Finite Group Databasebeta

← #10691263

Subgroups — #10691263

Order 768.

578 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 111 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#82199C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic, complemented
3C2×C3×#324C211yesmaximal, characteristic, complemented
4#69944C212yesmaximal
5#67676C212yesmaximal, complemented
6#9206—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#930C2211yescharacteristic
9C2×#929—21no—
10C2×C3×C22⋊C8C2212yes—
11#5213—21no—
12#5212C2211yescharacteristic
13#5212C2212yes—
14#5211—21no—
15C4×#217—21no—
16#4994—22no—
17#4991C412yescomplemented
18#4981C412yescomplemented
19#4980—24no—
20C3×#324—22no—
21#4470—24no—
22#2618—31no—
23#2313—31no—
24#1519—32no—
25C2×#324D311yescharacteristic, complemented
26#1289—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C22×D12C2×C411yescharacteristic
32C22×C3×C8—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4C2×C412yes—
36C2×C3×C4⋊C4D412yes—
37C2×#217—22no—
38C2×#217—21no—
39#930—22no—
40#930—22no—
41#929—44no—
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4—21no—
44C2×(C2×D3)⋊C4—22no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3—21no—
47C2×C4⋊Dic3C2×C411yescharacteristic
48C2×Dic3⋊C4—22no—
49C2×C4×Dic3—22no—
50C3×C22⋊C8—22no—
51C3×C22⋊C8—22no—
52C3⋊#95—22no—
53C3⋊#95—24no—
54C3⋊#95—22no—
55C3⋊#95C2×C412yes—
56C3⋊#95D414yes—
57C2×#123—61no—
58C2×#121—31no—
59C22×C4⋊C4C2×D311yescharacteristic
60C2×C22⋊C8C2×D312yes—
61#398—32no—
62#396—62no—
63#394—32no—
64#387—64no—
65#386—61no—
66#381—61no—
67#380—31no—
68#380—32no—
69C4×C22⋊C4—61no—
70#327—64no—
71#324—22no—
72C24×C3C2×D411yescharacteristic
73C23×D3—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—21no—
80C22×C3×C4—81no—
81C22×C3×C4—81no—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22×C411yescharacteristic
85C22×C3×C4C2×D412yes—
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4112yes—
88C2×#67—41no—
89C2×#67—41no—
90C2×#67—42no—
91C2×#67—42no—
92C22×Dic3—21no—
93C22×Dic3—22no—
94C22×Dic3—24no—
95C2×D12—22no—
96C2×D12—81no—
97C2×C3×C8—42no—
98C2×C3×C8—82no—
99C3×C4⋊C4—24no—
100C3×C4⋊C4—24no—
101#217—42no—
102#217—42no—
103#217—42no—
104(C2×D3)⋊C4—44no—
105(C2×D3)⋊C4—42no—
106(C2×D3)⋊C4—82no—
107C4⋊Dic3—22no—
108C4⋊Dic3—82no—
109C4⋊Dic3—81no—
110Dic3⋊C4—44no—
111C4×Dic3—82no—
112C22×D4—31no—
113C22×D4—61no—
114C23×C4D1211yescharacteristic
115C23×C4D1211yescharacteristic
116C23×C4C22×D311yescharacteristic
117C22×C8—22no—
118#123—124no—
119#121—62no—
120#121—62no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—62no—
125C2×C4⋊C4—62no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4C4×D312yes—
128C2×C4⋊C4#6712yes—
129C2×C22⋊C4—61no—
130C2×C22⋊C4—62no—
131C2×C22⋊C4—62no—
132C2×C22⋊C4—61no—
133C2×C22⋊C4—62no—
134C2×C42—62no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137#95—32no—
138#95—34no—
139#95—62no—
140#95—64no—
141#95—62no—
142C23×C3—41no—
143C23×C3—41no—
144C23×C3C2×C22⋊C411yescharacteristic
145C23×C3#12011yescharacteristic
146C23×C3#12311yescharacteristic
147C23×C3#13512yes—
148C23×C3AGL(1,ℤ/8)12yes—
149C22×D3—42no—
150C22×D3—42no—
151C22×D3—41no—
152C22×D3—41no—
153C22×D3—41no—
