Finite Group Databasebeta

← #10692977

Subgroups — #10692977

Order 768.

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

normal subgroups only

This group has 556 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×#2487C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3#71619C211yesmaximal, characteristic
4#71554C212yesmaximal, complemented
5#67625C212yesmaximal, complemented
6#9473—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C3×#400C412yescomplemented
10C3×#399C412yescomplemented
11C3×#389—21no—
12C3×#386—21no—
13C3×#385—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#2487D311yescharacteristic
26#1824—32no—
27#1799—31no—
28#1289—32no—
29C22×C3×D4D411yescharacteristic
30C23×C3×C4—21no—
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4C2×C412yes—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C22⋊C4—21no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4D411yescharacteristic
46C2×C3×C42D411yescharacteristic
47C2×#217—22no—
48C2×C4⋊Dic3—22no—
49C2×#208—22no—
50C22×C3⋊C8D414yes—
51C3×#95—22no—
52C3×#95—22no—
53C3×#95C2×C412yes—
54#834—42no—
55C3⋊#95—22no—
56#830—22no—
57#830—22no—
58#830—22no—
59C4⋊(C3⋊C8)—22no—
60#803—22no—
61C2×C4×D4C2×D311yescharacteristic
62C22×C4⋊C4C2×D311yescharacteristic
63#435—64no—
64#432—61no—
65C2×#99—61no—
66#407—61no—
67C2×C22⋊C8—32no—
68#400Dic312yescomplemented
69#399Dic312yescomplemented
70#389—21no—
71#386—21no—
72#385—21no—
73#380—21no—
74#380—32no—
75#380C2×D311yescharacteristic
76#342—61no—
77#340—64no—
78C24×C3C22⋊C411yescharacteristic
79C24×C3C2×D411yescharacteristic
80C2×C3×D4—22no—
81C2×C3×D4—21no—
82C2×C3×D4—22no—
83C2×C3×D4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—42no—
89C22×C3×C4—42no—
90C22×C3×C4—42no—
91C22×C3×C4—82no—
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22⋊C411yescharacteristic
94C22×C3×C4C22⋊C411yescharacteristic
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D412yes—
97C22×C3×C4C2×D411yescharacteristic
98C22×C3×C4#4112yes—
99C22×Dic3—42no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—24no—
102C3×C4⋊C4—42no—
103C3×C4⋊C4—41no—
104C3×C4⋊C4—41no—
105C3×C4⋊C4—82no—
106C3×C22⋊C4—22no—
107C3×C22⋊C4—22no—
108C3×C22⋊C4—41no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—82no—
111C3×C42—21no—
112C3×C42—21no—
113#217—42no—
114#217—42no—
115C4⋊Dic3—44no—
116#208—82no—
117C2×C3⋊C8—24no—
118C2×C3⋊C8—44no—
119C22×D4#6711yescharacteristic
120C23×C4—21no—
121C23×C4#6711yescharacteristic
122C23×C4C22×D311yescharacteristic
123C2×M16—62no—
124C22×C8—34no—
125C4×D4—22no—
126C4×D4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4C2×Dic312yes—
134C2×C4⋊C4C2×Dic311yescharacteristic
135C2×C22⋊C4—21no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—62no—
138C2×C22⋊C4C2×Dic311yescharacteristic
139C2×C22⋊C4#6711yescharacteristic
140C2×C42#6711yescharacteristic
141C4⋊C8—62no—
142#99—122no—
143C22⋊C8—62no—
144C22⋊C8—62no—
145C22⋊C8—62no—
146#97—62no—
147#95—22no—
148#95—22no—
149#95—62no—
150#95C2×Dic312yes—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—41no—
155C23×C3—41no—
156C23×C3—41no—
157C23×C3—41no—
158C23×C3C2×C22⋊C411yescharacteristic
159C23×C3#12112yes—
160C23×C3AGL(1,ℤ/8)11yescharacteristic
161C23×C3#13712yes—
162C23×C3#13711yescharacteristic
163C3×D4—42no—
164C3×D4—42no—
165C2×C3×C4—24no—
166C2×C3×C4—22no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—41no—
173C2×C3×C4—42no—
174C2×C3×C4—42no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—42no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—81no—
184C2×C3×C4—84no—
185C2×C3×C4C4×D412yes—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12311yescharacteristic
188C2×Dic3—82no—
189C2×Dic3—82no—
190C2×Dic3—82no—
191C3⋊C8—44no—
192C24#21711yescharacteristic
193C24C2×#6711yescharacteristic
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—21no—
197C2×D4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—82no—
206C22×C4—122no—
207C22×C4#21711yescharacteristic
208C22×C4#21711yescharacteristic
209C22×C4#21711yescharacteristic
210C22×C4D3×D412yes—
211C22×C4#23712yes—
212C22×C4C22×Dic311yescharacteristic
213C22×C4C2×#6711yescharacteristic
214M16—242no—
215C2×C8—64no—
216C2×C8—124no—
217C4⋊C4—22no—
218C4⋊C4—24no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—82no—
223C4⋊C4—124no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—41no—
227C22⋊C4—41no—
228C22⋊C4—82no—
229C22⋊C4—122no—
230C22⋊C4—122no—
231C42—21no—
232C42—21no—
233C22×C3—22no—
234C22×C3—21no—
235C22×C3—22no—
236C22×C3—21no—
237C22×C3—41no—
238C22×C3—42no—
239C22×C3—41no—
240C22×C3—42no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—41no—
244C22×C3—81no—
245C22×C3#38611yescharacteristic
246C22×C3#41711yescharacteristic
247C22×C3#41911yescharacteristic
248C22×C3#46812yes—
249C22×C3#47012yes—
250C3×C4—21no—
251C3×C4—22no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—41no—
255C3×C4—82no—
256C3×C4—82no—
257C3×C4#45812yes—
258Dic3—162no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#93912yes—
267C23#94212yes—
268C23#94911yescharacteristic
269C23#95111yescharacteristic
270C23C2×#21711yescharacteristic
271D4—42no—
272D4—42no—
273C2×C4—22no—
274C2×C4—24no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—81no—
292C2×C4—84no—
293C2×C4—242no—
294C2×C4—242no—
295C2×C4—242no—
296C2×C4D4×Dic312yes—
297C2×C4#93511yescharacteristic
298C2×C4#93711yescharacteristic
299C8—124no—
300C2×C3—22no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—22no—
304C2×C3—81no—
305C2×C3#189011yescharacteristic
306C2×C3#189411yescharacteristic
307C2×C3#191011yescharacteristic
308C22—21no—
309C22—22no—
310C22—21no—
311C22—22no—
312C22—42no—
313C22—41no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22#504912yes—
321C22#505212yes—
322C22#522011yescharacteristic
323C22#522511yescharacteristic
324C22#523211yescharacteristic
325C4—21no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—82no—
331C4—82no—
332C4—482no—
333C4#505312yes—
334C3#947311yescharacteristic, complemented, Sylow
335C2—21no—
336C2—22no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#7169711yescharacteristic
341C2#7170411yescharacteristic
342C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2487C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2487D3
hypocenterC3#9473
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2487

Complements

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

NH ≅ G/Ntype
4. #71554C2semidirect
5. #67625C2semidirect
9. C3×#400C4semidirect
10. C3×#399C4semidirect
68. #400Dic3semidirect
69. #399Dic3semidirect
334. C3#9473semidirect

Maximal subgroups

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

subgroupindex
C3×#24872
#823022
#716192
#715542
#676252
#94733