Finite Group Databasebeta

← #10708564

Subgroups — #10708564

Order 768.

318 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 53 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#80855C211yesmaximal, characteristic
2#71088C211yesmaximal, characteristic, complemented
3#69116C211yesmaximal, characteristic, complemented
4#68701C211yesmaximal, characteristic, complemented
5C3×#1339C211yesmaximal, characteristic, complemented
6#67532C211yesmaximal, characteristic, complemented
7#67481C211yesmaximal, characteristic, complemented
8#12307—31nomaximal, Sylow
9#5596—21no—
10C3×#469C2211yescharacteristic
11#5150—21no—
12#5135C2211yescharacteristic
13#5125C2211yescharacteristic
14#5118C2211yescharacteristic
15#5039—21no—
16#4870—21no—
17#4786—21no—
18#4761—21no—
19C3×#349C2211yescharacteristic
20C3×H(3,ℤ/4)C2211yescharacteristic
21#4563—21no—
22#4553C2211yescharacteristic
23#4533—21no—
24#4532—21no—
25#4516—21no—
26#4513—21no—
27#4487—21no—
28#3253—31no—
29#2238—31no—
30#2238—31no—
31#2102—31no—
32#1339—31no—
33#1339D311yescharacteristic, complemented
34#1327—31no—
35C2×#237—41no—
36C2×C3×D8D411yescharacteristic
37C2×C3×M16C2311yescharacteristic
38C3×#121—21no—
39C3×#117—21no—
40#940—21no—
41#939—41no—
42D4×Dic3—21no—
43C2×#213—21no—
44#930—21no—
45C2×C4×Dic3D411yescharacteristic
46C2×#208—21no—
47C2×#208D411yescharacteristic
48C2×D24D411yescharacteristic
49#893—21no—
50#888—21no—
51#885—41no—
52#67⋊C4—41no—
53C3×M32—21no—
54C3×#106D411yescharacteristic
55C3×#102—21no—
56#832—21no—
57#821—21no—
58#819—21no—
59#819—21no—
60#817D411yescharacteristic
61Dic3⋊C8—21no—
62Dic3⋊C8—21no—
63#812—21no—
64#809—21no—
65#808—21no—
66#532—61no—
67#476—61no—
68#476—61no—
69#469—31no—
70#469—61no—
71#469C2×D311yescharacteristic
72#459—61no—
73#459—61no—
74#430—31no—
75#423—31no—
76#365—61no—
77#365—61no—
78#361—61no—
79#361—61no—
80#349—31no—
81#349C2×D311yescharacteristic
82#344—61no—
83#339—61no—
84H(3,ℤ/4)C2×D311yescharacteristic
85C2×C3×D4—21no—
86C2×C3×D4—41no—
87C22×C3×C4C2×D411yescharacteristic
88C2×#67—41no—
89C2×#67—41no—
90C2×#67—41no—
91C22×Dic3—21no—
92C22×Dic3—41no—
93#237—84no—
94C2×D12—21no—
95C2×C4×D3—41no—
96C2×Dic6—41no—
97C3×D8—41no—
98C3×M16—41no—
99C2×C3×C8C2×D411yescharacteristic
100C2×C3×C8C2×D411yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4—41no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C42—41no—
106#217—41no—
107#213—81no—
108(C2×D3)⋊C4—41no—
109(C2×D3)⋊C4—41no—
110C4⋊Dic3—21no—
111Dic3⋊C4—41no—
112C4×Dic3—21no—
113C4×Dic3SD1611yescharacteristic
114C4×Dic3SD1611yescharacteristic
115#208—41no—
116#208—41no—
117#208—41no—
118C2×C3⋊C8—21no—
119C2×C3⋊C8—41no—
120D24—41no—
121C3×C16—41no—
122C3⋊C16—41no—
123C2×#41—121no—
124C2×D8—31no—
125C2×D8—61no—
126C2×D8D1211yescharacteristic
127C2×M16—31no—
128C2×M16—61no—
129C2×M16C22×D311yescharacteristic
130C4⋊Q8—61no—
131#127—61no—
132#124—121no—
133#121—21no—
134#121—61no—
135#121—121no—
136C4×D4—61no—
137C4×D4—61no—
138C4×D4—121no—
139#117—21no—
140C2×C42—31no—
141M32—21no—
142M32—61no—
143#108—61no—
144#108—61no—
145#106—31no—
146#106—61no—
147#106D1211yescharacteristic
148C4⋊C8—61no—
149C4⋊C8—61no—
