C22×C3B(4,𝔽2)12yes—
230C22×C3#48512yes—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—44no—
234C3×C4—42no—
235C3×C4—44no—
236C3×C4—42no—
237Dic3—84no—
238C23—22no—
239C23—22no—
240C23#93511yescharacteristic
241C23#93711yescharacteristic
242C23#94212yes—
243C23#95012yes—
244C23C2×#21711yescharacteristic
245Q8—28no—
246Q8—48no—
247Q8—44no—
248C2×C4—24no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—42no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—48no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—44no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—124no—
271C2×C4—124no—
272C2×C4—124no—
273C2×C4D4×Dic312yes—
274C2×C4#93912yes—
275C8—242no—
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—22no—
279C2×C3#160314yes—
280C2×C3#187311yescharacteristic
281C2×C3#188611yescharacteristic
282C2×C3#188911yescharacteristic
283C22—24no—
284C22—24no—
285C22—21no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22#503212yes—
290C22#504512yes—
291C22#504712yes—
292C22#509512yes—
293C22#522011yescharacteristic
294C22#522511yescharacteristic
295C22#523311yescharacteristic
296C4—22no—
297C4—22no—
298C4—44no—
299C4—42no—
300C4—44no—
301C4—42no—
302C4—244no—
303C3#901411yescharacteristic, complemented, Sylow
304C2—21no—
305C2—21no—
306C2—22no—
307C2#6970514yes—
308C2#7166911yescharacteristic
309C2#7169311yescharacteristic
310C2#7178411yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#395C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#395D3
hypocenterC3#9014
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
11. C3×#395C4semidirect
61. #395Dic3semidirect
303. C3#9014semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3952
#821552
#721322
C2×#45282
#676122
#90143