Finite Group Databasebeta

← #11713231

Subgroups — #11713231

Order 768.

589 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 63 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×#548⋊C3C211yesmaximal, characteristic
2C22×#859C211yesmaximal, characteristic, complemented
3#73151C212yesmaximal, complemented
4#73148C211yesmaximal, characteristic, complemented
5#73059C212yesmaximal, complemented
6#24036—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C23×SL(2,𝔽3)C2211yescharacteristic
9#548⋊C3—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11C2×#859—22no—
12C2×#859C2212yes—
13C2×#859C2212yes—
14#4946—41nomaximal
15C2×#548D311yescharacteristic
16C2×#481—31no—
17C22×#103—31no—
18#2487—31no—
19#2042—32no—
20#2035—32no—
21#1893—31no—
22#1869—31no—
23#1865—32no—
24#1788—32no—
25#1787—31no—
26C23×Dic3—41no—
27C2×#41⋊C3—21no—
28C2×#41⋊C3—22no—
29C22×SL(2,𝔽3)—22no—
30C22×SL(2,𝔽3)D412yes—
31C22×SL(2,𝔽3)C2311yescharacteristic
32C2×C3×C22⋊C4—41no—
33C2×#217—41no—
34C2×Dic3⋊C4—42no—
35C4×SL(2,𝔽3)—21no—
36#859—22no—
37#859—22no—
38C3⋊#95—42no—
39C22×#41C2×D311yescharacteristic
40C23×Q8C2×D311yescharacteristic
41C23×C8—31no—
42#548—22no—
43#548—62no—
44C2×#124—31no—
45C2×C22⋊Q8—31no—
46C2×#121—31no—
47C2×C4×Q8C2×D311yescharacteristic
48C2×C4×D4—31no—
49C22×C4⋊C4—31no—
50#481—64no—
51C2×#106—32no—
52C2×#103—32no—
53C2×#103—32no—
54C2×#103—32no—
55C2×#103—32no—
56C2×#103—62no—
57C2×#103—62no—
58C2×#102—32no—
59C2×#102—32no—
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—
72C22×C3×C4—41no—
73C22×C3×C4—41no—
74C22×Dic3—42no—
75C22×Dic3—42no—
76C22×Dic3—82no—
77C22×Dic3—84no—
78#41⋊C3—41no—
79C2×SL(2,𝔽3)—22no—
80C2×SL(2,𝔽3)C2×D411yescharacteristic
81C2×SL(2,𝔽3)#4112yes—
82C3×C22⋊C4—82no—
83#217—44no—
84Dic3⋊C4—84no—
85C2×#41—22no—
86C2×#41—21no—
87C2×#41—61no—
88C2×#41—62no—
89C22×Q8—22no—
90C22×Q8—31no—
91C22×Q8—32no—
92C22×Q8—62no—
93C22×Q8#6712yes—
94C22×Q8C22×D311yescharacteristic
95C22×D4—31no—
96C22×D4—31no—
97C23×C4—31no—
98C23×C4—31no—
99C23×C4—61no—
100C22×C8—32no—
101C22×C8—32no—
102C22×C8—62no—
103C22×C8—122no—
104#124—62no—
105#124—62no—
106C22⋊Q8—62no—
107C22⋊Q8—62no—
108#121—34no—
109#121—62no—
110C4×Q8—21no—
111C4×Q8—61no—
112C4×D4—62no—
113C4×D4—62no—
114C2×C4⋊C4—31no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—32no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4—62no—
122C2×C22⋊C4—31no—
123C2×C22⋊C4—61no—
124C2×C22⋊C4—61no—
125C2×C22⋊C4—62no—
126C2×C22⋊C4S411yescharacteristic
127C2×C42—31no—
128#106—64no—
129#103—62no—
130#103—62no—
131#103—64no—
132#103—62no—
133#103—62no—
134#103—122no—
135#102—64no—
136#102—122no—
137C22⋊C8—62no—
138C22⋊C8—122no—
139#95—61no—
140#95—62no—
141#95—62no—
142C23×C3—42no—
143C23×C3—42no—
144C23×C3—42no—
145C23×C3—41no—
146C23×C3—82no—
147C23×C3—82no—
148C2×C3×C4—81no—
149C2×C3×C4—82no—
150C2×C3×C4—81no—
151C2×C3×C4—82no—
152C2×Dic3—44no—
153C2×Dic3—44no—
154C2×Dic3—82no—
155C2×Dic3—82no—
156C2×Dic3—82no—
157C2×Dic3—82no—
158C2×Dic3—88no—