150#102—21no—
151#102—61no—
152#102—61no—
153#102—61no—
154C23×C3—41no—
155C23×C3AGL(1,ℤ/8)11yescharacteristic
156C22×D3—41no—
157C3×D4—41no—
158C3×D4—41no—
159C3×D4—82no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4—41no—
163C2×C3×C4—81no—
164C2×C3×C4#12011yescharacteristic
165C2×C3×C4C2×SD1611yescharacteristic
166#67—82no—
167#67—82no—
168#67—82no—
169C2×Dic3—21no—
170C2×Dic3—21no—
171C2×Dic3—42no—
172C2×Dic3—41no—
173C2×Dic3—82no—
174D12—41no—
175D12—41no—
176C4×D3—82no—
177Dic6—82no—
178C3×C8—21no—
179C3×C8—21no—
180C3⋊C8—41no—
181C3⋊C8—81no—
182#41—244no—
183C2×Q8—121no—
184C2×D4—21no—
185C2×D4—41no—
186C2×D4—61no—
187C2×D4—121no—
188C2×D4—121no—
189C2×D4—121no—
190C22×C4—61no—
191C22×C4—121no—
192C22×C4—121no—
193C22×C4D3×D411yescharacteristic
194D8—41no—
195D8—121no—
196D8—241no—
197M16—41no—
198M16—121no—
199M16—121no—
200M16—121no—
201C2×C8—61no—
202C2×C8—121no—
203C2×C8C2×D1211yescharacteristic
204C2×C8D3×D411yescharacteristic
205C4⋊C4—21no—
206C4⋊C4—41no—
207C4⋊C4—61no—
208C4⋊C4—121no—
209C22⋊C4—41no—
210C22⋊C4—41no—
211C22⋊C4—121no—
212C22⋊C4—121no—
213C22⋊C4—121no—
214C42—31no—
215C42—31no—
216C42—41no—
217C42—61no—
218C16—41no—
219C16—121no—
220C22×C3—21no—
221C22×C3—41no—
222C22×C3—81no—
223C22×C3—81no—
224C22×C3—81no—
225C22×C3#45011yescharacteristic
226C2×D3—81no—
227C2×D3—81no—
228C2×D3—81no—
229C3×C4—21no—
230C3×C4—81no—
231C3×C4—161no—
232C3×C4#45411yescharacteristic
233C3×C4#45811yescharacteristic
234Dic3—22no—
235Dic3—41no—
236Dic3—81no—
237C23—41no—
238C23—121no—
239C23#91111yescharacteristic
240Q8—242no—
241D4—41no—
242D4—41no—
243D4—82no—
244D4—121no—
245D4—121no—
246D4—242no—
247D4—242no—
248D4—242no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—41no—
252C2×C4—61no—
253C2×C4—61no—
254C2×C4—81no—
255C2×C4—122no—
256C2×C4—121no—
257C2×C4—242no—
258C2×C4—242no—
259C2×C4#88211yescharacteristic
260C2×C4D3×SD1611yescharacteristic
261C8—21no—
262C8—21no—
263C8—121no—
264C8—241no—
265C2×C3—21no—
266C2×C3—41no—
267C2×C3—161no—
268C2×C3#161611yescharacteristic
269D3—161no—
270C22—21no—
271C22—41no—
272C22—81no—
273C22—81no—
274C22—81no—
275C22—241no—
276C22—241no—
277C22—241no—
278C22#477611yescharacteristic
279C4—21no—
280C4—62no—
281C4—81no—
282C4—121no—
283C4—161no—
284C4—241no—
285C4#474211yescharacteristic
286C4#482711yescharacteristic
287C3#1230711yescharacteristic, complemented, Sylow
288C2—21no—
289C2—41no—
290C2—161no—
291C2—481no—
292C2#6829911yescharacteristic
subgroupG/H
centerC2#68299
commutator, cosocleC2×C3×M16C23
FrattiniC2×M16C22×D3
FittingC3×#1339C2
socleC2×C3#1616
prime radical#80855C2
hypercenter#1339D3
hypocenterC3#12307
LCSG ▷ C2×C3×M16 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #1339

Complements

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

NH ≅ G/Ntype
2. #71088C2semidirect
3. #69116C2semidirect
4. #68701C2semidirect
5. C3×#1339C2semidirect
6. #67532C2semidirect
7. #67481C2semidirect
33. #1339D3semidirect
287. C3#12307semidirect

Maximal subgroups

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

subgroupindex
#808552
#710882
#691162
#687012
C3×#13392
#675322
#674812
#123073