Finite Group Databasebeta

← #11714045

Subgroups — #11714045

Order 768.

669 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 669 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C22×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#5421C211yesmaximal, characteristic
3#73148C211yesmaximal, characteristic, complemented
4#73065C212yesmaximal, complemented
5#73059C212yesmaximal, complemented
6#33005—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2211yescharacteristic
8C22×GL(2,𝔽3)C2211yescharacteristic
9C2×#929—41nomaximal
10C2×C4×SL(2,𝔽3)—22no—
11C2×C4×SL(2,𝔽3)C2212yes—
12#5421—22no—
13C2×#859C2211yescharacteristic
14C2×#859C2212yes—
15C22×C4×Q8D311yescharacteristic, complemented
16C2×#481—31no—
17C2×#448—31no—
18#2854—31no—
19#1880—32no—
20#1880—32no—
21#1865—32no—
22#1865—32no—
23#1824—31no—
24#1817—31no—
25#1787—31no—
26C23×C3×C4—41no—
27C22×#67—41no—
28C22×SL(2,𝔽3)—22no—
29C22×SL(2,𝔽3)D412yes—
30C22×SL(2,𝔽3)C2311yescharacteristic
31C2×GL(2,𝔽3)—21no—
32C2×GL(2,𝔽3)—22no—
33C2×#217—41no—
34#929—48no—
35C2×(C2×D3)⋊C4—42no—
36C2×Dic3⋊C4—42no—
37C4×SL(2,𝔽3)—22no—
38#859—21no—
39#859—22no—
40C23×Q8C2×D311yescharacteristic
41C22×SD16—31no—
42C22×SD16—31no—
43C2×#123—61no—
44C2×#121—31no—
45C2×C4×Q8—22no—
46C2×C4×Q8—32no—
47C2×C4×Q8—62no—
48C2×C4×Q8C2×D312yes—
49C22×C4⋊C4—31no—
50C22×C42—31no—
51#481—64no—
52#448—62no—
53#448—62no—
54C2×#106—32no—
55C2×#103—31no—
56C2×#103—32no—
57C2×#103—31no—
58C2×#103—32no—
59C2×#102—32no—
60C2×C22⋊C8—31no—
61C2×C22⋊C8—31no—
62#390—32no—
63#389—32no—
64#388—62no—
65#386—61no—
66#380—31no—
67#340—32no—
68#340—32no—
69#336—32no—
70#336—32no—
71C24×C3—41no—
72C23×D3—41no—
73C22×C3×C4—44no—
74C22×C3×C4—42no—
75C22×C3×C4—82no—
76C22×C3×C4—82no—
77C2×#67—44no—
78C2×#67—82no—
79C2×#67—82no—
80C22×Dic3—41no—
81C22×Dic3—42no—
82C2×SL(2,𝔽3)—22no—
83C2×SL(2,𝔽3)C2×D411yescharacteristic
84C2×SL(2,𝔽3)#4112yes—
85GL(2,𝔽3)—41no—
86#217—44no—
87(C2×D3)⋊C4—48no—
88Dic3⋊C4—48no—
89C22×Q8—22no—
90C22×Q8—31no—
91C22×Q8—32no—
92C22×Q8—62no—
93C22×Q8#6712yes—
94C22×Q8C22×D311yescharacteristic
95C22×D4—31no—
96C22×D4—61no—
97C23×C4—31no—
98C23×C4—31no—
99C23×C4S411yescharacteristic
100C2×SD16—62no—
101C2×SD16—64no—
102C2×SD16—62no—
103C2×SD16—61no—
104C2×SD16—62no—
105C2×SD16—61no—
106C22×C8—31no—
107C22×C8—31no—
108C22×C8—32no—
109#123—124no—
110#121—62no—
111#121—62no—
112C4×Q8—22no—
113C4×Q8—64no—
114C4×Q8—62no—
115C4×Q8—64no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—34no—
118C2×C4⋊C4—31no—
119C2×C4⋊C4—32no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4—62no—
122C2×C4⋊C4—62no—
123C2×C22⋊C4—61no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4—61no—
126C2×C22⋊C4—62no—
127C2×C42—32no—
128C2×C42—32no—
129C2×C42—62no—
130C2×C42—62no—
131#106—64no—
132#103—62no—
133#103—64no—
134#103—62no—
135#103—61no—
136#103—62no—
137#103—61no—
138#102—64no—
139C22⋊C8—62no—
140C22⋊C8—62no—
141#95—61no—
142#95—62no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3—41no—
147C23×C3—82no—
148C23×C3—82no—
149C22×D3—42no—
150C22×D3—81no—
151C22×D3—82no—
152C22×D3—81no—
153C22×D3—82no—
154C2×C3×C4—44no—
155C2×C3×C4—82no—
156C2×C3×C4—82no—
157C2×C3×C4—88no—
158#67—88no—
159C2×Dic3—42no—
160C2×Dic3—44no—
161C2×Dic3—82no—
162C2×Dic3—81no—
163C2×Dic3—82no—
164C2×Dic3—81no—
165SL(2,𝔽3)#12311yescharacteristic, complemented
