Finite Group Databasebeta

← #11713028

Subgroups — #11713028

Order 768.

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

normal subgroups only

This group has 465 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×#5404C211yesmaximal, characteristic
2#85364C212yesmaximal
3#2913⋊C3C211yesmaximal, characteristic, complemented
4#73126C211yesmaximal, characteristic, complemented
5#73075C212yesmaximal, complemented
6#22691—31nomaximal, Sylow
7C2×#985C2211yescharacteristic
8C2×#131⋊C3C2211yescharacteristic
9#520⋊C3C2212yes—
10#5426C2212yes—
11C2×GU(2,𝔽3)C2211yescharacteristic
12#5404C414yescomplemented
13C4×GL(2,𝔽3)—22no—
14#5388—22no—
15D3×#107—41nomaximal
16C2×#443—31no—
17#2952—32no—
18#2913—31no—
19#2913D311yescharacteristic, complemented
20#1951—32no—
21#1860—32no—
22#1860—32no—
23#1846—31no—
24#1816—31no—
25#1816—31no—
26#1772—31no—
27C2×#41⋊C3C2311yescharacteristic
28#985C2×C412yes—
29#985Q812yes—
30C2×GL(2,𝔽3)D411yescharacteristic
31C2×#226D411yescharacteristic
32C2×D3×C8—41no—
33D3×C4⋊C4—42no—
34#131⋊C3C2×C412yescomplemented
35C4×SL(2,𝔽3)—22no—
36GU(2,𝔽3)C2×C412yescomplemented
37#859—22no—
38C3×#107—41no—
39#818—41no—
40#807—42no—
41C2×#135—31no—
42C2×#131—31no—
43C2×#131C2×D311yescharacteristic
44#520—32no—
45#520C2×D312yes—
46#443—34no—
47#443—34no—
48#442—62no—
49#441—62no—
50C4×SD16—62no—
51C4×SD16—62no—
52C2×#108—31no—
53C2×#108—31no—
54#428—32no—
55#428—32no—
56#428—62no—
57#427—31no—
58#427—61no—
59C2×#107—31no—
60C2×#107—31no—
61C2×#107—61no—
62C2×#106—32no—
63C2×#104—31no—
64C2×#104—31no—
65#417—32no—
66#417—32no—
67#405—32no—
68C2×C4×C8—31no—
69#382—32no—
70#343—32no—
71#343—32no—
72#339—32no—
73C2×C4×D3—41no—
74C2×C4×D3—82no—
75#41⋊C3C22×C411yescharacteristic
76#41⋊C3C2×Q811yescharacteristic
77C2×SL(2,𝔽3)C2×D411yescharacteristic
78GL(2,𝔽3)C4⋊C412yes—
79#226C4⋊C412yes—
80C2×C3×C8—41no—
81C3×C4⋊C4—42no—
82C4⋊Dic3—42no—
83Dic3⋊C4—82no—
84C2×C3⋊C8—41no—
85D3×C8—44no—
86C2×#41—31no—
87C2×#41C22×D311yescharacteristic
88#135—32no—
89#135—32no—
90#135—32no—
91#135—32no—
92C2×Q16—31no—
93C2×SD16—31no—
94C2×SD16—31no—
95C2×D8—31no—
96#131—32no—
97#131—34no—
98#131—62no—
99#131C4×D312yescomplemented
100C2×M16—31no—
101C2×M16—31no—
102C2×M16—32no—
103C2×M16—61no—
104C22×C8—31no—
105C22×C8—31no—
106C22×C8—61no—
107C4×Q8—22no—
108C4×Q8—62no—
109C4×D4—62no—
110C4×D4—62no—
111C4×D4—122no—
112#117—32no—
113#117—62no—
114C2×C4⋊C4—32no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—62no—
117C2×C42—31no—
118#108—34no—
119#108—62no—
120#107—31no—
121#107—61no—
122#107—61no—
123#107—61no—
124#107—62no—
125#107S411yescharacteristic
126#106—32no—
127#106—62no—
128#106—62no—
129#106—62no—
130#104—34no—
131#104—32no—
132#104—32no—
133#103—62no—
134#103—62no—
135#102—62no—
136#102—62no—
137#97—62no—
138C4×C8—34no—
139C4×C8—62no—
140#95—122no—
141C22×D3—41no—
142C2×C3×C4—41no—
143C2×C3×C4—82no—
