Finite Group Databasebeta

← #11711811

Subgroups — #11711811

Order 768.

399 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#3482⋊C3C211yesmaximal, characteristic
2#73072C211yesmaximal, characteristic, complemented
3#73068C211yesmaximal, characteristic, complemented
4#73056C212yesmaximal, complemented
5#73055C212yesmaximal, complemented
6#17525—31nomaximal, Sylow
7#557⋊C3C411yescharacteristic
8D4×SL(2,𝔽3)C411yescharacteristic
9#548⋊C3C412yes—
10C4×#41⋊C3C2211yescharacteristic
11C2×C4×SL(2,𝔽3)C2212yes—
12#5012—41nomaximal
13SL(2,𝔽3)⋊C8C2211yescharacteristic
14SL(2,𝔽3)⋊C8C2212yes—
15SL(2,𝔽3)⋊C8C2211yescharacteristic
16#3482D311yescharacteristic
17#2991—31no—
18#2922—31no—
19#1495—32no—
20#1493—31no—
21#1489—31no—
22#1487—32no—
23#1485—31no—
24#1482—32no—
25#1479—32no—
26#1476—31no—
27C2×#41⋊C3C2×C411yescharacteristic
28C22×SL(2,𝔽3)C2×C412yes—
29C3×C4×D4—41no—
30C2×#208—42no—
31C4×SL(2,𝔽3)C2×C412yes—
32C4×SL(2,𝔽3)C2×C411yescharacteristic
33C4×SL(2,𝔽3)C2311yescharacteristic
34#830—42no—
35C4⋊(C3⋊C8)—41no—
36C4×C3⋊C8—41no—
37#557Dic311yescharacteristic
38#556—31no—
39#552—32no—
40D4×Q8Dic311yescharacteristic
41#548Dic312yes—
42#547—31no—
43#533—31no—
44#529—32no—
45C4×#41—31no—
46C4×#41C2×D311yescharacteristic
47C2×C4×Q8C2×D312yes—
48C8×Q8—61no—
49#436—62no—
50#436—61no—
51#433—62no—
52#431—32no—
53#431—32no—
54#431—31no—
55#431—32no—
56#424—61no—
57#423—32no—
58#423—31no—
59#408—61no—
60C4×M16—31no—
61C2×C4×C8—31no—
62Q8⋊C8—32no—
63Q8⋊C8—32no—
64Q8⋊C8—32no—
65Q8⋊C8—32no—
66Q8⋊C8—31no—
67Q8⋊C8—31no—
68Q8⋊C8—32no—
69D4⋊C8—31no—
70D4⋊C8—31no—
71D4⋊C8—32no—
72C2×C3×D4—41no—
73C22×C3×C4—42no—
74#41⋊C3—21no—
75C2×SL(2,𝔽3)—22no—
76C2×SL(2,𝔽3)C22×C411yescharacteristic
77C3×C4⋊C4—41no—
78C3×C22⋊C4—42no—
79C3×C42—41no—
80#208—84no—
81C2×C3⋊C8—41no—
82C2×C3⋊C8—41no—
83C2×C3⋊C8—42no—
84C2×#41—31no—
85C2×#41C2×Dic311yescharacteristic
86C22×Q8C2×Dic312yes—
87C2×M16—61no—
88C2×M16—61no—
89C2×M16—62no—
90C22×C8—61no—
91C4⋊Q8—31no—
92C4⋊Q8—31no—
93#126—62no—
94#125—32no—
95#125—61no—
96#124—32no—
97#123—62no—
98C22⋊Q8—32no—
99C22⋊Q8—32no—
100C22⋊Q8—61no—
101#121—32no—
102C4×Q8—31no—
103C4×Q8—31no—
104C4×Q8—32no—
105C4×Q8—32no—
106C4×Q8C2×Dic312yes—
107C4×Q8C2×Dic311yescharacteristic
108C4×Q8C22×D311yescharacteristic
109C4×D4—31no—
110C4×D4—32no—
111C4×D4—32no—
112C4×D4—31no—
113C4×D4—31no—
114C4×D4S411yescharacteristic
115#117—31no—
116#117—32no—
117#117—31no—
118#117—61no—
119C2×C4⋊C4—32no—
120C2×C42—32no—
121C2×C42—31no—
122C4⋊C8—31no—
123C4⋊C8—32no—
124C4⋊C8—31no—
125C4⋊C8—62no—
126C4⋊C8—62no—
127C4⋊C8—62no—
128C4⋊C8—61no—
129C22⋊C8—62no—
130C22⋊C8—62no—
131C22⋊C8—61no—
132C22⋊C8—62no—
133#97—61no—
134C4×C8—31no—
135C4×C8—31no—
136C4×C8—32no—
137C4×C8—61no—
138C4×C8—62no—
139C4×C8—61no—
140C23×C3—42no—
