Finite Group Databasebeta

← #10715596

Subgroups — #10715596

Order 768.

336 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 67 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#83393C211yesmaximal, characteristic
2C3×#2127C211yesmaximal, characteristic
3#71861C211yesmaximal, characteristic
4#67540C212yesmaximal, complemented
5#67539C212yesmaximal, complemented
6#13610—31nomaximal, Sylow
7C3×#535C2211yescharacteristic
8#5518C411yescharacteristic, complemented
9C2×C3×#99—21no—
10#5240—21no—
11#5239—21no—
12#5218—21no—
13#5076C2211yescharacteristic
14#5075—21no—
15#5071C2211yescharacteristic
16#5060C411yescharacteristic, complemented
17#5056C411yescharacteristic, complemented
18#5054—21no—
19#5000C411yescharacteristic, complemented
20C3×#354—21no—
21C3×#353C2212yes—
22#4545—21no—
23#4545C2212yes—
24#3222—31no—
25#2128—31no—
26#2127D311yescharacteristic
27#1412—32no—
28#1409—32no—
29C2×C3×#41D411yescharacteristic
30C22×C3×D4D411yescharacteristic
31C3×#127D411yescharacteristic
32C3×#127C2311yescharacteristic
33C3×#124C2×C411yescharacteristic
34C3×#124D411yescharacteristic
35C3×#121—41no—
36C3×#120—41no—
37C3×#117C2×C411yescharacteristic
38C2×C3×C22⋊C4—41no—
39C2×#216—21no—
40C2×#214—21no—
41C2×#214—21no—
42C2×#214—21no—
43#926—41no—
44#924—21no—
45#889—41no—
46#879—41no—
47#870C2×C411yescharacteristic, complemented
48#869C2×C411yescharacteristic, complemented
49C4×Dic6—21no—
50C3×#99—22no—
51C3×#99—41no—
52#835—21no—
53#834—22no—
54#834—41no—
55#832—21no—
56#832—21no—
57#832—21no—
58#830—21no—
59C4⋊(C3⋊C8)—21no—
60#803C2×C411yescharacteristic, complemented
61C4×C3⋊C8C2×C411yescharacteristic, complemented
62#541—31no—
63#535C2×D311yescharacteristic
64#497—31no—
65#492—31no—
66#490—31no—
67#488—31no—
68#464—61no—
69#460—61no—
70#452—61no—
71#450—61no—
72#433—31no—
73#410—61no—
74C2×#99—21no—
75#354—21no—
76#353—32no—
77#353—61no—
78#353C2×D312yes—
79C24×C3—21no—
80C3×#41—81no—
81C2×C3×Q8C22⋊C411yescharacteristic
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—41no—
85C2×C3×D4—41no—
86C2×C3×D4—81no—
87C2×C3×D4C22⋊C411yescharacteristic
88C2×C3×D4C2×D411yescharacteristic
89C2×C3×D4C2×D411yescharacteristic
90C22×C3×C4—21no—
91C22×C3×C4—81no—
92C22×C3×C4C22⋊C411yescharacteristic
93C2×Dic6—21no—
94C3×C4⋊C4—21no—
95C3×C22⋊C4—21no—
96C3×C22⋊C4—41no—
97C3×C22⋊C4—41no—
98C3×C22⋊C4—42no—
99C3×C22⋊C4—81no—
100C3×C42C22⋊C411yescharacteristic
101C3×C42C22×C411yescharacteristic
102#217—41no—
103#217—42no—
104#216—81no—
105#214—81no—
106#214—81no—
107#214—81no—
108C4⋊Dic3—21no—
109C4⋊Dic3—41no—
110C4⋊Dic3—81no—
111Dic3⋊C4—41no—
112Dic3⋊C4—41no—
113Dic3⋊C4—41no—
114C4×Dic3—41no—
115C4×Dic3—81no—
116C2×C3⋊C8—21no—
117C2×C3⋊C8—21no—
118C2×#41D1211yescharacteristic
119C22×D4#6711yescharacteristic
120C2×Q16—61no—
121C2×SD16—61no—
122C2×SD16—61no—
123C2×SD16—61no—
124C4⋊Q8—31no—
125#127D1211yescharacteristic
126#127C22×D311yescharacteristic
127#126—121no—
128#125—31no—
129#125—121no—
130#124C4×D311yescharacteristic
131#124#6711yescharacteristic
132C22⋊Q8—61no—
133#121—41no—
134#120—41no—
135C4×Q8—61no—
136#117—121no—
137#117C4×D311yescharacteristic
138C2×C22⋊C4—41no—
139C4⋊C8—61no—
140#103—61no—
141#102—61no—
142#102—61no—
143#102—61no—
144#99—22no—
145#99—41no—
146#99—62no—
147#99—121no—
148C22⋊C8—61no—
149#97—31no—