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—42no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4—41no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—81no—
167C2×C3×C4C4×D412yes—
168C2×C3×C4#12112yes—
169#67—82no—
170#67—82no—
171C2×Dic3—44no—
172C2×Dic3—42no—
173C2×Dic3—44no—
174C2×Dic3—42no—
175C2×Dic3—41no—
176C2×Dic3—41no—
177C2×Dic3—41no—
178C2×Dic3—42no—
179C2×Dic3—44no—
180D12—82no—
181C3×C8—82no—
182C24—61no—
183C24C2×D1211yescharacteristic
184C2×D4—62no—
185C2×D4—121no—
186C2×D4—121no—
187C2×D4—122no—
188C2×D4—122no—
189C2×D4—241no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—21no—
196C22×C4—64no—
197C22×C4—62no—
198C22×C4—61no—
199C22×C4—81no—
200C22×C4—81no—
201C22×C4(C2×D3)⋊C412yes—
202C22×C4(C2×D3)⋊C412yes—
203C22×C4C2×C4×D311yescharacteristic
204C22×C4D3×D412yes—
205C22×C4#23712yes—
206C22×C4C2×#6711yescharacteristic
207C2×C8—42no—
208C2×C8—82no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—62no—
212C4⋊C4—124no—
213C4⋊C4—242no—
214C4⋊C4—241no—
215C22⋊C4—124no—
216C22⋊C4—122no—
217C22⋊C4—122no—
218C22⋊C4—122no—
219C22⋊C4—122no—
220C22⋊C4—242no—
221C42—242no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—42no—
229C22×C3—42no—
230C22×C3#38611yescharacteristic
231C22×C3#41611yescharacteristic
232C22×C3#41811yescharacteristic
233C22×C3#44912yes—
234C22×C3#48212yes—
235C2×D3—82no—
236C2×D3—82no—
237C2×D3—81no—
238C2×D3—81no—
239C2×D3—81no—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—42no—
243C3×C4—42no—
244Dic3—82no—
245Dic3—84no—
246Dic3—81no—
247C23—41no—
248C23—41no—
249C23—122no—
250C23—122no—
251C23—121no—
252C23—121no—
253C23—121no—
254C23#88211yescharacteristic
255C23#88611yescharacteristic
256C23#90412yes—
257C23#90812yes—
258C23C2×(C2×D3)⋊C411yescharacteristic
259D4—242no—
260D4—242no—
261D4—242no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—81no—
275C2×C4—124no—
276C2×C4—122no—
277C2×C4—124no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—124no—
283C2×C4—121no—
284C2×C4#67⋊C412yes—
285C2×C4#93912yes—
286C8—82no—
287C2×C3—21no—
288C2×C3—22no—
289C2×C3—21no—
290C2×C3—22no—
291C2×C3#187511yescharacteristic
292C2×C3#191211yescharacteristic
293C2×C3#191311yescharacteristic
294D3—161no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—42no—
302C22—42no—
303C22—242no—
304C22—242no—
305C22—241no—
306C22—241no—
307C22—241no—
308C22#473412yes—
309C22#473712yes—
310C22#495811yescharacteristic
311C22#511311yescharacteristic
312C22#513111yescharacteristic
313C4—22no—
314C4—22no—
315C4—42no—
316C4—42no—
317C4—242no—
318C4—244no—
319C4—241no—
320C3#920611yescharacteristic, complemented, Sylow
321C2—21no—
322C2—22no—
323C2—21no—
324C2—22no—
325C2—481no—
326C2#6993311yescharacteristic
327C2#7002411yescharacteristic
328C2#7010511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC22×C3#386
prime radicalC2×#930C22
cosocleC23×C3×C4C23
hypercenterC2×#324D3
hypocenterC3#9206
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
3. C2×C3×#324C2semidirect
5. #67676C2semidirect
17. #4991C4semidirect
18. #4981C4semidirect
25. C2×#324D3semidirect
320. C3#9206semidirect

Maximal subgroups

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

subgroupindex
#821992
#821552
C2×C3×#3242
#699442
#676762
#92063