Finite Group Databasebeta

← #10708481

Subgroups — #10708481

Order 768.

316 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 61 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#80748C211yesmaximal, characteristic
2#71079C211yesmaximal, characteristic, complemented
3#69096C211yesmaximal, characteristic
4#68839C211yesmaximal, characteristic, complemented
5C3×#1348C211yesmaximal, characteristic, complemented
6#67524C211yesmaximal, characteristic, complemented
7#67485C211yesmaximal, characteristic
8#12293—31nomaximal, Sylow
9#5624—21no—
10C3×#467C2211yescharacteristic
11#5185—21no—
12#5115C2211yescharacteristic
13C2×#818C2211yescharacteristic
14C2×Dic3⋊C8C2211yescharacteristic
15#5048—21no—
16#4984—21no—
17#4891—21no—
18#4819—21no—
19#4796—21no—
20C3×C22⋊C16C2211yescharacteristic
21C3×#340C2211yescharacteristic
22#4564—21no—
23#4549C2211yescharacteristic
24#4529—21no—
25#4509—21no—
26#4505—21no—
27#4489—21no—
28#3250—31no—
29#2236—31no—
30#2233—31no—
31#2087—31no—
32#1348D311yescharacteristic, complemented
33#1347—31no—
34#1322—31no—
35C2×C3×Q16D411yescharacteristic
36C22×C3×C8C2311yescharacteristic
37C3×C22⋊Q8—21no—
38C2×C3×C4⋊C4—21no—
39#947—21no—
40Q8×Dic3—21no—
41C2×#215—21no—
42#930—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—41no—
45C2×C4×Dic3D411yescharacteristic
46C22×C3⋊C8D411yescharacteristic
47C2×D24D411yescharacteristic
48#898—41no—
49#893—21no—
50#892—41no—
51#890—21no—
52#886—41no—
53#67⋊C4—41no—
54C2×C3×C16—21no—
55C3×#107D411yescharacteristic
56C3×#103—21no—
57#835—21no—
58C3⋊#95—41no—
59#821—21no—
60#818—21no—
61#818—21no—
62#818D411yescharacteristic
63Dic3⋊C8—21no—
64Dic3⋊C8—21no—
65C2×C3⋊C16—21no—
66#809—21no—
67#808—21no—
68#529—61no—
69#479—61no—
70#478—61no—
71#467C2×D311yescharacteristic
72#466—31no—
73#465—61no—
74#464—61no—
75#463—61no—
76C2×#107—31no—
77C2×C4⋊C8—31no—
78#389—61no—
79#367—61no—
80#366—61no—
81#358—61no—
82#357—61no—
83C22⋊C16—31no—
84C22⋊C16C2×D311yescharacteristic
85#340—61no—
86#340C2×D311yescharacteristic
87C2×C3×Q8—21no—
88C22×C3×C4—41no—
89C22×C3×C4C2×D411yescharacteristic
90C2×#67—41no—
91C22×Dic3—41no—
92C22×Dic3D811yescharacteristic
93C22×Dic3D811yescharacteristic
94C2×D12—21no—
95C2×C4×D3—41no—
96C3×Q16—41no—
97C2×C3×C8—41no—
98C2×C3×C8C2×D411yescharacteristic
99C2×C3×C8C2×D411yescharacteristic
100C3×C4⋊C4—22no—
101C3×C4⋊C4—21no—
102C3×C4⋊C4—41no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—41no—
105#215—81no—
106(C2×D3)⋊C4—41no—
107(C2×D3)⋊C4—41no—
108(C2×D3)⋊C4—41no—
109(C2×D3)⋊C4—41no—
110C4⋊Dic3—21no—
111C4⋊Dic3—41no—
112C4⋊Dic3—41no—
113C4⋊Dic3—41no—
114C4⋊Dic3—41no—
115C4⋊Dic3—82no—
116Dic3⋊C4—41no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—41no—
120C2×C3⋊C8—21no—
121C2×C3⋊C8—41no—
122D24—41no—
123C3×C16—41no—
124C3⋊C16—41no—
125C2×Q16D1211yescharacteristic
126C2×SD16—61no—
127C2×D8—31no—
128C22×C8—31no—
129C22×C8C22×D311yescharacteristic
130#126—121no—
131#125—61no—
132#124—61no—
133#123—121no—
134C22⋊Q8—21no—
135#121—61no—
136C4×Q8—61no—
137C4×D4—61no—
138C4×D4—121no—
139#117—121no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—61no—
142C2×C4⋊C4—121no—
143C2×C42—31no—
144C2×C16—21no—
145C2×C16—61no—
146#107—31no—
147#107—61no—
148#107—61no—
149#107D1211yescharacteristic
150#106—61no—
151C4⋊C8—61no—
152C4⋊C8—61no—
