Finite Group Databasebeta

← #11713236

Subgroups — #11713236

Order 768.

536 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 57 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×#5426C211yesmaximal, characteristic, complemented
3#73151C212yesmaximal, complemented
4#73151C211yesmaximal, characteristic, complemented
5#73062C212yesmaximal, complemented
6#24046—31nomaximal, Sylow
7C22×#41⋊C3C2211yescharacteristic
8C22×#41⋊C3C2211yescharacteristic
9C4⋊(#41⋊C3)—22no—
10C2×C4×SL(2,𝔽3)C2211yescharacteristic
11#5426—22no—
12C2×#859C2212yes—
13C2×#859C2212yes—
14#4986—41nomaximal
15C2×#550D311yescharacteristic
16C2×#480—31no—
17C2×#417—31no—
18#2487—31no—
19#2041—32no—
20#2033—32no—
21#1895—31no—
22#1870—31no—
23#1863—32no—
24#1788—32no—
25#1788—31no—
26C2×#41⋊C3—21no—
27C2×#41⋊C3—22no—
28C2×#41⋊C3—22no—
29C2×#41⋊C3D412yes—
30C22×SL(2,𝔽3)C2311yescharacteristic
31C2×C3×C4⋊C4—41no—
32C2×C4⋊Dic3—41no—
33C4×SL(2,𝔽3)—21no—
34#859—22no—
35#859—22no—
36C3⋊#95—42no—
37C3⋊#95—42no—
38C3⋊#95—41no—
39C22×#41C2×D311yescharacteristic
40C22×#41C2×D311yescharacteristic
41C22×M16—31no—
42#550—22no—
43#550—62no—
44C2×#127—31no—
45C2×#121—31no—
46C2×#121—31no—
47C2×C4×Q8C2×D311yescharacteristic
48C2×C4×D4—31no—
49C22×C4⋊C4—31no—
50#480—64no—
51C2×#107—32no—
52#417—62no—
53#417—62no—
54C2×#103—32no—
55C2×#103—32no—
56C2×#102—32no—
57C2×#102—32no—
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—
71C22×C3×C4—41no—
72C22×C3×C4—41no—
73C22×C3×C4—41no—
74C22×Dic3—42no—
75C22×Dic3—42no—
76#41⋊C3—22no—
77#41⋊C3—41no—
78C2×SL(2,𝔽3)C2×D411yescharacteristic
79C2×SL(2,𝔽3)#4112yes—
80C3×C4⋊C4—82no—
81C4⋊Dic3—82no—
82C2×#41—22no—
83C2×#41—22no—
84C2×#41—21no—
85C2×#41—32no—
86C2×#41—61no—
87C2×#41—62no—
88C2×#41—62no—
89C2×#41#6712yes—
90C22×Q8C22×D311yescharacteristic
91C22×D4—31no—
92C22×D4—31no—
93C22×D4—31no—
94C22×D4—61no—
95C23×C4—31no—
96C23×C4—31no—
97C23×C4—61no—
98C2×M16—62no—
99C2×M16—122no—
100C22×C8—32no—
101C22×C8—32no—
102#127—62no—
103#127—62no—
104#121—34no—
105#121—62no—
106#121—62no—
107#121—62no—
108C4×Q8—21no—
109C4×Q8—61no—
110C4×D4—62no—
111C4×D4—62no—
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—62no—
119C2×C4⋊C4S411yescharacteristic
120C2×C22⋊C4—31no—
121C2×C22⋊C4—31no—
122C2×C22⋊C4—61no—
123C2×C22⋊C4—62no—
124C2×C42—31no—
125#107—64no—
126#103—62no—
127#103—62no—
128#103—62no—
129#103—62no—
130#102—64no—
131#102—64no—
132#102—122no—
133#102—122no—
134C22⋊C8—62no—
135C22⋊C8—122no—
136#95—62no—
137#95—61no—
138#95—62no—
139C23×C3—41no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—81no—
143C2×C3×C4—82no—
144C2×C3×C4—81no—
145C2×C3×C4—81no—
146C2×C3×C4—82no—
147C2×Dic3—82no—
148C2×Dic3—82no—
149C2×Dic3—82no—
150C2×Dic3—82no—
151C2×Dic3—82no—
152C2×Dic3—82no—
153SL(2,𝔽3)#12311yescharacteristic
154C24—31no—
155C24—31no—
156#41—22no—
157#41—41no—
158#41—62no—