150C4×C8—31no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—41no—
155C23×C3—41no—
156C23×C3—161no—
157C23×C3#9911yescharacteristic
158C3×Q8—41no—
159C3×D4—41no—
160C3×D4—41no—
161C3×D4—41no—
162C3×D4—81no—
163C3×D4—81no—
164C3×D4—161no—
165C3×D4—161no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—41no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—82no—
173C2×C3×C4—162no—
174C2×C3×C4#9911yescharacteristic
175C2×C3×C4C2×C22⋊C411yescharacteristic
176C2×Dic3—41no—
177C2×Dic3—41no—
178C2×Dic3—82no—
179Dic6—41no—
180Dic6—41no—
181C3⋊C8—41no—
182C3⋊C8—41no—
183C24—21no—
184#41—81no—
185C2×Q8—61no—
186C2×Q8(C2×D3)⋊C411yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—81no—
192C2×D4(C2×D3)⋊C411yescharacteristic
193C2×D4C2×D1211yescharacteristic
194C2×D4C2×#6711yescharacteristic
195C22×C4—21no—
196C22×C4—81no—
197C22×C4(C2×D3)⋊C411yescharacteristic
198Q16—241no—
199SD16—241no—
200SD16—241no—
201SD16—241no—
202C2×C8—61no—
203C2×C8—61no—
204C4⋊C4—21no—
205C4⋊C4—61no—
206C4⋊C4—121no—
207C4⋊C4—121no—
208C4⋊C4—121no—
209C4⋊C4—121no—
210C4⋊C4—241no—
211C22⋊C4—21no—
212C22⋊C4—41no—
213C22⋊C4—41no—
214C22⋊C4—42no—
215C22⋊C4—81no—
216C22⋊C4—121no—
217C22⋊C4—122no—
218C42—121no—
219C42—241no—
220C42(C2×D3)⋊C411yescharacteristic
221C42C2×C4×D311yescharacteristic
222C22×C3—21no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—81no—
228C22×C3—81no—
229C22×C3—81no—
230C22×C3—82no—
231C22×C3—161no—
232C22×C3C2×#9911yescharacteristic
233C3×C4—21no—
234C3×C4—21no—
235C3×C4—41no—
236C3×C4—41no—
237C3×C4—41no—
238C3×C4—162no—
239Dic3—81no—
240Dic3—81no—
241Dic3—162no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—161no—
248C23#80611yescharacteristic
249Q8—41no—
250Q8—121no—
251Q8—121no—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—81no—
256D4—81no—
257D4—161no—
258D4—161no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—82no—
266C2×C4—121no—
267C2×C4—121no—
268C2×C4—162no—
269C2×C4—242no—
270C2×C4#80611yescharacteristic
271C2×C4C2×(C2×D3)⋊C411yescharacteristic
272C8—121no—
273C8—121no—
274C2×C3—21no—
275C2×C3—41no—
276C2×C3—81no—
277C2×C3—81no—
278C2×C3—81no—
279C2×C3#212711yescharacteristic
280C22—21no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—81no—
286C22—81no—
287C22—81no—
288C22—82no—
289C22—161no—
290C22C2×#80611yescharacteristic
291C4—21no—
292C4—21no—
293C4—41no—
294C4—41no—
295C4—41no—
296C4—162no—
297C4—241no—
298C4—241no—
299C4—482no—
300C3#1361011yescharacteristic, complemented, Sylow
301C2—21no—
302C2—41no—
303C2—81no—
304C2—81no—
305C2—81no—
306C2#7001911yescharacteristic
subgroupG/H
centerC2#70019
commutatorC3×C42C22×C4
Frattini#127C22×D3
FittingC3×#2127C2
socleC2×C3#2127
prime radicalC3×#535C22
cosocleC3×#127C23
hypercenter#2127D3
hypocenterC3#13610
LCSG ▷ C3×C42 ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22 ◁ C22×C4 ◁ #535 ◁ #2127

Complements

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

NH ≅ G/Ntype
4. #67540C2semidirect
5. #67539C2semidirect
8. #5518C4semidirect
16. #5060C4semidirect
17. #5056C4semidirect
19. #5000C4semidirect
47. #870C2×C4semidirect
48. #869C2×C4semidirect
60. #803C2×C4semidirect
61. C4×C3⋊C8C2×C4semidirect
300. C3#13610semidirect

Maximal subgroups

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

subgroupindex
#833932
C3×#21272
#718612
#675402
#675392
#136103