Finite Group Databasebeta

← #11713233

Subgroups — #11713233

Order 768.

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

normal subgroups only

This group has 666 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
1C2×D4×SL(2,𝔽3)C211yesmaximal, characteristic
2C22×#859C211yesmaximal, characteristic, complemented
3#73148C211yesmaximal, characteristic, complemented
4#73148C212yesmaximal, complemented
5#73059C212yesmaximal, complemented
6#24041—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2211yescharacteristic
8C23×SL(2,𝔽3)C2211yescharacteristic
9C2×#935—41nomaximal
10D4×SL(2,𝔽3)—22no—
11C2×C4×SL(2,𝔽3)C2211yescharacteristic
12C2×#859—22no—
13C2×#859C2212yes—
14C2×#859C2212yes—
15C2×D4×Q8D311yescharacteristic
16C2×#483—31no—
17C22×#103—31no—
18#2486—31no—
19#2043—32no—
20#2035—32no—
21#1892—31no—
22#1869—31no—
23#1865—32no—
24#1787—32no—
25#1787—31no—
26C22×C3×D4—41no—
27C23×Dic3—41no—
28C22×SL(2,𝔽3)—22no—
29C22×SL(2,𝔽3)—21no—
30C22×SL(2,𝔽3)—22no—
31C22×SL(2,𝔽3)D412yes—
32C22×SL(2,𝔽3)C2311yescharacteristic
33C2×#217—41no—
34C2×#217—42no—
35#935—48no—
36C2×Dic3⋊C4—42no—
37C4×SL(2,𝔽3)—21no—
38#859—22no—
39#859—22no—
40C23×Q8C2×D311yescharacteristic
41C23×Q8C2×D311yescharacteristic
42C23×C8—31no—
43D4×Q8—22no—
44D4×Q8—62no—
45C2×C4⋊Q8—31no—
46C2×#123—61no—
47C2×C22⋊Q8—31no—
48C2×C22⋊Q8—31no—
49C2×C4×Q8C2×D311yescharacteristic
50C2×C4×D4—31no—
51C22×C4⋊C4—31no—
52#483—64no—
53C2×#106—32no—
54C2×#103—32no—
55C2×#103—32no—
56C2×#103—32no—
57C2×#103—32no—
58C2×#103—32no—
59C2×#103—32no—
60C2×#103—62no—
61C2×#103—62no—
62C2×C22⋊C8—31no—
63C2×C22⋊C8—32no—
64#398—61no—
65#397—61no—
66#389—32no—
67#381—61no—
68#380—31no—
69#380—32no—
70#340—32no—
71#336—32no—
72C24×C3—41no—
73C24×C3—41no—
74C2×C3×D4—44no—
75C2×C3×D4—82no—
76C2×C3×D4—82no—
77C22×C3×C4—41no—
78C22×Dic3—42no—
79C22×Dic3—42no—
80C22×Dic3—44no—
81C22×Dic3—82no—
82C22×Dic3—82no—
83C2×SL(2,𝔽3)—22no—
84C2×SL(2,𝔽3)—41no—
85C2×SL(2,𝔽3)C2×D411yescharacteristic
86C2×SL(2,𝔽3)#4112yes—
87#217—44no—
88#217—48no—
89Dic3⋊C4—48no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—21no—
93C22×Q8—31no—
94C22×Q8—31no—
95C22×Q8—32no—
96C22×Q8—61no—
97C22×Q8—62no—
98C22×Q8—62no—
99C22×Q8#6712yes—
100C22×Q8C22×D311yescharacteristic
101C22×D4S411yescharacteristic
102C23×C4—31no—
103C23×C4—31no—
104C23×C4—61no—
105C22×C8—32no—
106C22×C8—32no—
107C22×C8—62no—
108C22×C8—122no—
109C4⋊Q8—62no—
110C4⋊Q8—62no—
111#123—124no—
112C22⋊Q8—34no—
113C22⋊Q8—62no—
114C22⋊Q8—62no—
115C22⋊Q8—62no—
116C4×Q8—21no—
117C4×Q8—61no—
118C4×D4—62no—
119C4×D4—62no—
120C2×C4⋊C4—31no—
121C2×C4⋊C4—31no—
122C2×C4⋊C4—32no—
123C2×C4⋊C4—32no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—62no—
126C2×C4⋊C4—62no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—62no—
129C2×C22⋊C4—31no—
130C2×C22⋊C4—31no—
131C2×C22⋊C4—61no—
132C2×C22⋊C4—62no—
133C2×C42—31no—
134#106—64no—
135#103—62no—
136#103—62no—
137#103—64no—
138#103—64no—
139#103—62no—
140#103—62no—
141#103—122no—
142#103—122no—
143C22⋊C8—62no—
144C22⋊C8—122no—
145#95—62no—
146#95—61no—
147#95—62no—
148C23×C3—42no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—42no—
152C23×C3—41no—
153C23×C3—82no—
154C23×C3—82no—
155C23×C3—81no—
156C23×C3—82no—
157C23×C3—81no—
158C23×C3—82no—
159C3×D4—88no—
160C2×C3×C4—42no—
161C2×C3×C4—81no—
162C2×C3×C4—81no—
163C2×Dic3—44no—
