Finite Group Databasebeta

← #11713010

Subgroups — #11713010

Order 768.

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

normal subgroups only

This group has 466 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
1M16×S4C212yesmaximal
2C2×#5400C211yesmaximal, characteristic
3A4×#108C211yesmaximal, characteristic, complemented
4#73114C212yesmaximal, complemented
5#73105C211yesmaximal, characteristic, complemented
6#22668—31nomaximal, Sylow
7C2×C4×S4C2211yescharacteristic
8A4×M16C2212yes—
9C2×C8×A4C2211yescharacteristic
10#5409C2212yes—
11C2×A4⋊C8C2211yescharacteristic
12#5400—22no—
13#5400C414yes—
14C8×S4—22no—
15#4893—41nomaximal
16D4×M16—32no—
17C2×#435—31no—
18C2×#430—31no—
19C22×#108D311yescharacteristic, complemented
20#1950—32no—
21#1855—32no—
22#1855—32no—
23#1849—31no—
24#1811—31no—
25#1810—31no—
26#1778—31no—
27C22×S4Q811yescharacteristic
28C2×C4×A4C2311yescharacteristic
29C2×SL(2,ℤ/4)Q811yescharacteristic
30C4×S4C2×C412yes—
31C4×S4D412yes—
32D3×M16—42no—
33C2×ΓL(1,𝔽25)—41no—
34C8×A4—22no—
35C8×A4C2×C412yes—
36A4⋊C8—22no—
37A4⋊C8C2×C412yes—
38C3×#108—41no—
39#822—42no—
40#819—41no—
41C22×M16—31no—
42C22×M16—32no—
43C22×M16C2×D312yes—
44C23×C8C2×D311yescharacteristic
45C2×C4×D4—31no—
46#436—62no—
47#435—34no—
48#435—34no—
49#435—62no—
50#435—62no—
51C8×D4—62no—
52#430—62no—
53#430—61no—
54#430—61no—
55C2×#108—31no—
56C2×#108—31no—
57C2×#108—32no—
58C2×#108—31no—
59C2×#108—31no—
60C2×#108—32no—
61C2×#108—32no—
62#423—32no—
63C2×C4⋊C8—31no—
64#407—32no—
65#407—32no—
66C2×C22⋊C8—31no—
67C2×C22⋊C8—31no—
68C4×M16—32no—
69C2×#97—31no—
70#341—32no—
71#341—32no—
72#341—32no—
73C22×A4C2×Q811yescharacteristic
74C2×S4—41no—
75C2×S4C4⋊C412yes—
76C2×C4×D3—41no—
77C4×A4C22×C411yescharacteristic
78C4×A4C2×D411yescharacteristic
79SL(2,ℤ/4)C4⋊C412yes—
80C3×M16—42no—
81C2×C3×C8—41no—
82#208—42no—
83C2×C3⋊C8—41no—
84ΓL(1,𝔽25)—44no—
85ΓL(1,𝔽25)—82no—
86D3×C8—82no—
87C22×D4—31no—
88C23×C4—31no—
89C23×C4C22×D311yescharacteristic
90C2×M16—34no—
91C2×M16—32no—
92C2×M16—32no—
93C2×M16—32no—
94C2×M16—32no—
95C2×M16—62no—
96C2×M16—61no—
97C2×M16—62no—
98C2×M16—62no—
99C2×M16—61no—
100C2×M16—62no—
101C2×M16—62no—
102C2×M16—122no—
103C22×C8—22no—
104C22×C8—31no—
105C22×C8—31no—
106C22×C8—32no—
107C22×C8—31no—
108C22×C8—31no—
109C22×C8—62no—
110C22×C8C4×D312yes—
111C4×D4—32no—
112C4×D4—32no—
113C4×D4—32no—
114C4×D4—32no—
115C2×C4⋊C4—31no—
116C2×C22⋊C4—31no—
117C2×C22⋊C4—31no—
118C2×C42—31no—
119#108—32no—
120#108—31no—
121#108—61no—
122#108—62no—
123#108—61no—
124#108—62no—
125#108—62no—
126#108—61no—
127#108—122no—
128#108S411yescharacteristic, complemented
129C4⋊C8—34no—
130C4⋊C8—62no—
131C4⋊C8—62no—
132C22⋊C8—34no—
133C22⋊C8—32no—
134C22⋊C8—32no—
135C22⋊C8—62no—
136C22⋊C8—62no—
137C22⋊C8—62no—
138C22⋊C8—62no—
139#97—34no—
140#97—62no—
141C4×C8—62no—
142C22×D3—41no—
143C2×A4—21no—
144C2×A4C2×C4⋊C411yescharacteristic
145S4—22no—
146C2×C3×C4—41no—
147C2×Dic3—41no—
148C4×D3—42no—
