Finite Group Databasebeta

← #11714044

Subgroups — #11714044

Order 768.

641 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 641 conjugacy classes of subgroups — too many to draw. Only its 67 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#73062C212yesmaximal, complemented
6#33003—31nomaximal, Sylow
7C23×SL(2,𝔽3)C2211yescharacteristic
8C22×GL(2,𝔽3)C2211yescharacteristic
9C2×C4×SL(2,𝔽3)—22no—
10C2×C4×SL(2,𝔽3)C2212yes—
11#5421—22no—
12C2×#859C2211yescharacteristic
13C2×#859C2212yes—
14#5213—41nomaximal
15C22×C4×Q8D311yescharacteristic, complemented
16C2×#480—31no—
17C2×#448—31no—
18#2854—31no—
19#1880—32no—
20#1880—32no—
21#1863—32no—
22#1863—32no—
23#1823—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—
34C2×(C2×D3)⋊C4—42no—
35C4×SL(2,𝔽3)—22no—
36#859—21no—
37#859—22no—
38C3⋊#95—42no—
39C23×Q8C2×D311yescharacteristic
40C22×SD16—31no—
41C22×SD16—31no—
42C2×#123—61no—
43C2×#121—31no—
44C2×C4×Q8—22no—
45C2×C4×Q8—32no—
46C2×C4×Q8—62no—
47C2×C4×Q8C2×D312yes—
48C22×C4⋊C4—31no—
49C22×C42—31no—
50#480—64no—
51#448—62no—
52#448—62no—
53C2×#107—32no—
54C2×#103—31no—
55C2×#103—32no—
56C2×#103—31no—
57C2×#103—32no—
58C2×#102—32no—
59C2×C22⋊C8—31no—
60C2×C22⋊C8—31no—
61#390—32no—
62#389—32no—
63#388—62no—
64#386—61no—
65#380—31no—
66#340—32no—
67#340—32no—
68#336—32no—
69#336—32no—
70C24×C3—41no—
71C23×D3—41no—
72C22×C3×C4—42no—
73C22×C3×C4—84no—
74C22×C3×C4—82no—
75C2×#67—44no—
76C2×#67—82no—
77C2×#67—82no—
78C22×Dic3—41no—
79C22×Dic3—42no—
80C2×SL(2,𝔽3)—22no—
81C2×SL(2,𝔽3)C2×D411yescharacteristic
82C2×SL(2,𝔽3)#4112yes—
83GL(2,𝔽3)—41no—
84#217—82no—
85(C2×D3)⋊C4—84no—
86C22×Q8—22no—
87C22×Q8—31no—
88C22×Q8—32no—
89C22×Q8—62no—
90C22×Q8#6712yes—
91C22×Q8C22×D311yescharacteristic
92C22×D4—31no—
93C22×D4—61no—
94C23×C4—31no—
95C23×C4—31no—
96C23×C4S411yescharacteristic
97C2×SD16—62no—
98C2×SD16—64no—
99C2×SD16—62no—
100C2×SD16—61no—
101C2×SD16—62no—
102C2×SD16—61no—
103C22×C8—31no—
104C22×C8—31no—
105C22×C8—32no—
106#123—124no—
107#121—62no—
108#121—62no—
109C4×Q8—22no—
110C4×Q8—64no—
111C4×Q8—62no—
112C4×Q8—64no—
113C2×C4⋊C4—32no—
114C2×C4⋊C4—34no—
115C2×C4⋊C4—31no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—61no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—62no—
120C2×C22⋊C4—61no—
121C2×C22⋊C4—62no—
122C2×C22⋊C4—61no—
123C2×C22⋊C4—62no—
124C2×C42—32no—
125C2×C42—32no—
126C2×C42—62no—
127C2×C42—62no—
128#107—64no—
129#103—62no—
130#103—64no—
131#103—62no—
132#103—61no—
133#103—62no—
134#103—61no—
135#102—64no—
136C22⋊C8—62no—
137C22⋊C8—62no—
138#95—61no—
139#95—62no—
140C23×C3—42no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—41no—
144C23×C3—82no—
145C23×C3—82no—
146C22×D3—42no—
147C22×D3—82no—
148C22×D3—81no—
149C22×D3—82no—
150C22×D3—81no—
151C2×C3×C4—44no—
152C2×C3×C4—82no—
153C2×C3×C4—82no—
154C2×C3×C4—88no—
155#67—88no—
156C2×Dic3—42no—
157C2×Dic3—81no—
158C2×Dic3—81no—
159C2×Dic3—82no—
160C2×Dic3—82no—
161C2×Dic3—82no—
162SL(2,𝔽3)#12311yescharacteristic, complemented
