Finite Group Databasebeta

← #11713702

Subgroups — #11713702

Order 768.

574 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 59 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×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#5389C211yesmaximal, characteristic, complemented
3#73148C212yesmaximal, complemented
4#73148C212yesmaximal, complemented
5#73063C211yesmaximal, characteristic, complemented
6#25894—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2212yes—
8D4×SL(2,𝔽3)—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×#859C2211yescharacteristic
11C2×#859C2211yescharacteristic
12C2×#859C2212yes—
13#5389—22no—
14#5222—41nomaximal
15C2×D4×Q8D311yescharacteristic
16C2×#491—31no—
17C2×#479—31no—
18#2562—31no—
19#2036—32no—
20#2035—32no—
21#1981—31no—
22#1955—31no—
23#1787—32no—
24#1787—32no—
25#1763—31no—
26C22×C3×D4—41no—
27C22×SL(2,𝔽3)—22no—
28C22×SL(2,𝔽3)—22no—
29C22×SL(2,𝔽3)—22no—
30C22×SL(2,𝔽3)C2311yescharacteristic
31C2×#217—42no—
32C2×#217—42no—
33C2×Dic3⋊C4—41no—
34C4×SL(2,𝔽3)D412yes—
35#859—21no—
36#859—21no—
37#859—22no—
38C3⋊#95—41no—
39C23×Q8C2×D312yes—
40D4×Q8—22no—
41D4×Q8—62no—
42C2×C4⋊Q8—31no—
43C2×#125—31no—
44C2×C22⋊Q8—32no—
45C2×C4×Q8C2×D311yescharacteristic
46C2×C4×D4—31no—
47#491—64no—
48#479—62no—
49#479—62no—
50C2×#107—31no—
51C2×#106—31no—
52C2×C4⋊C8—31no—
53C2×#103—32no—
54C2×#103—32no—
55C2×#103—31no—
56C2×#103—31no—
57C2×#103—32no—
58C2×#103—32no—
59C2×#103—32no—
60C2×C22⋊C8—32no—
61C2×C22⋊C8—32no—
62C2×#97—31no—
63#400—61no—
64#394—61no—
65#388—62no—
66#380—32no—
67#380—32no—
68C4×C4⋊C4—31no—
69#340—32no—
70C24×C3—42no—
71C2×C3×D4—82no—
72C2×C3×D4—84no—
73C22×C3×C4—41no—
74C22×Dic3—41no—
75C22×Dic3—42no—
76C22×Dic3—41no—
77C2×SL(2,𝔽3)—42no—
78C2×SL(2,𝔽3)C2×D411yescharacteristic
79C2×SL(2,𝔽3)#4112yes—
80#217—84no—
81#217—84no—
82Dic3⋊C4—82no—
83C22×Q8—22no—
84C22×Q8—22no—
85C22×Q8—22no—
86C22×Q8—32no—
87C22×Q8—62no—
88C22×Q8—62no—
89C22×Q8—62no—
90C22×Q8C22×D311yescharacteristic
91C22×D4S411yescharacteristic
92C23×C4—32no—
93C22×C8—32no—
94C22×C8—31no—
95C22×C8—31no—
96C4⋊Q8—34no—
97C4⋊Q8—62no—
98#125—62no—
99#125—62no—
100C22⋊Q8—64no—
101C22⋊Q8—64no—
102C4×Q8—32no—
103C4×Q8#6712yes—
104C4×D4—62no—
105C4×D4—62no—
106C2×C4⋊C4—31no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—31no—
110C2×C4⋊C4—32no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4—61no—
113C2×C4⋊C4—61no—
114C2×C22⋊C4—32no—
115C2×C22⋊C4—62no—
116C2×C22⋊C4—62no—
117C2×C42—31no—
118C2×C42—61no—
119#107—62no—
120#106—62no—
121C4⋊C8—62no—
122#103—62no—
123#103—61no—
124#103—61no—
125#103—64no—
126#103—64no—
127#103—62no—
128#103—61no—
129#103—61no—
130#103—122no—
131#103—122no—
132C22⋊C8—122no—
133C22⋊C8—122no—
134#97—62no—
135#95—62no—
136#95—62no—
137#95—61no—
138C23×C3—44no—
139C23×C3—41no—
140C23×C3—82no—
141C23×C3—82no—
142C23×C3—82no—
143C23×C3—82no—
144C23×C3—82no—
145C23×C3—82no—
146C3×D4—88no—
147C2×C3×C4—42no—
148C2×C3×C4—82no—
149C2×Dic3—42no—
150C2×Dic3—81no—
151C2×Dic3—81no—
152C2×Dic3—81no—