153#103—21no—
154#103—61no—
155#102—61no—
156#102—61no—
157#95—121no—
158C23×C3C2×D811yescharacteristic
159C22×D3—41no—
160C3×Q8—41no—
161C3×Q8—41no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—42no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—81no—
168C2×C3×C4—81no—
169C2×C3×C4#12011yescharacteristic
170C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
171#67—81no—
172#67—81no—
173C2×Dic3—21no—
174C2×Dic3—21no—
175C2×Dic3—21no—
176C2×Dic3—21no—
177C2×Dic3—41no—
178C2×Dic3—41no—
179C2×Dic3—81no—
180C2×Dic3—81no—
181C2×Dic3—81no—
182D12—41no—
183D12—41no—
184C4×D3—82no—
185C3×C8—21no—
186C3×C8—21no—
187C3⋊C8—41no—
188C2×Q8—21no—
189C2×D4—61no—
190C2×D4—121no—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—41no—
194C22×C4—121no—
195C22×C4—121no—
196C22×C4D3×D411yescharacteristic
197Q16—41no—
198SD16—241no—
199D8—121no—
200C2×C8—41no—
201C2×C8—61no—
202C2×C8—121no—
203C2×C8C2×D1211yescharacteristic
204C2×C8D3×D411yescharacteristic
205C4⋊C4—22no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—41no—
209C4⋊C4—61no—
210C4⋊C4—121no—
211C4⋊C4—121no—
212C4⋊C4—121no—
213C4⋊C4—121no—
214C4⋊C4—121no—
215C4⋊C4—242no—
216C22⋊C4—41no—
217C22⋊C4—121no—
218C22⋊C4—121no—
219C22⋊C4—121no—
220C22⋊C4—121no—
221C42—61no—
222C42—61no—
223C42—121no—
224C16—41no—
225C16—121no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3#44711yescharacteristic
229C22×C3#50811yescharacteristic
230C22×C3#51011yescharacteristic
231C2×D3—81no—
232C2×D3—81no—
233C2×D3—81no—
234C3×C4—21no—
235C3×C4—82no—
236C3×C4—81no—
237C3×C4—81no—
238C3×C4#45611yescharacteristic
239C3×C4#45811yescharacteristic
240Dic3—41no—
241Dic3—41no—
242Dic3—81no—
243Dic3—161no—
244C23—121no—
245C23D3×D811yescharacteristic
246Q8—41no—
247Q8—41no—
248D4—121no—
249D4—121no—
250D4—241no—
251D4—241no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—61no—
258C2×C4—61no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4—121no—
264C2×C4—121no—
265C2×C4—242no—
266C2×C4—241no—
267C2×C4—241no—
268C2×C4—241no—
269C2×C4#88211yescharacteristic
270C2×C4#91411yescharacteristic
271C8—21no—
272C8—21no—
273C8—121no—
274C2×C3—21no—
275C2×C3—21no—
276C2×C3#161011yescharacteristic
277C2×C3#218911yescharacteristic
278C2×C3#219011yescharacteristic
279D3—161no—
280C22—21no—
281C22—21no—
282C22—241no—
283C22—241no—
284C22—241no—
285C22#477311yescharacteristic
286C22#491611yescharacteristic
287C22#491811yescharacteristic
288C4—21no—
289C4—82no—
290C4—81no—
291C4—81no—
292C4—121no—
293C4—121no—
294C4—241no—
295C4—481no—
296C4#474211yescharacteristic
297C4#482611yescharacteristic
298C3#1229311yescharacteristic, complemented, Sylow
299C2—21no—
300C2—21no—
301C2—481no—
302C2#6829711yescharacteristic
303C2#6922611yescharacteristic
304C2#6925211yescharacteristic
subgroupG/H
centerC22#4773
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#1348C2
socleC22×C3#447
prime radical#5115C22
hypercenter#1348D3
hypocenterC3#12293
LCSG ▷ C22×C3×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
2. #71079C2semidirect
4. #68839C2semidirect
5. C3×#1348C2semidirect
6. #67524C2semidirect
32. #1348D3semidirect
298. C3#12293semidirect

Maximal subgroups

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

subgroupindex
#807482
#710792
#690962
#688392
C3×#13482
#675242
#674852
#122933