Finite Group Databasebeta

← #10708482

Subgroups — #10708482

Order 768.

302 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 57 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#80706C211yesmaximal, characteristic
2C3×#2233C211yesmaximal, characteristic, complemented
3#70626C211yesmaximal, characteristic, complemented
4#70599C211yesmaximal, characteristic, complemented
5#67775C211yesmaximal, characteristic, complemented
6#67527C211yesmaximal, characteristic, complemented
7#67525C211yesmaximal, characteristic, complemented
8#12294—31nomaximal, Sylow
9#5619—21no—
10C3×#466C2211yescharacteristic
11C2×C3×#107C2211yescharacteristic
12#5150—21no—
13#5115C2211yescharacteristic
14#5102C2211yescharacteristic
15#5039—21no—
16#4964—21no—
17#4891—21no—
18#4786—21no—
19#4766—21no—
20C3×#367—21no—
21C3×#357—21no—
22C3×C22⋊C16C2211yescharacteristic
23#4552C2211yescharacteristic
24#4549C2211yescharacteristic
25#4530—21no—
26#4491—21no—
27#4489—21no—
28#3259—31no—
29#2233D311yescharacteristic, complemented
30#2232—31no—
31#2089—31no—
32#1349—31no—
33#1349—31no—
34#1323—31no—
35C2×C3×D8D411yescharacteristic
36C22×C3×C8C2311yescharacteristic
37C3×#121—21no—
38C2×C3×C4⋊C4—21no—
39#940—21no—
40#939—41no—
41#938—41no—
42D4×Dic3—21no—
43C2×#213—21no—
44#930—21no—
45#926D411yescharacteristic
46C2×#208—21no—
47C2×#208D411yescharacteristic
48C2×D24D411yescharacteristic
49#893—21no—
50#890—21no—
51#886—41no—
52#883—41no—
53C2×C3×C16—21no—
54C3×#107—21no—
55C3×#107—21no—
56C3×#107D411yescharacteristic
57C3×#102—21no—
58#832—21no—
59#821—21no—
60#818D411yescharacteristic
61Dic3⋊C8—21no—
62Dic3⋊C8—21no—
63C2×C3⋊C16—21no—
64#809—21no—
65#808—21no—
66#808—41no—
67#807—41no—
68#540—61no—
69#478—61no—
70#478—61no—
71#469—61no—
72#466—31no—
73#466C2×D311yescharacteristic
74#459—61no—
75#459—61no—
76#428—61no—
77C2×#107C2×D311yescharacteristic
78#424—31no—
79#367—21no—
80#366—61no—
81#357—21no—
82#357—61no—
83C22⋊C16—31no—
84C22⋊C16C2×D311yescharacteristic
85#343—61no—
86#341—31no—
87C2×C3×D4—21no—
88C2×C3×D4—41no—
89C22×C3×C4—41no—
90C22×C3×C4C2×D411yescharacteristic
91C2×#67—41no—
92C2×#67—41no—
93C2×#67—41no—
94C22×Dic3—41no—
95C2×D12—21no—
96C2×C4×D3—41no—
97C3×D8—41no—
98C2×C3×C8—41no—
99C2×C3×C8C2×D411yescharacteristic
100C2×C3×C8C2×D411yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—41no—
105#217—21no—
106#217—41no—
107#213—81no—
108(C2×D3)⋊C4—41no—
109(C2×D3)⋊C4—41no—
110C4⋊Dic3—21no—
111C4⋊Dic3—41no—
112C4⋊Dic3D811yescharacteristic
113C4⋊Dic3D811yescharacteristic
114Dic3⋊C4—41no—
115C4×Dic3—21no—
116#208—22no—
117#208—41no—
118#208—41no—
119C2×C3⋊C8—21no—
120C2×C3⋊C8—41no—
121D24—41no—
122C3×C16—41no—
123C3⋊C16—41no—
124C2×D8—31no—
125C2×D8—61no—
126C2×D8#6711yescharacteristic
127C2×M16—31no—
128C2×M16—61no—
129C22×C8C22×D311yescharacteristic
130#127—61no—
131#125—61no—
132#123—121no—
133#123—121no—
134#121—21no—
135#121—61no—
136#121—121no—
137#121—121no—
138C4×D4—61no—
139C4×D4—61no—
140#117—31no—
141C2×C4⋊C4—21no—
142C2×C16—21no—
143C2×C16—61no—
144#107—21no—
145#107—21no—
146#107—31no—
147#107—121no—
148#107#6711yescharacteristic
149#106—61no—
150#106—121no—
151C4⋊C8—61no—
152C4⋊C8—61no—