153C2×Dic3—82no—
154C2×Dic3—82no—
155C2×Dic3—82no—
156C2×Dic3—82no—
157SL(2,𝔽3)#12411yescharacteristic
158C24C2×S412yes—
159C2×Q8—31no—
160C2×Q8—32no—
161C2×Q8—34no—
162C2×Q8—42no—
163C2×Q8—62no—
164C2×Q8—62no—
165C2×Q8—62no—
166C2×Q8—62no—
167C2×Q8—62no—
168C2×Q8—124no—
169C2×Q8—122no—
170C2×Q8—122no—
171C2×Q8—122no—
172C2×Q8#23712yes—
173C2×Q8C2×#6711yescharacteristic
174C2×D4—24no—
175C2×D4—22no—
176C22×C4—31no—
177C22×C4—31no—
178C22×C4—32no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4—62no—
182C22×C4—62no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4—61no—
187C22×C4—61no—
188C22×C4C2×S411yescharacteristic
189C2×C8—62no—
190C2×C8—62no—
191C2×C8—62no—
192C2×C8—62no—
193C2×C8—61no—
194C2×C8—61no—
195C2×C8—62no—
196C4⋊C4—32no—
197C4⋊C4—61no—
198C4⋊C4—64no—
199C4⋊C4—61no—
200C4⋊C4—61no—
201C4⋊C4—61no—
202C4⋊C4—61no—
203C4⋊C4—62no—
204C4⋊C4—62no—
205C4⋊C4—62no—
206C4⋊C4—122no—
207C4⋊C4—122no—
208C22⋊C4—64no—
209C22⋊C4—124no—
210C22⋊C4—242no—
211C42—32no—
212C42—32no—
213C42—241no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3—41no—
219C22×C3—84no—
220C22×C3—84no—
221C22×C3—84no—
222C22×C3—162no—
223C22×C3—162no—
224C22×C3—162no—
225C22×C3—162no—
226C3×C4—82no—
227Dic3—82no—
228Dic3—161no—
229Dic3—162no—
230C23—22no—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—22no—
235C23—22no—
236C23#98314yes—
237C23C22×S411yescharacteristic
238Q8—32no—
239Q8—31no—
240Q8—61no—
241Q8—64no—
242Q8—64no—
243Q8—122no—
244Q8—122no—
245Q8#93611yescharacteristic
246D4—28no—
247C2×C4—22no—
248C2×C4—32no—
249C2×C4—32no—
250C2×C4—31no—
251C2×C4—31no—
252C2×C4—32no—
253C2×C4—61no—
254C2×C4—61no—
255C2×C4—62no—
256C2×C4—62no—
257C2×C4—62no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4—122no—
261C2×C4—122no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—122no—
265C2×C4—122no—
266C2×C4—121no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—121no—
270C2×C4GL(2,ℤ/4)12yes—
271C8—121no—
272C8—121no—
273C8—122no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—42no—
277C2×C3—42no—
278C2×C3—41no—
279C2×C3—164no—
280C22—24no—
281C22—24no—
282C22—24no—
283C22—42no—
284C22—42no—
285C22—42no—
286C22—42no—
287C22#591412yes—
288C22C2×#98312yes—
289C22#592611yescharacteristic
290C22#592711yescharacteristic
291C22C2×GL(2,ℤ/4)11yescharacteristic
292C4—22no—
293C4—31no—
294C4—32no—
295C4—31no—
296C4—62no—
297C4—122no—
298C4—241no—
299C4—241no—
300C4—242no—
301C3—41noSylow
302C2—44no—
303C2#8539412yes—
304C2#8540112yes—
305C2#8549211yescharacteristic
306C2#8550711yescharacteristic
307C2#8553811yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×D4×Q8D3
prime radicalC2×D4×SL(2,𝔽3)C2
hypercenterC22×D4S4
hypocenterSL(2,𝔽3)#124
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
2. C2×#5389C2semidirect
3. #73148C2semidirect
4. #73148C2semidirect
5. #73063C2semidirect

Maximal subgroups

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

subgroupindex
C2×D4×SL(2,𝔽3)2
C2×#53892
#731482
#730632
#258943
#52224