Finite Group Databasebeta

← #10692982

Subgroups — #10692982

Order 768.

576 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 123 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×#2486C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4#71553C212yesmaximal, complemented
5#67625C212yesmaximal, complemented
6#9474—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×C4×D4C2211yescharacteristic
9C22×C3×C4⋊C4C2211yescharacteristic
10C3×#398C412yescomplemented
11C3×#397C412yescomplemented
12C3×#389—21no—
13C3×#381—21no—
14C3×#380—21no—
15C3×#380C2211yescharacteristic
16C2×#834—21no—
17#5212C2212yes—
18#5207—21no—
19C2×#830C2212yes—
20#5011—24no—
21#4926—21no—
22#4529—24no—
23#4526—21no—
24#2973—31no—
25#2486D311yescharacteristic
26#1823—32no—
27#1799—31no—
28#1289—32no—
29C22×C3×D4D411yescharacteristic
30C23×C3×C4—21no—
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C3×#123—44no—
34C3×C4×D4—22no—
35C3×C4×D4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C412yes—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4D411yescharacteristic
47C2×C3×C42D411yescharacteristic
48C2×#217—22no—
49C2×C4⋊Dic3—22no—
50C2×#208—22no—
51C22×C3⋊C8D414yes—
52C3×#95—22no—
53C3×#95—22no—
54C3×#95C2×C412yes—
55#834—42no—
56C3⋊#95—22no—
57#830—22no—
58#830—22no—
59#830—22no—
60C4⋊(C3⋊C8)—22no—
61#803—22no—
62C2×#123—21no—
63C2×C4×D4C2×D311yescharacteristic
64C22×C4⋊C4C2×D311yescharacteristic
65#435—64no—
66#432—61no—
67C2×#99—61no—
68#407—61no—
69C2×C22⋊C8—32no—
70#398Dic312yescomplemented
71#397Dic312yescomplemented
72#389—21no—
73#381—21no—
74#380—21no—
75#380—32no—
76#380C2×D311yescharacteristic
77#342—61no—
78#340—64no—
79C24×C3C22⋊C411yescharacteristic
80C24×C3C2×D411yescharacteristic
81C2×C3×D4—22no—
82C2×C3×D4—21no—
83C2×C3×D4—22no—
84C2×C3×D4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—42no—
93C22×C3×C4—41no—
94C22×C3×C4—41no—
95C22×C3×C4C22⋊C411yescharacteristic
96C22×C3×C4C22⋊C411yescharacteristic
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D412yes—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4112yes—
102C22×Dic3—42no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—24no—
105C3×C4⋊C4—42no—
106C3×C4⋊C4—44no—
107C3×C4⋊C4—41no—
108C3×C4⋊C4—41no—
109C3×C22⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—44no—
112C3×C22⋊C4—42no—
113C3×C42—21no—
114C3×C42—21no—
115#217—42no—
116#217—42no—
117C4⋊Dic3—44no—
118#208—82no—
119C2×C3⋊C8—24no—
120C2×C3⋊C8—44no—
121C22×D4#6711yescharacteristic
122C23×C4—21no—
123C23×C4#6711yescharacteristic
124C23×C4C22×D311yescharacteristic
125C2×M16—62no—
126C22×C8—34no—
127#123—44no—
128C4×D4—22no—
129C4×D4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—62no—
136C2×C4⋊C4C2×Dic312yes—
137C2×C4⋊C4C2×Dic311yescharacteristic
138C2×C22⋊C4—22no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4C2×Dic311yescharacteristic
142C2×C22⋊C4#6711yescharacteristic
143C2×C42#6711yescharacteristic
144C4⋊C8—62no—
145#99—122no—
146C22⋊C8—62no—
147C22⋊C8—62no—
148C22⋊C8—62no—
149#97—62no—
150#95—22no—
151#95—22no—
152#95—62no—
153#95C2×Dic312yes—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3—21no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3C2×C22⋊C411yescharacteristic
162C23×C3#12112yes—
163C23×C3AGL(1,ℤ/8)12yes—
164C23×C3AGL(1,ℤ/8)11yescharacteristic
165C23×C3#13711yescharacteristic
166C3×D4—42no—
167C3×D4—42no—
168C2×C3×C4—24no—
169C2×C3×C4—22no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—41no—
176C2×C3×C4—42no—
177C2×C3×C4—42no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—42no—
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×C4—82no—
188C2×C3×C4—82no—
189C2×C3×C4C4×D412yes—
190C2×C3×C4#12011yescharacteristic
191C2×C3×C4#12311yescharacteristic
192C2×Dic3—82no—
193C2×Dic3—82no—
194C2×Dic3—82no—
195C3⋊C8—44no—
196C24#21711yescharacteristic
197C24C2×#6711yescharacteristic
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—122no—
213C22×C4#21711yescharacteristic
214C22×C4#21711yescharacteristic
215C22×C4#21711yescharacteristic
216C22×C4D3×D412yes—
217C22×C4#23712yes—
218C22×C4C22×Dic311yescharacteristic
219C22×C4C2×#6711yescharacteristic
220M16—242no—
221C2×C8—64no—
222C2×C8—124no—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—124no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—44no—
233C22⋊C4—42no—
234C22⋊C4—122no—
235C22⋊C4—122no—
236C42—21no—
237C42—21no—
238C22×C3—22no—
239C22×C3—21no—
240C22×C3—22no—
241C22×C3—21no—
242C22×C3—42no—
243C22×C3—41no—
244C22×C3—41no—
245C22×C3—42no—
246C22×C3—41no—
247C22×C3—41no—
248C22×C3—41no—
249C22×C3—81no—
250C22×C3#38611yescharacteristic
251C22×C3#41711yescharacteristic
252C22×C3#41811yescharacteristic
253C22×C3#46812yes—
254C22×C3#46912yes—
255C3×C4—21no—
256C3×C4—22no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—82no—
261C3×C4—82no—
262C3×C4#45812yes—
263Dic3—162no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#93212yes—
272C23#93912yes—
273C23#94911yescharacteristic
274C23#95111yescharacteristic
275C23C2×#21711yescharacteristic
276D4—42no—
277D4—42no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—81no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—242no—
300C2×C4—242no—
301C2×C4—242no—
302C2×C4D4×Dic312yes—
303C2×C4#93511yescharacteristic
304C2×C4#93711yescharacteristic
305C8—124no—
306C2×C3—22no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—22no—
310C2×C3—81no—
311C2×C3#189011yescharacteristic
312C2×C3#189511yescharacteristic
313C2×C3#190911yescharacteristic
314C22—21no—
315C22—22no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22#503912yes—
327C22#504212yes—
328C22#521511yescharacteristic
329C22#522011yescharacteristic
330C22#523211yescharacteristic
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—82no—
337C4—82no—
338C4—482no—
339C4#505312yes—
340C3#947411yescharacteristic, complemented, Sylow
341C2—21no—
342C2—22no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#7168211yescharacteristic
347C2#7170411yescharacteristic
348C2#7186511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2486C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2486D3
hypocenterC3#9474
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2486

Complements

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

NH ≅ G/Ntype
4. #71553C2semidirect
5. #67625C2semidirect
10. C3×#398C4semidirect
11. C3×#397C4semidirect
70. #398Dic3semidirect
71. #397Dic3semidirect
340. C3#9474semidirect

Maximal subgroups

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

subgroupindex
C3×#24862
#823022
#716192
#715532
#676252
#94743