144C2×Dic3—41no—
145C2×Dic3—82no—
146C4×D3—42no—
147C4×D3—42no—
148C4×D3—164no—
149SL(2,𝔽3)C2×C4⋊C411yescharacteristic
150C3×C8—42no—
151C3⋊C8—42no—
152#41—31no—
153#41—31no—
154#41—32no—
155#41—32no—
156#41—61no—
157#41—61no—
158#41C2×C4×D311yescharacteristic
159#41D3×Q811yescharacteristic
160C2×Q8—31no—
161C2×Q8D3×D411yescharacteristic
162C2×D4—31no—
163C2×D4—31no—
164C2×D4—61no—
165C22×C4—31no—
166C22×C4—61no—
167C22×C4—61no—
168C22×C4—62no—
169C22×C4—62no—
170C22×C4—122no—
171Q16—32no—
172Q16—32no—
173SD16—32no—
174SD16—32no—
175SD16—32no—
176SD16—32no—
177D8—32no—
178D8—32no—
179M16—32no—
180M16—32no—
181M16—61no—
182M16—64no—
183M16—61no—
184M16—62no—
185M16—62no—
186C2×C8—32no—
187C2×C8—31no—
188C2×C8—31no—
189C2×C8—32no—
190C2×C8—31no—
191C2×C8—32no—
192C2×C8—32no—
193C2×C8—61no—
194C2×C8—64no—
195C2×C8—62no—
196C2×C8C2×S411yescharacteristic
197C4⋊C4—32no—
198C4⋊C4—62no—
199C4⋊C4—62no—
200C4⋊C4—62no—
201C4⋊C4—122no—
202C4⋊C4C2×S412yes—
203C22⋊C4—62no—
204C22⋊C4—122no—
205C42—32no—
206C42—62no—
207C42—61no—
208C42—122no—
209C22×C3—41no—
210C2×D3—42no—
211C2×D3—42no—
212C2×D3—42no—
213C3×C4—41no—
214C3×C4—41no—
215C3×C4—162no—
216Dic3—42no—
217Dic3—162no—
218C23—31no—
219C23—61no—
220Q8—31no—
221Q8—32no—
222Q8—61no—
223Q8D3×C4⋊C411yescharacteristic
224D4—31no—
225D4—31no—
226D4—32no—
227D4—61no—
228D4—61no—
229D4—62no—
230D4—62no—
231C2×C4—22no—
232C2×C4—31no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—61no—
238C2×C4—62no—
239C2×C4—62no—
240C2×C4—62no—
241C2×C4—62no—
242C2×C4—121no—
243C2×C4—121no—
244C2×C4—122no—
245C2×C4—242no—
246C2×C4—242no—
247C2×C4—482no—
248C2×C4C22×S411yescharacteristic
249C8—32no—
250C8—32no—
251C8—32no—
252C8—62no—
253C8—122no—
254C8C4×S412yes—
255C2×C3—41no—
256C2×C3—41no—
257C2×C3—41no—
258D3—44no—
259C22—31no—
260C22—31no—
261C22—61no—
262C22—61no—
263C22—62no—
264C22—121no—
265C22—121no—
266C22D4×S411yescharacteristic
267C4—31no—
268C4—31no—
269C4—42no—
270C4—62no—
271C4—122no—
272C4—122no—
273C4—242no—
274C4C2×C4×S411yescharacteristic
275C4Q8×S411yescharacteristic
276C3—41noSylow
277C2—61no—
278C2—61no—
279C2—122no—
280C2C4⋊C4×S411yescharacteristic
281C2#8546611yescharacteristic
282C2#8547311yescharacteristic
subgroupG/H
center, socleC22D4×S4
commutator#41⋊C3C22×C4
FrattiniC2×C4C22×S4
Fitting#2913D3
prime radicalC2×#985C22
cosocleC2×#41⋊C3C23
hypercenter#107S4
hypocenterSL(2,𝔽3)C2×C4⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ #107

Complements

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

NH ≅ G/Ntype
3. #2913⋊C3C2semidirect
4. #73126C2semidirect
5. #73075C2semidirect
12. #5404C4semidirect
19. #2913D3semidirect
34. #131⋊C3C2×C4semidirect
36. GU(2,𝔽3)C2×C4semidirect
99. #131C4×D3semidirect

Maximal subgroups

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

subgroupindex
C2×#54042
#853642
#2913⋊C32
#731262
#730752
#226913
D3×#1074