Finite Group Databasebeta

← #11713725

Subgroups — #11713725

Order 768.

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

normal subgroups only

This group has 554 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
1C2×#557⋊C3C211yesmaximal, characteristic, complemented
2C2×C4⋊GL(2,𝔽3)C211yesmaximal, characteristic
3#73065C212yesmaximal, complemented
4#73065C212yesmaximal, complemented
5#73063C211yesmaximal, characteristic, complemented
6#25944—31nomaximal, Sylow
7C22×GL(2,𝔽3)C2212yes—
8#557⋊C3—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×#859C2212yes—
13C4⋊GL(2,𝔽3)—22no—
14#4991—41nomaximal
15C2×#557D311yescharacteristic, complemented
16C2×#490—31no—
17C2×#460—31no—
18#2548—31no—
19#2025—32no—
20#2023—32no—
21#1989—31no—
22#1985—31no—
23#1880—32no—
24#1880—32no—
25#1763—31no—
26C22×D12—41no—
27C22×SL(2,𝔽3)C2311yescharacteristic
28C2×GL(2,𝔽3)—22no—
29C2×GL(2,𝔽3)—22no—
30C2×GL(2,𝔽3)—22no—
31C2×C3×C4⋊C4—41no—
32C2×(C2×D3)⋊C4—42no—
33C2×(C2×D3)⋊C4—42no—
34C4×SL(2,𝔽3)—21no—
35C4×SL(2,𝔽3)—21no—
36C4×SL(2,𝔽3)D412yes—
37#859—22no—
38C3⋊#95—41no—
39C22×SD16—32no—
40#557—22no—
41#557—62no—
42C2×C4⋊Q8—31no—
43C2×#127—31no—
44C2×#125—31no—
45C2×#125—31no—
46C2×#124—61no—
47C2×C4×Q8—31no—
48C2×C4×Q8C2×D311yescharacteristic
49C2×C4×Q8C2×D311yescharacteristic
50C2×C4×Q8C2×D311yescharacteristic
51#490—64no—
52#460—62no—
53#460—62no—
54C2×C4⋊C8—31no—
55C2×#103—32no—
56C2×#102—32no—
57C2×#102—32no—
58C2×#102—32no—
59C2×#97—31no—
60#394—61no—
61#394—61no—
62#390—32no—
63#390—32no—
64#390—61no—
65C4×C4⋊C4—31no—
66#340—32no—
67#340—32no—
68#340—32no—
69#336—32no—
70#336—32no—
71C23×D3—42no—
72C22×C3×C4—41no—
73C22×C3×C4—41no—
74C22×C3×C4—41no—
75C22×Dic3—42no—
76C2×D12—82no—
77C2×D12—84no—
78C2×SL(2,𝔽3)C2×D411yescharacteristic
79C2×SL(2,𝔽3)#4112yes—
80GL(2,𝔽3)—42no—
81C3×C4⋊C4—82no—
82(C2×D3)⋊C4—84no—
83(C2×D3)⋊C4—84no—
84C22×Q8—31no—
85C22×Q8C22×D311yescharacteristic
86C22×D4—32no—
87C22×D4—61no—
88C22×D4—61no—
89C2×SD16—62no—
90C2×SD16—62no—
91C2×SD16—62no—
92C2×SD16—62no—
93C2×SD16—62no—
94C2×SD16—62no—
95C22×C8—32no—
96C22×C8—32no—
97C4⋊Q8—62no—
98C4⋊Q8—62no—
99#127—62no—
100#127—62no—
101#125—34no—
102#125—62no—
103#125—62no—
104#125—62no—
105#124—124no—
106C4×Q8—21no—
107C4×Q8—21no—
108C4×Q8—32no—
109C4×Q8—62no—
110C4×Q8—62no—
111C4×Q8—61no—
112C4×Q8—61no—
113C4×Q8#6712yes—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—32no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4S411yescharacteristic
122C2×C22⋊C4—62no—
123C2×C22⋊C4—62no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4—62no—
126C2×C42—31no—
127C2×C42—31no—
128C2×C42—31no—
129C2×C42—61no—
130C4⋊C8—62no—
131#103—62no—
132#103—62no—
133#102—64no—
134#102—64no—
135#102—64no—
136#97—62no—
137#95—61no—
138C23×C3—41no—
139C22×D3—44no—
140C22×D3—82no—
141C22×D3—82no—
142C22×D3—82no—
143C22×D3—82no—
144C22×D3—82no—
145C22×D3—82no—
146C2×C3×C4—42no—
147C2×C3×C4—42no—
148C2×C3×C4—81no—
149C2×C3×C4—82no—
150C2×C3×C4—81no—
151C2×C3×C4—81no—
152C2×C3×C4—82no—
153C2×Dic3—82no—