159#41—64no—
160#41—122no—
161#41—122no—
162#41—122no—
163#41—121no—
164C2×Q8—31no—
165C2×Q8—32no—
166C2×Q8#23712yes—
167C2×Q8C2×#6711yescharacteristic
168C2×D4—32no—
169C2×D4—32no—
170C2×D4—62no—
171C2×D4—62no—
172C2×D4—62no—
173C2×D4—62no—
174C2×D4—62no—
175C2×D4—61no—
176C2×D4—61no—
177C2×D4—64no—
178C2×D4—62no—
179C2×D4—121no—
180C2×D4—242no—
181C22×C4—31no—
182C22×C4—32no—
183C22×C4—32no—
184C22×C4—31no—
185C22×C4—62no—
186C22×C4—62no—
187C22×C4—62no—
188C22×C4—61no—
189C22×C4—61no—
190C22×C4—62no—
191C22×C4—62no—
192C22×C4—62no—
193C22×C4—122no—
194C22×C4—242no—
195C22×C4C2×S411yescharacteristic
196C22×C4C2×S411yescharacteristic
197C22×C4C2×S411yescharacteristic
198M16—124no—
199C2×C8—62no—
200C2×C8—62no—
201C2×C8—62no—
202C2×C8—62no—
203C2×C8—62no—
204C2×C8—62no—
205C4⋊C4—22no—
206C4⋊C4—32no—
207C4⋊C4—61no—
208C4⋊C4—62no—
209C4⋊C4—62no—
210C4⋊C4—62no—
211C4⋊C4—62no—
212C4⋊C4—122no—
213C4⋊C4—242no—
214C22⋊C4—62no—
215C22⋊C4—62no—
216C22⋊C4—122no—
217C22⋊C4—242no—
218C42—61no—
219C42—61no—
220C22×C3—42no—
221C22×C3—41no—
222C22×C3—41no—
223C22×C3—42no—
224C22×C3—41no—
225C3×C4—82no—
226C3×C4—82no—
227C3×C4—161no—
228Dic3—162no—
229Dic3—162no—
230C23—32no—
231C23—62no—
232C23—61no—
233C23—61no—
234C23—62no—
235C23—62no—
236C23—61no—
237C23—61no—
238C23—62no—
239C23—61no—
240C23C22×S411yescharacteristic
241Q8—32no—
242Q8—31no—
243Q8—61no—
244Q8#93511yescharacteristic
245D4—62no—
246D4—62no—
247D4—64no—
248D4—124no—
249D4—122no—
250D4—121no—
251D4—121no—
252D4—244no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—32no—
259C2×C4—32no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—64no—
265C2×C4—62no—
266C2×C4—61no—
267C2×C4—121no—
268C2×C4—121no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4—122no—
276C2×C4—244no—
277C2×C4#98412yes—
278C2×C4GL(2,ℤ/4)12yes—
279C8—122no—
280C8—122no—
281C2×C3—41no—
282C2×C3—41no—
283C2×C3—42no—
284C2×C3—42no—
285C2×C3—41no—
286C22—62no—
287C22—62no—
288C22—62no—
289C22—122no—
290C22—122no—
291C22—121no—
292C22—121no—
293C22—122no—
294C22—122no—
295C22—121no—
296C22#591412yes—
297C22C2×#98411yescharacteristic
298C22#592311yescharacteristic
299C22#592512yes—
300C22C2×GL(2,ℤ/4)11yescharacteristic
301C4—22no—
302C4—22no—
303C4—31no—
304C4—32no—
305C4—31no—
306C4—41no—
307C4—121no—
308C4—242no—
309C4—242no—
310C3—41noSylow
311C2—122no—
312C2—241no—
313C2#8540612yes—
314C2#8541312yes—
315C2#8549111yescharacteristic
316C2#8551711yescharacteristic
317C2#8553011yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingC2×#550D3
prime radicalC2×C4⋊(#41⋊C3)C2
hypercenterC2×C4⋊C4S4
hypocenterSL(2,𝔽3)#123
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#5426C2semidirect
3. #73151C2semidirect
4. #73151C2semidirect
5. #73062C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊(#41⋊C3)2
C2×#54262
#731512
#730622
#240463
#49864