141C3×D4—82no—
142C2×C3×C4—41no—
143C2×C3×C4—42no—
144C2×C3×C4—41no—
145C2×C3×C4—41no—
146C2×C3×C4—82no—
147C2×C3×C4—82no—
148SL(2,𝔽3)#13111yescharacteristic
149C3⋊C8—42no—
150C3⋊C8—81no—
151C3⋊C8—82no—
152#41—21no—
153#41—61no—
154#41—122no—
155C2×Q8—22no—
156C2×Q8—31no—
157C2×Q8—32no—
158C2×Q8—62no—
159C2×Q8C22×Dic311yescharacteristic
160C2×D4—31no—
161C2×D4—32no—
162C2×D4SL(2,ℤ/4)11yescharacteristic
163C22×C4—32no—
164C22×C4—32no—
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4C2×S412yes—
169M16—122no—
170M16—242no—
171M16—241no—
172C2×C8—61no—
173C2×C8—61no—
174C2×C8—62no—
175C2×C8—61no—
176C2×C8—61no—
177C2×C8—62no—
178C2×C8—122no—
179C4⋊C4—31no—
180C4⋊C4—32no—
181C4⋊C4—31no—
182C4⋊C4—32no—
183C4⋊C4—31no—
184C4⋊C4—32no—
185C4⋊C4—62no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4SL(2,ℤ/4)11yescharacteristic
189C22⋊C4—32no—
190C22⋊C4—32no—
191C22⋊C4—61no—
192C22⋊C4—62no—
193C22⋊C4SL(2,ℤ/4)12yes—
194C42—31no—
195C42—32no—
196C42—31no—
197C42—32no—
198C42—31no—
199C42—32no—
200C42C2×S411yescharacteristic
201C22×C3—41no—
202C22×C3—82no—
203C22×C3—82no—
204C22×C3—82no—
205C3×C4—41no—
206C3×C4—41no—
207C3×C4—42no—
208C3×C4—82no—
209C3×C4—81no—
210C23—31no—
211C23C2×SL(2,ℤ/4)12yes—
212Q8—31no—
213Q8—62no—
214Q8—62no—
215Q8—62no—
216Q8#94811yescharacteristic
217D4—22no—
218D4—61no—
219D4—61no—
220D4—122no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—31no—
226C2×C4—32no—
227C2×C4—32no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4—62no—
231C2×C4—62no—
232C2×C4—61no—
233C2×C4—61no—
234C2×C4—62no—
235C2×C4—122no—
236C2×C4C2×SL(2,ℤ/4)12yes—
237C2×C4C2×SL(2,ℤ/4)11yescharacteristic
238C2×C4C2×SL(2,ℤ/4)11yescharacteristic
239C2×C4C22×S411yescharacteristic
240C8—121no—
241C8—121no—
242C8—122no—
243C8—121no—
244C8—122no—
245C8—121no—
246C2×C3—41no—
247C2×C3—41no—
248C2×C3—41no—
249C2×C3—162no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—61no—
254C22—61no—
255C22—61no—
256C22C22×SL(2,ℤ/4)11yescharacteristic
257C4—21no—
258C4—22no—
259C4—31no—
260C4—31no—
261C4—61no—
262C4—62no—
263C4—62no—
264C4—61no—
265C4#542312yes—
266C4#592611yescharacteristic
267C4#592711yescharacteristic
268C3—41noSylow
269C2—42no—
270C2—121no—
271C2C2×#542311yescharacteristic
272C2#8551111yescharacteristic
273C2#8553411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×S4
commutatorC2×SL(2,𝔽3)C22×C4
Fitting#3482D3
socleC22C22×SL(2,ℤ/4)
prime radical#3482⋊C3C2
cosocleC4×SL(2,𝔽3)C23
hypercenterC4×D4S4
hypocenterSL(2,𝔽3)#131
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #73072C2semidirect
3. #73068C2semidirect
4. #73056C2semidirect
5. #73055C2semidirect

Maximal subgroups

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

subgroupindex
#3482⋊C32
#730722
#730682
#730562
#730552
#175253
#50124