Finite Group Databasebeta

← #11714014

Subgroups — #11714014

Order 768.

591 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 83 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
2C2×#5391C211yesmaximal, characteristic, complemented
3#73148C212yesmaximal
4#73062C212yesmaximal, complemented
5C4⋊#859C211yesmaximal, characteristic, complemented
6#32957—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2211yescharacteristic
8C2×C4×SL(2,𝔽3)—22no—
9C2×C4×SL(2,𝔽3)C2211yescharacteristic
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×#859C2212yes—
12C2×#859C2212yes—
13#5391—22no—
14#5212—41nomaximal
15C22×C4×Q8D311yescharacteristic
16C2×#477—31no—
17C2×#477—31no—
18#2857—31no—
19#1863—32no—
20#1863—32no—
21#1852—32no—
22#1803—31no—
23#1803—31no—
24#1797—31no—
25#1787—32no—
26C23×C3×C4—41no—
27C22×SL(2,𝔽3)—21no—
28C22×SL(2,𝔽3)—21no—
29C22×SL(2,𝔽3)Q812yescomplemented
30C22×SL(2,𝔽3)C2311yescharacteristic
31C2×#217—42no—
32C2×C4⋊Dic3—41no—
33C2×C4⋊Dic3—41no—
34C4×SL(2,𝔽3)—21no—
35C4×SL(2,𝔽3)D412yes—
36#859—22no—
37#859—22no—
38C3⋊#95—42no—
39C23×Q8C2×D311yescharacteristic
40C2×C4⋊Q8—31no—
41C2×C22⋊Q8—61no—
42C2×C4×Q8—22no—
43C2×C4×Q8—31no—
44C2×C4×Q8—31no—
45C2×C4×Q8—62no—
46C2×C4×Q8C2×D311yescharacteristic
47C2×C4×Q8C2×D311yescharacteristic
48C22×C4⋊C4—31no—
49C22×C42—31no—
50#477—62no—
51#477—64no—
52#477—62no—
53C2×#107—32no—
54C2×#107—32no—
55C2×C4⋊C8—31no—
56C2×C4⋊C8—31no—
57C2×#103—32no—
58C2×#103—32no—
59C2×#103—32no—
60C2×#103—32no—
61C2×C22⋊C8—32no—
62#391—61no—
63#389—32no—
64#389—61no—
65#384—31no—
66#380—32no—
67#380—61no—
68#340—32no—
69#340—32no—
70#336—32no—
71C24×C3—41no—
72C22×C3×C4—44no—
73C22×C3×C4—41no—
74C22×C3×C4—41no—
75C22×C3×C4—82no—
76C22×C3×C4—82no—
77C22×Dic3—42no—
78C22×Dic3—42no—
79C2×SL(2,𝔽3)—22no—
80C2×SL(2,𝔽3)C2×D411yescharacteristic
81C2×SL(2,𝔽3)C2×Q811yescharacteristic
82C2×SL(2,𝔽3)#4111yescharacteristic
83#217—84no—
84C4⋊Dic3—82no—
85C4⋊Dic3—82no—
86C22×Q8—21no—
87C22×Q8—21no—
88C22×Q8—31no—
89C22×Q8—32no—
90C22×Q8—61no—
91C22×Q8—61no—
92C22×Q8—61no—
93C22×Q8Dic612yes—
94C22×Q8C22×D311yescharacteristic
95C23×C4—31no—
96C23×C4—31no—
97C23×C4S411yescharacteristic
98C22×C8—32no—
99C22×C8—32no—
100C4⋊Q8—62no—
101C4⋊Q8—62no—
102C22⋊Q8—124no—
103C4×Q8—21no—
104C4×Q8—32no—
105C4×Q8—34no—
106C4×Q8—62no—
107C4×Q8—62no—
108C4×Q8—61no—
109C4×Q8—62no—
110C4×Q8#6712yes—
111C2×C4⋊C4—31no—
112C2×C4⋊C4—31no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—62no—
116C2×C4⋊C4—62no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—61no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4—61no—
122C2×C22⋊C4—62no—
123C2×C42—31no—
124C2×C42—31no—
125C2×C42—31no—
126C2×C42—31no—
127C2×C42—62no—
128C2×C42—62no—
129#107—64no—
130#107—64no—
131C4⋊C8—62no—
132C4⋊C8—62no—
133#103—62no—
134#103—64no—
135#103—62no—
136#103—64no—
137#103—62no—
138#103—62no—
139C22⋊C8—64no—
140#95—62no—
141#95—62no—
142C23×C3—42no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3—41no—
146C23×C3—81no—
147C23×C3—81no—
148C23×C3—81no—
149C23×C3—81no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—88no—
155C2×C3×C4—81no—
156C2×C3×C4—81no—
157C2×Dic3—82no—
158C2×Dic3—82no—
159C2×Dic3—82no—
160C2×Dic3—82no—
161C2×Dic3—82no—
162C2×Dic3—82no—
163SL(2,𝔽3)C22⋊Q811yescharacteristic, complemented