149C4×D3—42no—
150C3×C8—42no—
151C3×C8—82no—
152C3⋊C8—42no—
153C3⋊C8—82no—
154C24—31no—
155C24D3×Q811yescharacteristic
156C2×D4—32no—
157C2×D4—32no—
158C2×D4—121no—
159C2×D4—121no—
160C22×C4—31no—
161C22×C4—31no—
162C22×C4—31no—
163C22×C4—31no—
164C22×C4—32no—
165C22×C4—32no—
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—61no—
170C22×C4—61no—
171C22×C4—61no—
172C22×C4—61no—
173C22×C4C2×C4×D311yescharacteristic
174C22×C4D3×D411yescharacteristic
175M16—32no—
176M16—62no—
177M16—64no—
178M16—64no—
179M16—62no—
180M16—62no—
181M16—62no—
182M16—62no—
183M16—62no—
184M16—122no—
185M16—122no—
186M16C2×S412yes—
187C2×C8—31no—
188C2×C8—32no—
189C2×C8—32no—
190C2×C8—32no—
191C2×C8—32no—
192C2×C8—32no—
193C2×C8—62no—
194C2×C8—61no—
195C2×C8—61no—
196C2×C8—61no—
197C2×C8—61no—
198C2×C8—61no—
199C2×C8—62no—
200C2×C8—62no—
201C2×C8—62no—
202C2×C8—122no—
203C2×C8—122no—
204C2×C8C2×S411yescharacteristic
205C4⋊C4—32no—
206C4⋊C4—32no—
207C22⋊C4—32no—
208C22⋊C4—32no—
209C22⋊C4—32no—
210C22⋊C4—32no—
211C42—32no—
212C42—32no—
213C22×C3—41no—
214C2×D3—42no—
215C2×D3—161no—
216A4#43011yescharacteristic, complemented
217C3×C4—41no—
218C3×C4—41no—
219Dic3—42no—
220C23—21no—
221C23—31no—
222C23—31no—
223C23—32no—
224C23—61no—
225C23—61no—
226C23—61no—
227C23—121no—
228C23—121no—
229C23D3×C4⋊C411yescharacteristic
230D4—62no—
231D4—62no—
232D4—122no—
233C2×C4—31no—
234C2×C4—31no—
235C2×C4—31no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—31no—
239C2×C4—31no—
240C2×C4—32no—
241C2×C4—32no—
242C2×C4—32no—
243C2×C4—61no—
244C2×C4—62no—
245C2×C4—62no—
246C2×C4—61no—
247C2×C4—61no—
248C2×C4—62no—
249C2×C4—62no—
250C2×C4—121no—
251C2×C4—121no—
252C2×C4C22×S411yescharacteristic
253C8—22no—
254C8—32no—
255C8—62no—
256C8—62no—
257C8—62no—
258C8—122no—
259C8—122no—
260C8C4×S412yes—
261C2×C3—41no—
262C2×C3—81no—
263D3—82no—
264C22—31no—
265C22—31no—
266C22—31no—
267C22—61no—
268C22—61no—
269C22—61no—
270C22—61no—
271C22—62no—
272C22—62no—
273C22—62no—
274C22—241no—
275C22—241no—
276C22#489311yescharacteristic, complemented
277C22Q8×S411yescharacteristic
278C4—31no—
279C4—31no—
280C4—62no—
281C4—62no—
282C4—62no—
283C4C2×C4×S411yescharacteristic
284C4D4×S411yescharacteristic
285C3—41noSylow
286C2—21no—
287C2—31no—
288C2—31no—
289C2—61no—
290C2—122no—
291C2C4⋊C4×S411yescharacteristic
subgroupG/H
centerC4D4×S4
commutatorC4×A4C22×C4
FrattiniC2×C4C22×S4
FittingC22×#108D3
socleC23D3×C4⋊C4
prime radicalC22×S4Q8
cosocleC2×C4×A4C23
hypercenter#108S4
hypocenterA4#430
LCSG ▷ C4×A4 ▷ C2×A4 ▷ A4
UCSC1 ◁ C4 ◁ C2×C4 ◁ #108

Complements

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

NH ≅ G/Ntype
3. A4×#108C2semidirect
4. #73114C2semidirect
5. #73105C2semidirect
19. C22×#108D3semidirect
128. #108S4semidirect
216. A4#430semidirect
276. C22#4893semidirect

Maximal subgroups

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

subgroupindex
M16×S42
C2×#54002
A4×#1082
#731142
#731052
#226683
#48934