Finite Group Databasebeta

← #10708779

Subgroups — #10708779

Order 768.

346 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 75 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×#3334C211yesmaximal, characteristic, complemented
2#71381C211yesmaximal, characteristic, complemented
3#70889C211yesmaximal, characteristic, complemented
4#70847C211yesmaximal, characteristic, complemented
5#67801C211yesmaximal, characteristic, complemented
6#67763C211yesmaximal, characteristic, complemented
7D4⋊(C3⋊C16)C211yesmaximal, characteristic, complemented
8#12340—31nomaximal, Sylow
9C3×#547—21no—
10C3×#500C2211yescharacteristic
11C3×#480—22no—
12C3×#474—21no—
13C3×#466—21no—
14C3×C4×D8C2211yescharacteristic
15C3×C8×D4C2211yescharacteristic
16C2×C3×#107—21no—
17C2×#826—21no—
18#5015—21no—
19#4695C2211yescharacteristic
20#4491—22no—
21#4491—21no—
22#4485—21no—
23C4⋊(C3⋊C16)C2211yescharacteristic
24C4×C3⋊C16C2211yescharacteristic
25#4458C2211yescharacteristic
26#3334D311yescharacteristic, complemented
27#2240—31no—
28#2206—31no—
29#1667—31no—
30#1364—31no—
31#1331—31no—
32D4⋊C16—31no—
33C2×C3×#41—41no—
34C2×C3×D8D411yescharacteristic
35C22×C3×C8—21no—
36C3×C4⋊Q8—21no—
37C3×#123—42no—
38C3×C22⋊Q8—41no—
39C3×#121—22no—
40C3×C4×D4—21no—
41C3×C4×D4D411yescharacteristic
42C2×C3×C4⋊C4—22no—
43C2×#213—41no—
44C2×D24—21no—
45C2×D24—21no—
46#874—21no—
47C3×#107—22no—
48C3×#107—21no—
49C3×#107D411yescharacteristic
50C3×#107D412yes—
51C3×C4⋊C8D411yescharacteristic
52C3×#102—22no—
53C3×#102—21no—
54C3×C22⋊C8—21no—
55C3×C4×C8C2311yescharacteristic
56#826—42no—
57C2×C3⋊C16—21no—
58C2×C3⋊C16—21no—
59#809—41no—
60C4⋊(C3⋊C8)—21no—
61#547—21no—
62C2×D16—61no—
63#500C2×D311yescharacteristic
64#493—31no—
65#480—22no—
66#474—21no—
67#466—21no—
68#459—61no—
69C4×D8C2×D311yescharacteristic
70C8×D4C2×D311yescharacteristic
71C2×#107—21no—
72C4⋊C16—31no—
73#357—61no—
74#357—62no—
75C4×C16—31no—
76#335—31no—
77#331—61no—
78C3×#41—82no—
79C3×#41—82no—
80C2×C3×Q8—41no—
81C2×C3×D4—21no—
82C2×C3×D4—42no—
83C2×C3×D4D811yescharacteristic
84C22×C3×C4—21no—
85C22×C3×C4—42no—
86C22×C3×C4—41no—
87C2×D12—41no—
88C2×D12—41no—
89C2×D12—41no—
90C3×D8—21no—
91C3×D8—21no—
92C2×C3×C8—21no—
93C2×C3×C8—41no—
94C2×C3×C8—41no—
95C2×C3×C8C2×D411yescharacteristic
96C2×C3×C8C2×D411yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—42no—
100C3×C4⋊C4—41no—
101C3×C4⋊C4D811yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—42no—
104C3×C22⋊C4—41no—
105C3×C42C2×D411yescharacteristic
106#213—82no—
107C2×C3⋊C8—41no—
108D24—42no—
109D24—41no—
110D24—41no—
111C3⋊C16—41no—
112C3⋊C16—41no—
113C2×#41—41no—
114C2×D8—61no—
115C2×D8—61no—
116C2×D8—121no—
117C2×D8#6711yescharacteristic
118C22×C8—21no—
119C4⋊Q8—21no—
120#127—61no—
121#123—42no—
122C22⋊Q8—41no—
123#121—22no—
124C4×D4—21no—
125C4×D4#6711yescharacteristic
126C2×C4⋊C4—22no—
127D16—122no—
128C2×C16—61no—
129C2×C16—61no—
130#107—22no—
131#107—21no—
132#107#6711yescharacteristic
133#107#6712yes—
134C4⋊C8—61no—
135C4⋊C8#6711yescharacteristic
136#102—22no—
137#102—21no—
138#102—121no—
139C22⋊C8—21no—
140C4×C8C22×D311yescharacteristic
141C23×C3—21no—
142C23×C3—41no—
143C22×D3—81no—
144C3×Q8—81no—
145C3×Q8—81no—
146C3×D4—21no—
147C3×D4—41no—
148C3×D4—41no—
149C3×D4—84no—