159SL(2,𝔽3)#12311yescharacteristic
160C24—31no—
161C24C2×S411yescharacteristic
162#41—41no—
163#41—122no—
164#41—122no—
165#41—121no—
166C2×Q8—22no—
167C2×Q8—31no—
168C2×Q8—32no—
169C2×Q8—32no—
170C2×Q8—61no—
171C2×Q8—62no—
172C2×Q8—62no—
173C2×Q8—62no—
174C2×Q8—64no—
175C2×Q8—122no—
176C2×Q8#23712yes—
177C2×Q8C2×#6711yescharacteristic
178C2×D4—32no—
179C2×D4—62no—
180C2×D4—62no—
181C2×D4—61no—
182C2×D4—62no—
183C2×D4—62no—
184C2×D4—62no—
185C22×C4—31no—
186C22×C4—32no—
187C22×C4—32no—
188C22×C4—31no—
189C22×C4—31no—
190C22×C4—62no—
191C22×C4—61no—
192C22×C4—61no—
193C22×C4—62no—
194C22×C4—62no—
195C22×C4—62no—
196C22×C4—62no—
197C22×C4—62no—
198C22×C4—122no—
199C22×C4—242no—
200C22×C4C2×S411yescharacteristic
201C22×C4C2×S411yescharacteristic
202C2×C8—62no—
203C2×C8—62no—
204C2×C8—62no—
205C2×C8—62no—
206C2×C8—62no—
207C2×C8—62no—
208C2×C8—124no—
209C4⋊C4—32no—
210C4⋊C4—62no—
211C4⋊C4—61no—
212C4⋊C4—62no—
213C4⋊C4—62no—
214C4⋊C4—62no—
215C4⋊C4—62no—
216C4⋊C4—122no—
217C4⋊C4—242no—
218C22⋊C4—22no—
219C22⋊C4—62no—
220C22⋊C4—64no—
221C22⋊C4—122no—
222C22⋊C4—242no—
223C42—61no—
224C42—61no—
225C22×C3—42no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—42no—
229C22×C3—41no—
230C22×C3—44no—
231C22×C3—82no—
232C22×C3—84no—
233C22×C3—84no—
234C22×C3—82no—
235C3×C4—161no—
236C3×C4—161no—
237Dic3—84no—
238Dic3—84no—
239C23—22no—
240C23—22no—
241C23—62no—
242C23—61no—
243C23—61no—
244C23—62no—
245C23—61no—
246C23#98412yes—
247C23#98512yes—
248C23GL(2,ℤ/4)12yes—
249C23C22×S411yescharacteristic
250Q8—32no—
251Q8—31no—
252Q8—61no—
253Q8—62no—
254Q8—62no—
255Q8—122no—
256Q8#93511yescharacteristic
257D4—64no—
258D4—121no—
259D4—121no—
260D4—124no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—32no—
266C2×C4—32no—
267C2×C4—31no—
268C2×C4—31no—
269C2×C4—32no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4—64no—
273C2×C4—62no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4—244no—
287C8—122no—
288C8—122no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—42no—
292C2×C3—42no—
293C2×C3—41no—
294C2×C3—84no—
295C22—22no—
296C22—24no—
297C22—24no—
298C22—22no—
299C22—121no—
300C22—121no—
301C22—122no—
302C22—122no—
303C22—121no—
304C22#542514yes—
305C22#591412yes—
306C22C2×#98411yescharacteristic
307C22C2×#98511yescharacteristic
308C22#592612yes—
309C22C2×GL(2,ℤ/4)11yescharacteristic
310C4—31no—
311C4—32no—
312C4—31no—
313C4—41no—
314C4—41no—
315C4—62no—
316C4—121no—
317C4—242no—
318C4—242no—
319C3—41noSylow
320C2—24no—
321C2—241no—
322C2#8537712yes—
323C2#8538112yes—
324C2#8549111yescharacteristic
325C2C2×#542511yescharacteristic
326C2#8553711yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#548D3
prime radicalC2×#548⋊C3C2
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#859C2semidirect
3. #73151C2semidirect
4. #73148C2semidirect
5. #73059C2semidirect

Maximal subgroups

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

subgroupindex
C2×#548⋊C32
C22×#8592
#731512
#731482
#730592
#240363
#49464