Finite Group Databasebeta

← #10692919

Subgroups — #10692919

Order 768.

700 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 107 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#84065C211yesmaximal, characteristic
2#84062C211yesmaximal, characteristic
3C3×#1784C211yesmaximal, characteristic
4#69981C212yesmaximal
5#67682C212yesmaximal
6#9462—31nomaximal, Sylow
7C23×C3×Q8C2211yescharacteristic
8C2×#944C412yescomplemented
9C2×#926—21no—
10C2×#924C2211yescharacteristic
11C2×C3×#101—22no—
12C3×#407C2212yes—
13#5230—22no—
14#5229—24no—
15#5229C412yescomplemented
16#5211—21no—
17#5211C2212yes—
18#5210—21no—
19#5210—21no—
20C4×#217C2211yescharacteristic
21#4467—24no—
22#2434—31no—
23#2310—31no—
24#1784D311yescharacteristic
25#1511—32no—
26#1284—32no—
27C22×C3×Q8—22no—
28C22×C3×Q8—22no—
29C22×C3×Q8—44no—
30C22×C3×Q8C2×C412yes—
31C22×C3×Q8D412yes—
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4C2311yescharacteristic
35C22×Dic6C2×C411yescharacteristic
36C2×C3×M16—22no—
37C2×#217—21no—
38C2×#217—21no—
39C2×#217—22no—
40#944—44no—
41#926—44no—
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3C2×C411yescharacteristic
44#924—22no—
45#924—22no—
46C2×Dic3⋊C4—22no—
47C2×Dic3⋊C4—22no—
48C2×Dic3⋊C4—21no—
49C2×C4×Dic3—22no—
50C2×C4×Dic3C2×C412yes—
51C3×#101—44no—
52C3×C22⋊C8—22no—
53C3×C22⋊C8—22no—
54C3⋊#95—24no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57C3⋊#95—21no—
58C3⋊#95—22no—
59C3⋊#95D414yes—
60C23×Q8C2×D311yescharacteristic
61C2×C4⋊Q8—32no—
62C2×C22⋊Q8—31no—
63C2×#117—61no—
64C2×#101—22no—
65#407C2×D312yes—
66#395—62no—
67#391—32no—
68#391—64no—
69#385—61no—
70#385—61no—
71#381—32no—
72#381—61no—
73C4×C22⋊C4—31no—
74#328—64no—
75C24×C3C2×D411yescharacteristic
76C2×C3×Q8—24no—
77C2×C3×Q8—24no—
78C2×C3×Q8—24no—
79C2×C3×Q8—24no—
80C2×C3×Q8—42no—
81C2×C3×Q8—42no—
82C2×C3×Q8—44no—
83C2×C3×Q8—44no—
84C2×C3×Q8—48no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—24no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4C22⋊C412yes—
93C22×C3×C4C22⋊C412yes—
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4#4112yes—
98C22×Dic3—22no—
99C22×Dic3—21no—
100C22×Dic3—21no—
101C22×Dic3—24no—
102C2×Dic6—22no—
103C2×Dic6—42no—
104C3×M16—44no—
105C2×C3×C8—42no—
106#217—24no—
107#217—44no—
108#217—41no—
109#217—41no—
110C4⋊Dic3—22no—
111C4⋊Dic3—44no—
112Dic3⋊C4—44no—
113Dic3⋊C4—44no—
114Dic3⋊C4—42no—
115C4×Dic3—44no—
116C4×Dic3—42no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—31no—
120C22×Q8—44no—
121C22×Q8C4×D312yes—
122C22×Q8#6712yes—
123C23×C4D1211yescharacteristic
124C23×C4D1211yescharacteristic
125C23×C4C22×D311yescharacteristic
126C2×M16—22no—
127C4⋊Q8—124no—
128C22⋊Q8—62no—
129C22⋊Q8—62no—
130#117—124no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4—62no—
135C2×C4⋊C4—61no—
136C2×C22⋊C4—61no—
137C2×C22⋊C4—62no—
138C2×C22⋊C4—61no—
139C2×C42—32no—
140C2×C42—62no—
141#101—44no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144#95—34no—
145#95—64no—
146#95—62no—
147#95—62no—
148#95—62no—
149#95—61no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—41no—
153C23×C3C2×C22⋊C411yescharacteristic
154C23×C3#12011yescharacteristic
155C23×C3#12311yescharacteristic
156C3×Q8—48no—
157C3×Q8—48no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—24no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—22no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4#9914yes—
173C2×C3×C4C4×D412yes—
174C2×C3×C4#12112yes—
175C2×Dic3—24no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—44no—
179C2×Dic3—44no—
180C2×Dic3—42no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—42no—
184C2×Dic3—44no—
185Dic6—82no—
186C3×C8—82no—
187C24C2×D1211yescharacteristic
188C2×Q8—24no—
189C2×Q8—24no—
190C2×Q8—24no—
191C2×Q8—24no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—48no—
197C2×Q8—62no—
198C2×Q8—122no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—62no—
207C22×C4—64no—
208C22×C4—61no—
209C22×C4—61no—
210C22×C4(C2×D3)⋊C412yes—
211C22×C4(C2×D3)⋊C412yes—
212C22×C4C2×C4×D311yescharacteristic
213C22×C4D3×D412yes—
214C22×C4#23712yes—
215C22×C4C2×#6711yescharacteristic
216M16—44no—
217C2×C8—42no—
218C4⋊C4—62no—
219C4⋊C4—124no—
220C4⋊C4—124no—
221C4⋊C4—124no—
222C4⋊C4—122no—
223C22⋊C4—64no—
224C22⋊C4—124no—
225C22⋊C4—121no—
226C22⋊C4—121no—
227C42—122no—
228C42—124no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—44no—
237C22×C3#38611yescharacteristic
238C22×C3C2×#9912yes—
239C22×C3#45612yes—
240C3×C4—22no—
241C3×C4—22no—
242C3×C4—24no—
243C3×C4—42no—
244Dic3—44no—
245Dic3—84no—
246Dic3—81no—
247Dic3—81no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23#88211yescharacteristic
252C23#88611yescharacteristic
253C23C2×(C2×D3)⋊C411yescharacteristic
254Q8—48no—
255Q8—48no—
256Q8—242no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—64no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—124no—
275C2×C4—124no—
276C2×C4—122no—
277C2×C4—122no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—124no—
281C2×C4#80614yes—
282C2×C4#67⋊C412yes—
283C2×C4#93912yes—
284C8—82no—
285C2×C3—22no—
286C2×C3—22no—
287C2×C3—41no—
288C2×C3#188611yescharacteristic
289C2×C3#189811yescharacteristic
290C2×C3#191211yescharacteristic
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22—21no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22#475212yes—
300C22C2×#80612yes—
301C22#495811yescharacteristic
302C4—22no—
303C4—24no—
304C4—22no—
305C4—42no—
306C4—124no—
307C4—244no—
308C4—241no—
309C4—241no—
310C3#946211yescharacteristic, complemented, Sylow
311C2—22no—
312C2—22no—
313C2—41no—
314C2#6949611yescharacteristic
315C2#7010311yescharacteristic
316C2#7010511yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1784C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1784D3
hypocenterC3#9462
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1784

Complements

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

NH ≅ G/Ntype
8. C2×#944C4semidirect
15. #5229C4semidirect
310. C3#9462semidirect

Maximal subgroups

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

subgroupindex
#840652
#840622
C3×#17842
#699812
#676822
#94623