Finite Group Databasebeta

← #11713234

Subgroups — #11713234

Order 768.

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

normal subgroups only

This group has 628 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#73062C212yesmaximal, complemented
6#24042—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2211yescharacteristic
8C23×SL(2,𝔽3)C2211yescharacteristic
9D4×SL(2,𝔽3)—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×#859—22no—
12C2×#859C2212yes—
13C2×#859C2212yes—
14#5220—41nomaximal
15C2×D4×Q8D311yescharacteristic
16C2×#484—31no—
17C22×#103—31no—
18#2487—31no—
19#2043—32no—
20#2036—32no—
21#1892—31no—
22#1869—31no—
23#1863—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—
35C4×SL(2,𝔽3)—21no—
36#859—22no—
37#859—22no—
38C3⋊#95—42no—
39C23×Q8C2×D311yescharacteristic
40C23×Q8C2×D311yescharacteristic
41C23×C8—31no—
42D4×Q8—22no—
43D4×Q8—62no—
44C2×C4⋊Q8—31no—
45C2×C22⋊Q8—31no—
46C2×C22⋊Q8—31no—
47C2×C4×Q8C2×D311yescharacteristic
48C2×C4×D4—31no—
49C22×C4⋊C4—31no—
50#484—64no—
51C2×#107—32no—
52C2×#103—32no—
53C2×#103—32no—
54C2×#103—32no—
55C2×#103—32no—
56C2×#103—32no—
57C2×#103—32no—
58C2×#103—62no—
59C2×#103—62no—
60C2×C22⋊C8—31no—
61C2×C22⋊C8—32no—
62#400—61no—
63#399—61no—
64#389—32no—
65#386—61no—
66#385—61no—
67#380—31no—
68#380—32no—
69#340—32no—
70#336—32no—
71C24×C3—41no—
72C24×C3—41no—
73C2×C3×D4—44no—
74C2×C3×D4—82no—
75C2×C3×D4—82no—
76C22×C3×C4—41no—
77C22×Dic3—42no—
78C22×Dic3—42no—
79C22×Dic3—82no—
80C22×Dic3—84no—
81C2×SL(2,𝔽3)—22no—
82C2×SL(2,𝔽3)—41no—
83C2×SL(2,𝔽3)C2×D411yescharacteristic
84C2×SL(2,𝔽3)#4112yes—
85#217—82no—
86#217—84no—
87C22×Q8—22no—
88C22×Q8—22no—
89C22×Q8—21no—
90C22×Q8—31no—
91C22×Q8—31no—
92C22×Q8—32no—
93C22×Q8—61no—
94C22×Q8—62no—
95C22×Q8—62no—
96C22×Q8#6712yes—
97C22×Q8C22×D311yescharacteristic
98C22×D4S411yescharacteristic
99C23×C4—31no—
100C23×C4—31no—
101C23×C4—61no—
102C22×C8—32no—
103C22×C8—32no—
104C22×C8—62no—
105C22×C8—122no—
106C4⋊Q8—62no—
107C4⋊Q8—62no—
108C22⋊Q8—34no—
109C22⋊Q8—62no—
110C22⋊Q8—62no—
111C22⋊Q8—62no—
112C4×Q8—21no—
113C4×Q8—61no—
114C4×D4—62no—
115C4×D4—62no—
116C2×C4⋊C4—31no—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—32no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4—62no—
122C2×C4⋊C4—62no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—62no—
125C2×C22⋊C4—31no—
126C2×C22⋊C4—31no—
127C2×C22⋊C4—61no—
128C2×C22⋊C4—62no—
129C2×C42—31no—
130#107—64no—
131#103—62no—
132#103—62no—
133#103—64no—
134#103—64no—
135#103—62no—
136#103—62no—
137#103—122no—
138#103—122no—
139C22⋊C8—62no—
140C22⋊C8—122no—
141#95—62no—
142#95—61no—
143#95—62no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3—41no—
149C23×C3—82no—
150C23×C3—82no—
151C23×C3—82no—
152C23×C3—81no—
153C23×C3—82no—
154C23×C3—81no—
155C3×D4—88no—
156C2×C3×C4—42no—
157C2×C3×C4—81no—
158C2×C3×C4—81no—
159C2×Dic3—44no—
160C2×Dic3—82no—
161C2×Dic3—82no—
162C2×Dic3—82no—
163C2×Dic3—82no—
164C2×Dic3—82no—
165C2×Dic3—88no—
166SL(2,𝔽3)#12311yescharacteristic