163C24—61no—
164C24C2×S411yescharacteristic
165C2×Q8—22no—
166C2×Q8—32no—
167C2×Q8—31no—
168C2×Q8—32no—
169C2×Q8—34no—
170C2×Q8—61no—
171C2×Q8—62no—
172C2×Q8—64no—
173C2×Q8—62no—
174C2×Q8—64no—
175C2×Q8#23512yes—
176C2×Q8C2×#6711yescharacteristic
177C2×D4—61no—
178C2×D4—61no—
179C2×D4—62no—
180C2×D4—62no—
181C2×D4—122no—
182C2×D4—122no—
183C2×D4—122no—
184C22×C4—22no—
185C22×C4—24no—
186C22×C4—31no—
187C22×C4—32no—
188C22×C4—32no—
189C22×C4—31no—
190C22×C4—32no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—62no—
194C22×C4—62no—
195C22×C4—62no—
196C22×C4—62no—
197C22×C4—61no—
198C22×C4C2×S412yes—
199SD16—122no—
200SD16—121no—
201SD16—124no—
202SD16—121no—
203C2×C8—62no—
204C2×C8—61no—
205C2×C8—61no—
206C2×C8—62no—
207C2×C8—62no—
208C2×C8—62no—
209C2×C8—62no—
210C4⋊C4—34no—
211C4⋊C4—62no—
212C4⋊C4—64no—
213C4⋊C4—61no—
214C4⋊C4—62no—
215C4⋊C4—62no—
216C4⋊C4—61no—
217C4⋊C4—124no—
218C22⋊C4—122no—
219C22⋊C4—124no—
220C22⋊C4—122no—
221C22⋊C4—124no—
222C42—62no—
223C42—62no—
224C42—64no—
225C22×C3—42no—
226C22×C3—42no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—44no—
231C22×C3—82no—
232C22×C3—84no—
233C22×C3—84no—
234C22×C3—82no—
235C2×D3—82no—
236C2×D3—84no—
237C2×D3—161no—
238C2×D3—162no—
239C2×D3—161no—
240C3×C4—84no—
241Dic3—82no—
242Dic3—162no—
243C23—22no—
244C23—22no—
245C23—122no—
246C23—122no—
247C23—121no—
248C23—121no—
249C23—121no—
250C23#98512yes—
251C23#98612yes—
252C23GL(2,ℤ/4)12yes—
253C23C22×S411yescharacteristic
254Q8—32no—
255Q8—31no—
256Q8—34no—
257Q8—61no—
258Q8—62no—
259Q8—64no—
260Q8#92911yescharacteristic
261D4—121no—
262D4—121no—
263D4—122no—
264D4—244no—
265C2×C4—28no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—32no—
269C2×C4—32no—
270C2×C4—31no—
271C2×C4—31no—
272C2×C4—32no—
273C2×C4—62no—
274C2×C4—62no—
275C2×C4—64no—
276C2×C4—62no—
277C2×C4—62no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—124no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—121no—
284C2×C4—122no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4#98314yes—
288C8—122no—
289C8—121no—
290C8—121no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—42no—
294C2×C3—42no—
295C2×C3—41no—
296C2×C3—84no—
297D3—162no—
298C22—22no—
299C22—24no—
300C22—24no—
301C22—22no—
302C22—241no—
303C22—241no—
304C22—242no—
305C22—242no—
306C22—241no—
307C22#542714yes—
308C22#591212yes—
309C22C2×#98312yes—
310C22C2×#98511yescharacteristic
311C22C2×#98611yescharacteristic
312C22C2×GL(2,ℤ/4)11yescharacteristic
313C4—24no—
314C4—31no—
315C4—31no—
316C4—32no—
317C4—62no—
318C4—122no—
319C4—241no—
320C4—242no—
321C3—41noSylow
322C2—24no—
323C2—481no—
324C2#8532912yes—
325C2#8533212yes—
326C2#8548511yescharacteristic
327C2#8550711yescharacteristic
328C2C2×#542711yescharacteristic
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. #73062C2semidirect
15. C22×C4×Q8D3semidirect
162. SL(2,𝔽3)#123semidirect

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
#730622
#330033
#52134