166C24—61no—
167C24C2×S411yescharacteristic
168C2×Q8—22no—
169C2×Q8—32no—
170C2×Q8—31no—
171C2×Q8—32no—
172C2×Q8—34no—
173C2×Q8—61no—
174C2×Q8—62no—
175C2×Q8—64no—
176C2×Q8—62no—
177C2×Q8—64no—
178C2×Q8#23512yes—
179C2×Q8C2×#6711yescharacteristic
180C2×D4—61no—
181C2×D4—61no—
182C2×D4—62no—
183C2×D4—62no—
184C2×D4—122no—
185C2×D4—122no—
186C2×D4—122no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—31no—
190C22×C4—32no—
191C22×C4—32no—
192C22×C4—31no—
193C22×C4—32no—
194C22×C4—62no—
195C22×C4—62no—
196C22×C4—62no—
197C22×C4—62no—
198C22×C4—61no—
199C22×C4—122no—
200C22×C4GL(2,𝔽3)14yescomplemented
201C22×C4C2×S412yes—
202SD16—122no—
203SD16—121no—
204SD16—124no—
205SD16—121no—
206C2×C8—62no—
207C2×C8—61no—
208C2×C8—61no—
209C2×C8—62no—
210C2×C8—62no—
211C2×C8—62no—
212C2×C8—62no—
213C4⋊C4—34no—
214C4⋊C4—62no—
215C4⋊C4—64no—
216C4⋊C4—61no—
217C4⋊C4—62no—
218C4⋊C4—62no—
219C4⋊C4—61no—
220C4⋊C4—124no—
221C22⋊C4—122no—
222C22⋊C4—124no—
223C22⋊C4—122no—
224C22⋊C4—124no—
225C42—62no—
226C42—62no—
227C42—64no—
228C22×C3—42no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—42no—
232C22×C3—41no—
233C22×C3—44no—
234C22×C3—82no—
235C22×C3—84no—
236C22×C3—84no—
237C22×C3—82no—
238C2×D3—82no—
239C2×D3—84no—
240C2×D3—161no—
241C2×D3—162no—
242C2×D3—161no—
243C3×C4—84no—
244Dic3—84no—
245Dic3—82no—
246C23—22no—
247C23—22no—
248C23—122no—
249C23—121no—
250C23—121no—
251C23—122no—
252C23—121no—
253C23C2×GL(2,𝔽3)12yes—
254C23#98312yes—
255C23GL(2,ℤ/4)12yes—
256C23C22×S411yescharacteristic
257Q8—32no—
258Q8—31no—
259Q8—34no—
260Q8—61no—
261Q8—62no—
262Q8—64no—
263Q8#92911yescharacteristic, complemented
264D4—121no—
265D4—121no—
266D4—122no—
267D4—244no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—28no—
271C2×C4—32no—
272C2×C4—32no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—32no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—64no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—62no—
283C2×C4—124no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4—121no—
287C2×C4—121no—
288C2×C4—122no—
289C2×C4—121no—
290C2×C4C2×GL(2,𝔽3)14yes—
291C8—122no—
292C8—121no—
293C8—121no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—42no—
297C2×C3—42no—
298C2×C3—41no—
299C2×C3—84no—
300D3—162no—
301C22—22no—
302C22—24no—
303C22—24no—
304C22—22no—
305C22—241no—
306C22—241no—
307C22—242no—
308C22—242no—
309C22—241no—
310C22#542114yes—
311C22#591212yes—
312C22C22×GL(2,𝔽3)11yescharacteristic
313C22C2×#98311yescharacteristic
314C22#592312yes—
315C22C2×GL(2,ℤ/4)11yescharacteristic
316C4—24no—
317C4—31no—
318C4—31no—
319C4—32no—
320C4—62no—
321C4—122no—
322C4—241no—
323C4—242no—
324C3—41noSylow
325C2—24no—
326C2—481no—
327C2#8532512yes—
328C2#8533012yes—
329C2#8548511yescharacteristic
330C2C2×#542111yescharacteristic
331C2#8552211yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC22×C4×Q8D3
prime radicalC2×#5421C2
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4×SL(2,𝔽3)C2semidirect
3. #73148C2semidirect
4. #73065C2semidirect
5. #73059C2semidirect
15. C22×C4×Q8D3semidirect
165. SL(2,𝔽3)#123semidirect
200. C22×C4GL(2,𝔽3)semidirect
263. Q8#929semidirect

Maximal subgroups

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

subgroupindex
C22×C4×SL(2,𝔽3)2
C2×#54212
#731482
#730652
#730592
#330053
C2×#9294