Finite Group Databasebeta

← #10693012

Subgroups — #10693012

Order 768.

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

normal subgroups only

This group has 726 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
1C3×#2406C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic, complemented
3#72138C212yesmaximal
4#71552C211yesmaximal, characteristic, complemented
5#67628C212yesmaximal, complemented
6#9480—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×C3×#123—21no—
9C22×C3×C4⋊C4C2211yescharacteristic
10C3×#395C412yescomplemented
11C3×#394C412yescomplemented
12C3×#386—21no—
13C3×#385—22no—
14C3×#381C2211yescharacteristic
15C2×#837—24no—
16#5212—21no—
17#5211—21no—
18C4×#217C2212yes—
19#5207C2212yes—
20#4980—24no—
21#4528—22no—
22#2406D311yescharacteristic
23#2313—31no—
24#1789—32no—
25#1463—31no—
26#1284—32no—
27C2×C3×#41—22no—
28C2×C3×#41—22no—
29C2×C3×#41—44no—
30C22×C3×Q8C2×C411yescharacteristic
31C22×C3×D4—21no—
32C22×C3×D4C2×C411yescharacteristic
33C23×C3×C4—21no—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C3×#123—44no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4C2×C412yes—
42C2×C3×C4⋊C4D412yes—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×#217—22no—
47C2×C4⋊Dic3—22no—
48C2×C4×Dic3D414yes—
49C2×#208—22no—
50C3×#95—22no—
51C3×#95—21no—
52C3×#95C2×C412yes—
53#837—48no—
54C3⋊#95—24no—
55C3⋊#95—22no—
56C3⋊#95—22no—
57#830—22no—
58#830—22no—
59C22×#41C2×D311yescharacteristic
60C2×#123—21no—
61C22×C4⋊C4C2×D311yescharacteristic
62C2×#104—64no—
63#407—32no—
64#395Dic312yescomplemented
65#394Dic312yescomplemented
66#387—64no—
67#386—21no—
68#385—22no—
69#381—61no—
70#381C2×D311yescharacteristic
71#380—61no—
72C4×C22⋊C4—32no—
73#324—62no—
74C24×C3—21no—
75C24×C3C2×D411yescharacteristic
76C3×#41—28no—
77C3×#41—44no—
78C3×#41—44no—
79C3×#41—44no—
80C2×C3×Q8—22no—
81C2×C3×Q8—42no—
82C2×C3×D4—24no—
83C2×C3×D4—22no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—41no—
87C2×C3×D4—41no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4—42no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4C22⋊C412yes—
101C22×C3×C4C22⋊C412yes—
102C22×C3×C4C22×C411yescharacteristic
103C22×C3×C4C2×D412yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4112yes—
106C22×Dic3—24no—
107C3×C4⋊C4—24no—
108C3×C4⋊C4—24no—
109C3×C4⋊C4—44no—
110C3×C22⋊C4—44no—
111C3×C22⋊C4—44no—
112C3×C22⋊C4—42no—
113#217—44no—
114C4⋊Dic3—82no—
115C4×Dic3—44no—
116#208—44no—
117C2×C3⋊C8—42no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—44no—
121C22×Q8C2×Dic311yescharacteristic
122C22×D4—21no—
123C22×D4C2×Dic311yescharacteristic
124C23×C4—21no—
125C23×C4#6711yescharacteristic
126C23×C4#6711yescharacteristic
127C23×C4C22×D311yescharacteristic
128C2×M16—62no—
129#123—44no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—62no—
134C2×C4⋊C4C2×Dic312yes—
135C2×C4⋊C4#6712yes—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—62no—
140C2×C42—34no—
141#104—128no—
142C22⋊C8—62no—
143C22⋊C8—62no—
144#95—22no—
145#95—21no—
146#95—62no—
147#95—64no—
148#95—62no—
149#95C2×Dic312yes—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—22no—
153C23×C3—41no—
154C23×C3—41no—
155C23×C3—42no—
156C23×C3—41no—
157C23×C3—42no—
158C23×C3C2×C22⋊C411yescharacteristic
159C23×C3#12011yescharacteristic
160C23×C3#12311yescharacteristic
161C3×Q8—24no—
162C3×Q8—42no—
163C3×D4—24no—
164C3×D4—42no—
165C3×D4—44no—
166C3×D4—44no—
167C2×C3×C4—24no—
168C2×C3×C4—22no—
169C2×C3×C4—22no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—41no—
174C2×C3×C4—41no—
175C2×C3×C4—42no—
176C2×C3×C4—48no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—41no—
180C2×C3×C4—41no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—81no—
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12112yes—
189C2×Dic3—44no—
190C2×Dic3—44no—
191C2×Dic3—44no—
192C3⋊C8—82no—
193C24—21no—
194C24C2×#6711yescharacteristic
195#41—28no—
196#41—44no—
197#41—44no—
198#41—44no—
199C2×Q8—22no—
200C2×Q8—42no—
201C2×D4—24no—
202C2×D4—22no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—64no—
220C22×C4#21712yes—
221C22×C4#21712yes—
222C22×C4D3×D412yes—
223C22×C4#23712yes—
224C22×C4C22×Dic311yescharacteristic
225C22×C4C2×#6711yescharacteristic
226M16—124no—
227C2×C8—122no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—44no—
231C4⋊C4—242no—
232C22⋊C4—44no—
233C22⋊C4—44no—
234C22⋊C4—42no—
235C22⋊C4—124no—
236C42—124no—
237C22×C3—21no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—21no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—44no—
245C22×C3—44no—
246C22×C3—42no—
247C22×C3—82no—
248C22×C3—81no—
249C22×C3—81no—
250C22×C3#38611yescharacteristic
251C22×C3#42112yes—
252C22×C3B(4,𝔽2)12yes—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—42no—
256C3×C4—42no—
257C3×C4—42no—
258C3×C4—82no—
259Dic3—84no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—41no—
267C23—42no—
268C23#93511yescharacteristic
269C23#93711yescharacteristic
270C23C2×#21711yescharacteristic
271Q8—24no—
272Q8—42no—
273D4—24no—
274D4—42no—
275D4—44no—
276D4—44no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—48no—
287C2×C4—41no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—81no—
297C2×C4—124no—
298C2×C4—124no—
299C2×C4—124no—
300C2×C4D4×Dic312yes—
301C2×C4#93912yes—
302C8—242no—
303C2×C3—22no—
304C2×C3—22no—
305C2×C3—41no—
306C2×C3—82no—
307C2×C3#188911yescharacteristic
308C2×C3#189711yescharacteristic
309C2×C3#191211yescharacteristic
310C22—21no—
311C22—21no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—44no—
318C22—44no—
319C22—42no—
320C22—82no—
321C22—81no—
322C22—81no—
323C22#503212yes—
324C22#522011yescharacteristic
325C22#523612yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—42no—
330C4—42no—
331C4—82no—
332C4—244no—
333C3#948011yescharacteristic, complemented, Sylow
334C2—22no—
335C2—22no—
336C2—41no—
337C2—82no—
338C2#7166911yescharacteristic
339C2#7171211yescharacteristic
340C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2406C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#2406D3
hypocenterC3#9480
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
4. #71552C2semidirect
5. #67628C2semidirect
10. C3×#395C4semidirect
11. C3×#394C4semidirect
64. #395Dic3semidirect
65. #394Dic3semidirect
333. C3#9480semidirect

Maximal subgroups

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

subgroupindex
C3×#24062
#821552
#721382
#715522
#676282
#94803