164C24C2×S411yescharacteristic
165C2×Q8—22no—
166C2×Q8—31no—
167C2×Q8—31no—
168C2×Q8—32no—
169C2×Q8—31no—
170C2×Q8—32no—
171C2×Q8—32no—
172C2×Q8—61no—
173C2×Q8—61no—
174C2×Q8—61no—
175C2×Q8—64no—
176C2×Q8—62no—
177C2×Q8—62no—
178C2×Q8—62no—
179C2×Q8—122no—
180C2×Q8—122no—
181C2×Q8—122no—
182C2×Q8C2×Dic611yescharacteristic
183C2×Q8#23511yescharacteristic
184C2×Q8C2×#6711yescharacteristic
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—31no—
188C22×C4—32no—
189C22×C4—32no—
190C22×C4—31no—
191C22×C4—31no—
192C22×C4—31no—
193C22×C4—61no—
194C22×C4—61no—
195C22×C4—61no—
196C22×C4—61no—
197C22×C4—62no—
198C22×C4—62no—
199C22×C4—62no—
200C22×C4—62no—
201C22×C4—121no—
202C22×C4#22614yes—
203C22×C4C2×S411yescharacteristic
204C22×C4C2×S411yescharacteristic
205C2×C8—62no—
206C2×C8—62no—
207C2×C8—62no—
208C2×C8—62no—
209C2×C8—62no—
210C2×C8—62no—
211C4⋊C4—32no—
212C4⋊C4—32no—
213C4⋊C4—61no—
214C4⋊C4—61no—
215C4⋊C4—64no—
216C4⋊C4—62no—
217C4⋊C4—62no—
218C4⋊C4—62no—
219C4⋊C4—62no—
220C4⋊C4—122no—
221C4⋊C4—124no—
222C4⋊C4—122no—
223C4⋊C4—122no—
224C22⋊C4—124no—
225C42—32no—
226C42—32no—
227C42—61no—
228C42—61no—
229C42—62no—
230C42—62no—
231C22×C3—44no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—82no—
240C22×C3—82no—
241C22×C3—82no—
242C22×C3—82no—
243C22×C3—82no—
244C22×C3—82no—
245C3×C4—44no—
246C3×C4—82no—
247Dic3—162no—
248Dic3—162no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23A4⋊Q812yes—
254C23C2×#22612yes—
255C23#98612yes—
256C23C22×S411yescharacteristic
257Q8—31no—
258Q8—31no—
259Q8—31no—
260Q8—34no—
261Q8—61no—
262Q8—62no—
263Q8—62no—
264Q8—62no—
265Q8—244no—
266Q8#92411yescharacteristic
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—28no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—61no—
283C2×C4—61no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—61no—
288C2×C4—61no—
289C2×C4—124no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4C2×#22612yes—
297C2×C4C2×#22612yes—
298C2×C4#98312yes—
299C2×C4GL(2,ℤ/4)12yes—
300C8—122no—
301C8—122no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—41no—
308C2×C3—41no—
309C2×C3—84no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22#539114yescomplemented
317C22C2×A4⋊Q811yescharacteristic
318C22#591211yescharacteristic
319C22C22×#22611yescharacteristic
320C22C2×#98311yescharacteristic
321C22C2×#98611yescharacteristic
322C22#592311yescharacteristic
323C22C2×GL(2,ℤ/4)11yescharacteristic
324C4—22no—
325C4—31no—
326C4—31no—
327C4—31no—
328C4—31no—
329C4—62no—
330C4—62no—
331C4—121no—
332C4—242no—
333C4—242no—
334C4#541914yes—
335C3—41noSylow
336C2—24no—
337C2#8531811yescharacteristic
338C2C2×#539111yescharacteristic
339C2#8532211yescharacteristic
340C2#8533211yescharacteristic
341C2#8548011yescharacteristic
342C2C2×#541911yescharacteristic
343C2#8552311yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC22×C4×Q8D3
prime radicalC23×SL(2,𝔽3)C22
hypercenterC23×C4S4
hypocenterSL(2,𝔽3)C22⋊Q8
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
2. C2×#5391C2semidirect
4. #73062C2semidirect
5. C4⋊#859C2semidirect
29. C22×SL(2,𝔽3)Q8semidirect
163. SL(2,𝔽3)C22⋊Q8semidirect
316. C22#5391semidirect

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×#53912
#731482
#730622
C4⋊#8592
#329573
#52124