164C2×Dic3—44no—
165C2×Dic3—82no—
166C2×Dic3—82no—
167C2×Dic3—82no—
168C2×Dic3—82no—
169C2×Dic3—88no—
170SL(2,𝔽3)#12311yescharacteristic, complemented
171C24C2×S411yescharacteristic
172C24C2×S411yescharacteristic
173C2×Q8—22no—
174C2×Q8—31no—
175C2×Q8—32no—
176C2×Q8—32no—
177C2×Q8—32no—
178C2×Q8—41no—
179C2×Q8—61no—
180C2×Q8—62no—
181C2×Q8—62no—
182C2×Q8—62no—
183C2×Q8—62no—
184C2×Q8—61no—
185C2×Q8—62no—
186C2×Q8—64no—
187C2×Q8—122no—
188C2×Q8—122no—
189C2×Q8—122no—
190C2×Q8—121no—
191C2×Q8#23712yes—
192C2×Q8C2×#6711yescharacteristic
193C2×D4—22no—
194C2×D4—22no—
195C2×D4GL(2,𝔽3)14yes—
196C22×C4—31no—
197C22×C4—32no—
198C22×C4—32no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—62no—
203C22×C4—62no—
204C22×C4—62no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4—62no—
208C22×C4—62no—
209C22×C4—62no—
210C22×C4—122no—
211C22×C4—122no—
212C22×C4—122no—
213C22×C4C2×S411yescharacteristic
214C2×C8—62no—
215C2×C8—62no—
216C2×C8—62no—
217C2×C8—62no—
218C2×C8—62no—
219C2×C8—62no—
220C2×C8—124no—
221C4⋊C4—32no—
222C4⋊C4—62no—
223C4⋊C4—61no—
224C4⋊C4—64no—
225C4⋊C4—62no—
226C4⋊C4—62no—
227C4⋊C4—62no—
228C4⋊C4—62no—
229C4⋊C4—122no—
230C4⋊C4—124no—
231C22⋊C4—62no—
232C22⋊C4—62no—
233C22⋊C4—122no—
234C22⋊C4—124no—
235C42—61no—
236C42—61no—
237C22×C3—42no—
238C22×C3—41no—
239C22×C3—41no—
240C22×C3—42no—
241C22×C3—41no—
242C22×C3—44no—
243C22×C3—82no—
244C22×C3—84no—
245C22×C3—84no—
246C22×C3—82no—
247C22×C3—82no—
248C22×C3—84no—
249C22×C3—161no—
250C22×C3—162no—
251C22×C3—161no—
252C3×C4—82no—
253Dic3—84no—
254Dic3—84no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—21no—
259C23—22no—
260C23—21no—
261C23C2×GL(2,𝔽3)12yes—
262C23C2×GL(2,𝔽3)12yes—
263C23#98312yes—
264C23GL(2,ℤ/4)12yes—
265C23C22×S411yescharacteristic
266Q8—32no—
267Q8—31no—
268Q8—61no—
269Q8—62no—
270Q8—62no—
271Q8—64no—
272Q8—122no—
273Q8—121no—
274Q8—121no—
275Q8#93511yescharacteristic, complemented
276D4—28no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—31no—
282C2×C4—31no—
283C2×C4—32no—
284C2×C4—62no—
285C2×C4—62no—
286C2×C4—64no—
287C2×C4—62no—
288C2×C4—62no—
289C2×C4—62no—
290C2×C4—61no—
291C2×C4—61no—
292C2×C4—121no—
293C2×C4—121no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4—244no—
302C2×C4C2×GL(2,𝔽3)12yes—
303C8—122no—
304C8—122no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—42no—
308C2×C3—42no—
309C2×C3—41no—
310C2×C3—84no—
311C2×C3—162no—
312C22—22no—
313C22—24no—
314C22—24no—
315C22—22no—
316C22—22no—
317C22—24no—
318C22—41no—
319C22—42no—
320C22—41no—
321C22#542114yes—
322C22#591412yes—
323C22C22×GL(2,𝔽3)11yescharacteristic
324C22C2×#98311yescharacteristic
325C22#592712yes—
326C22C2×GL(2,ℤ/4)11yescharacteristic
327C4—22no—
328C4—31no—
329C4—32no—
330C4—31no—
331C4—62no—
332C4—121no—
333C4—121no—
334C4—242no—
335C4—242no—
336C3—41noSylow
337C2—24no—
338C2—42no—
339C2#8539412yes—
340C2#8539712yes—
341C2#8549111yescharacteristic
342C2C2×#542111yescharacteristic
343C2#8553511yescharacteristic
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)#123
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. C22×#859C2semidirect
3. #73148C2semidirect
4. #73148C2semidirect
5. #73059C2semidirect
170. SL(2,𝔽3)#123semidirect
275. Q8#935semidirect

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
C22×#8592
#731482
#730592
#240413
C2×#9354