150C3×D4D1612yescomplemented
151C2×C3×C4—21no—
152C2×C3×C4—41no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—84no—
157C2×C3×C4—81no—
158C2×C3×C4—81no—
159C2×C3×C4#12011yescharacteristic
160C2×C3×C4C2×D811yescharacteristic
161C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
162D12—81no—
163D12—81no—
164D12—81no—
165D12—81no—
166D12—161no—
167C3×C8—21no—
168C3×C8—41no—
169C3×C8#13511yescharacteristic
170C3×C8AGL(1,ℤ/8)11yescharacteristic
171C3⋊C8—81no—
172#41—82no—
173#41—82no—
174C2×Q8—41no—
175C2×D4—21no—
176C2×D4—42no—
177C2×D4—121no—
178C2×D4—121no—
179C2×D4—121no—
180C2×D4#21311yescharacteristic
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—41no—
184D8—21no—
185D8—21no—
186D8—122no—
187D8—121no—
188D8—121no—
189D8—242no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—121no—
194C2×C8C2×#6711yescharacteristic
195C2×C8C2×#6711yescharacteristic
196C4⋊C4—21no—
197C4⋊C4—22no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4#21311yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—42no—
203C22⋊C4—41no—
204C42C2×#6711yescharacteristic
205C16—121no—
206C16—121no—
207C22×C3—22no—
208C22×C3—41no—
209C22×C3—41no—
210C22×C3—81no—
211C22×C3—81no—
212C22×C3—81no—
213C22×C3#44711yescharacteristic
214C2×D3—161no—
215C2×D3—161no—
216C2×D3—161no—
217C3×C4—21no—
218C3×C4—41no—
219C3×C4—81no—
220C3×C4—82no—
221C3×C4#44911yescharacteristic
222C3×C4#45411yescharacteristic
223C3×C4C2×D1611yescharacteristic
224C3×C4#50911yescharacteristic
225C23—21no—
226C23—41no—
227C23—241no—
228Q8—81no—
229Q8—81no—
230D4—21no—
231D4—41no—
232D4—41no—
233D4—84no—
234D4—241no—
235D4—241no—
236D4—241no—
237D4—241no—
238D4—481no—
239D4#82612yescomplemented
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—84no—
246C2×C4—81no—
247C2×C4—81no—
248C2×C4C2×#21311yescharacteristic
249C2×C4#93211yescharacteristic
250C2×C4#95311yescharacteristic
251C8—21no—
252C8—41no—
253C8—241no—
254C8#94911yescharacteristic
255C8#95011yescharacteristic
256C2×C3—42no—
257C2×C3—161no—
258C2×C3#162511yescharacteristic
259C2×C3#218411yescharacteristic
260C2×C3#219311yescharacteristic
261D3—321no—
262C22—22no—
263C22—41no—
264C22—41no—
265C22—81no—
266C22—81no—
267C22—81no—
268C22—481no—
269C22—481no—
270C22—481no—
271C22#521711yescharacteristic
272C4—21no—
273C4—41no—
274C4—81no—
275C4—82no—
276C4C2×#82611yescharacteristic
277C4#514711yescharacteristic
278C4#523811yescharacteristic
279C4#524511yescharacteristic
280C3#1234011yescharacteristic, complemented, Sylow
281C2—42no—
282C2—161no—
283C2—961no—
284C2#7191011yescharacteristic
285C2#7203511yescharacteristic
286C2#7212411yescharacteristic
subgroupG/H
centerC22#5217
commutator, cosocleC3×C4×C8C23
FrattiniC4×C8C22×D3
FittingC3×#3334C2
socleC22×C3#447
hypercenter#3334D3
hypocenterC3#12340
LCSG ▷ C3×C4×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. C3×#3334C2semidirect
2. #71381C2semidirect
3. #70889C2semidirect
4. #70847C2semidirect
5. #67801C2semidirect
6. #67763C2semidirect
7. D4⋊(C3⋊C16)C2semidirect
26. #3334D3semidirect
150. C3×D4D16semidirect
239. D4#826semidirect
280. C3#12340semidirect

Maximal subgroups

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

subgroupindex
C3×#33342
#713812
#708892
#708472
#678012
#677632
D4⋊(C3⋊C16)2
#123403