167C24C2×S411yescharacteristic
168C24C2×S411yescharacteristic
169C2×Q8—22no—
170C2×Q8—31no—
171C2×Q8—32no—
172C2×Q8—32no—
173C2×Q8—32no—
174C2×Q8—41no—
175C2×Q8—61no—
176C2×Q8—62no—
177C2×Q8—62no—
178C2×Q8—62no—
179C2×Q8—62no—
180C2×Q8—61no—
181C2×Q8—62no—
182C2×Q8—64no—
183C2×Q8—122no—
184C2×Q8—122no—
185C2×Q8—122no—
186C2×Q8—121no—
187C2×Q8#23712yes—
188C2×Q8C2×#6711yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4#22614yes—
192C22×C4—31no—
193C22×C4—32no—
194C22×C4—32no—
195C22×C4—31no—
196C22×C4—31no—
197C22×C4—31no—
198C22×C4—62no—
199C22×C4—62no—
200C22×C4—62no—
201C22×C4—61no—
202C22×C4—61no—
203C22×C4—62no—
204C22×C4—62no—
205C22×C4—62no—
206C22×C4—122no—
207C22×C4—242no—
208C22×C4C2×S411yescharacteristic
209C2×C8—62no—
210C2×C8—62no—
211C2×C8—62no—
212C2×C8—62no—
213C2×C8—62no—
214C2×C8—62no—
215C2×C8—124no—
216C4⋊C4—32no—
217C4⋊C4—62no—
218C4⋊C4—61no—
219C4⋊C4—64no—
220C4⋊C4—62no—
221C4⋊C4—62no—
222C4⋊C4—62no—
223C4⋊C4—62no—
224C4⋊C4—122no—
225C4⋊C4—242no—
226C22⋊C4—62no—
227C22⋊C4—62no—
228C22⋊C4—122no—
229C22⋊C4—242no—
230C42—61no—
231C42—61no—
232C22×C3—42no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—42no—
236C22×C3—41no—
237C22×C3—44no—
238C22×C3—82no—
239C22×C3—84no—
240C22×C3—84no—
241C22×C3—82no—
242C22×C3—82no—
243C22×C3—84no—
244C22×C3—161no—
245C22×C3—162no—
246C22×C3—161no—
247C3×C4—82no—
248Dic3—84no—
249Dic3—162no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—21no—
254C23—22no—
255C23—21no—
256C23C2×#22612yes—
257C23C2×#22612yes—
258C23#98312yes—
259C23GL(2,ℤ/4)12yes—
260C23C22×S411yescharacteristic
261Q8—32no—
262Q8—31no—
263Q8—61no—
264Q8—62no—
265Q8—62no—
266Q8—64no—
267Q8—122no—
268Q8—121no—
269Q8—121no—
270Q8#93511yescharacteristic
271D4—28no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—32no—
275C2×C4—32no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—32no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—64no—
282C2×C4—62no—
283C2×C4—62no—
284C2×C4—62no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—122no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—244no—
297C2×C4C2×#22612yes—
298C8—122no—
299C8—122no—
300C2×C3—41no—
301C2×C3—42no—
302C2×C3—42no—
303C2×C3—41no—
304C2×C3—41no—
305C2×C3—84no—
306C2×C3—162no—
307C22—22no—
308C22—24no—
309C22—24no—
310C22—22no—
311C22—24no—
312C22—22no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22#541914yes—
317C22#591412yes—
318C22C22×#22611yescharacteristic
319C22C2×#98311yescharacteristic
320C22#592612yes—
321C22C2×GL(2,ℤ/4)11yescharacteristic
322C4—22no—
323C4—31no—
324C4—32no—
325C4—31no—
326C4—62no—
327C4—121no—
328C4—121no—
329C4—242no—
330C4—242no—
331C3—41noSylow
332C2—24no—
333C2—42no—
334C2#8539312yes—
335C2#8540112yes—
336C2#8549111yescharacteristic
337C2C2×#541911yescharacteristic
338C2#8553711yescharacteristic
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. #73062C2semidirect

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
#730622
#240423
#52204