153#102—21no—
154#102—61no—
155#102—61no—
156#102—61no—
157C23×C3—41no—
158C23×C3AGL(1,ℤ/8)11yescharacteristic
159C22×D3—41no—
160C3×D4—41no—
161C3×D4—41no—
162C3×D4—161no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—41no—
166C2×C3×C4—81no—
167C2×C3×C4—161no—
168C2×C3×C4#12011yescharacteristic
169C2×C3×C4C2×D811yescharacteristic
170#67—161no—
171#67—161no—
172#67—161no—
173C2×Dic3—21no—
174C2×Dic3—21no—
175C2×Dic3—41no—
176C2×Dic3—161no—
177D12—41no—
178D12—41no—
179C4×D3—161no—
180C3×C8—21no—
181C3×C8—21no—
182C3⋊C8—42no—
183C3⋊C8—41no—
184C2×D4—21no—
185C2×D4—41no—
186C2×D4—61no—
187C2×D4—121no—
188C2×D4—121no—
189C2×D4—121no—
190C22×C4—41no—
191C22×C4—121no—
192C22×C4—121no—
193C22×C4D3×D411yescharacteristic
194D8—41no—
195D8—121no—
196D8—241no—
197M16—62no—
198M16—121no—
199M16—121no—
200C2×C8—41no—
201C2×C8—61no—
202C2×C8—121no—
203C2×C8D3×D411yescharacteristic
204C2×C8C2×#6711yescharacteristic
205C4⋊C4—21no—
206C4⋊C4—31no—
207C4⋊C4—31no—
208C4⋊C4—41no—
209C4⋊C4—41no—
210C4⋊C4—61no—
211C4⋊C4—121no—
212C4⋊C4—121no—
213C22⋊C4—41no—
214C22⋊C4—61no—
215C22⋊C4—121no—
216C22⋊C4—121no—
217C22⋊C4—121no—
218C42—61no—
219C16—41no—
220C16—121no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3—81no—
225C22×C3—81no—
226C22×C3—81no—
227C22×C3#44711yescharacteristic
228C2×D3—81no—
229C2×D3—81no—
230C2×D3—81no—
231C3×C4—21no—
232C3×C4—81no—
233C3×C4—161no—
234C3×C4#45611yescharacteristic
235C3×C4#45811yescharacteristic
236Dic3—41no—
237Dic3—41no—
238Dic3—81no—
239C23—41no—
240C23—121no—
241C23#91111yescharacteristic
242D4—41no—
243D4—41no—
244D4—121no—
245D4—121no—
246D4—161no—
247D4—481no—
248D4—481no—
249D4—481no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—81no—
256C2×C4—121no—
257C2×C4—161no—
258C2×C4—481no—
259C2×C4—481no—
260C2×C4D3×D811yescharacteristic
261C2×C4#93711yescharacteristic
262C8—21no—
263C8—21no—
264C8—122no—
265C8—121no—
266C2×C3—41no—
267C2×C3—161no—
268C2×C3#161011yescharacteristic
269C2×C3#219211yescharacteristic
270C2×C3#219411yescharacteristic
271D3—161no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—81no—
276C22—81no—
277C22—81no—
278C22—241no—
279C22—241no—
280C22—241no—
281C22#515611yescharacteristic
282C4—21no—
283C4—81no—
284C4—121no—
285C4—121no—
286C4—161no—
287C4—241no—
288C4#505311yescharacteristic
289C4#520511yescharacteristic
290C3#1229411yescharacteristic, complemented, Sylow
291C2—41no—
292C2—161no—
293C2—481no—
294C2#7116711yescharacteristic
295C2#7149011yescharacteristic
296C2#7151711yescharacteristic
subgroupG/H
centerC22#5156
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#2233C2
socleC22×C3#447
prime radical#80706C2
hypercenter#2233D3
hypocenterC3#12294
LCSG ▷ C22×C3×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2233

Complements

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

NH ≅ G/Ntype
2. C3×#2233C2semidirect
3. #70626C2semidirect
4. #70599C2semidirect
5. #67775C2semidirect
6. #67527C2semidirect
7. #67525C2semidirect
29. #2233D3semidirect
290. C3#12294semidirect

Maximal subgroups

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

subgroupindex
#807062
C3×#22332
#706262
#705992
#677752
#675272
#675252
#122943