154C2×Dic3—82no—
155C2×Dic3—82no—
156D12—88no—
157SL(2,𝔽3)#12411yescharacteristic, complemented
158C24—62no—
159C2×Q8—32no—
160C2×Q8—31no—
161C2×Q8—62no—
162C2×Q8—62no—
163C2×Q8—121no—
164C2×Q8#23912yes—
165C2×Q8C2×#6711yescharacteristic
166C2×D4—62no—
167C2×D4—62no—
168C2×D4—62no—
169C2×D4—62no—
170C2×D4—62no—
171C2×D4—62no—
172C2×D4—122no—
173C2×D4—122no—
174C2×D4—122no—
175C2×D4—122no—
176C2×D4—242no—
177C22×C4—31no—
178C22×C4—31no—
179C22×C4—31no—
180C22×C4—31no—
181C22×C4—62no—
182C22×C4C2×S411yescharacteristic
183C22×C4C2×S411yescharacteristic
184C22×C4C2×S411yescharacteristic
185SD16—122no—
186SD16—122no—
187SD16—122no—
188SD16—122no—
189C2×C8—62no—
190C2×C8—62no—
191C2×C8—62no—
192C2×C8—62no—
193C2×C8—62no—
194C2×C8—62no—
195C4⋊C4—22no—
196C4⋊C4—32no—
197C4⋊C4—32no—
198C4⋊C4—32no—
199C4⋊C4—62no—
200C4⋊C4—61no—
201C4⋊C4—61no—
202C4⋊C4—61no—
203C4⋊C4—64no—
204C4⋊C4—62no—
205C4⋊C4—62no—
206C4⋊C4—62no—
207C22⋊C4—124no—
208C22⋊C4—124no—
209C22⋊C4—242no—
210C22⋊C4—242no—
211C42—32no—
212C42—32no—
213C42—61no—
214C42—61no—
215C42—61no—
216C42—61no—
217C42—122no—
218C22×C3—42no—
219C22×C3—41no—
220C22×C3—41no—
221C22×C3—42no—
222C22×C3—41no—
223C2×D3—84no—
224C2×D3—84no—
225C2×D3—84no—
226C2×D3—162no—
227C2×D3—162no—
228C2×D3—162no—
229C2×D3—162no—
230C3×C4—82no—
231C3×C4—82no—
232C3×C4—161no—
233Dic3—162no—
234C23—122no—
235C23—122no—
236C23—122no—
237C23—122no—
238C23—122no—
239C23—122no—
240C23—122no—
241C23C22×S411yescharacteristic
242Q8—32no—
243Q8—31no—
244Q8—61no—
245Q8—124no—
246Q8#94711yescharacteristic
247D4—122no—
248D4—122no—
249D4—122no—
250D4—122no—
251D4—244no—
252D4—244no—
253C2×C4—21no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—32no—
259C2×C4—32no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—32no—
263C2×C4—61no—
264C2×C4—61no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—61no—
269C2×C4—61no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4#98612yes—
274C2×C4GL(2,ℤ/4)12yes—
275C8—122no—
276C8—122no—
277C2×C3—41no—
278C2×C3—41no—
279C2×C3—41no—
280C2×C3—42no—
281C2×C3—42no—
282D3—164no—
283C22—242no—
284C22—242no—
285C22—242no—
286C22—242no—
287C22—242no—
288C22—242no—
289C22—242no—
290C22#591912yes—
291C22C2×#98611yescharacteristic
292C22#592311yescharacteristic
293C22#592612yes—
294C22C2×GL(2,ℤ/4)11yescharacteristic
295C4—22no—
296C4—22no—
297C4—31no—
298C4—32no—
299C4—31no—
300C4—41no—
301C4—62no—
302C4—121no—
303C4—121no—
304C4—242no—
305C3—41noSylow
306C2—482no—
307C2#8542112yes—
308C2#8542212yes—
309C2#8550811yescharacteristic
310C2#8552311yescharacteristic
311C2#8553711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#557D3
prime radicalC2×C4⋊GL(2,𝔽3)C2
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#557⋊C3C2semidirect
3. #73065C2semidirect
4. #73065C2semidirect
5. #73063C2semidirect
15. C2×#557D3semidirect
157. SL(2,𝔽3)#124semidirect

Maximal subgroups

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

subgroupindex
C2×#557⋊C32
C2×C4⋊GL(2,𝔽3)2
#730652
#730632
#259443
#49914