Finite Group Databasebeta

← #11713033

Subgroups — #11713033

Order 768.

437 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 437 conjugacy classes of subgroups — too many to draw. Only its 61 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#85452C212yesmaximal
2C2×#5405C211yesmaximal, characteristic
3#2916⋊C3C211yesmaximal, characteristic, complemented
4#73129C212yesmaximal, complemented
5#73120C211yesmaximal, characteristic, complemented
6#22695—31nomaximal, Sylow
7C2×#985C2211yescharacteristic
8#568⋊C3C2212yes—
9C2×#131⋊C3C2211yescharacteristic
10#5423C2212yes—
11C2×GU(2,𝔽3)C2211yescharacteristic
12#5405—22no—
13#5405C412yescomplemented
14#5405C412yescomplemented
15#5404—22no—
16#4893—41nomaximal
17#2957—32no—
18C2×#444—31no—
19#2916—31no—
20#2916D311yescharacteristic, complemented
21#1953—32no—
22#1861—32no—
23#1861—32no—
24#1849—31no—
25#1814—31no—
26#1814—31no—
27#1773—31no—
28C2×#41⋊C3C2311yescharacteristic
29#985C2×C412yes—
30#985D412yes—
31C2×GL(2,𝔽3)Q811yescharacteristic
32C2×#226Q811yescharacteristic
33D3×M16—42no—
34C2×ΓL(1,𝔽25)—41no—
35#131⋊C3—22no—
36#131⋊C3C2×C411yescharacteristic
37#131⋊C3C2×C411yescharacteristic
38GU(2,𝔽3)—22no—
39GU(2,𝔽3)C2×C412yes—
40C3×#108—41no—
41#822—42no—
42#819—41no—
43C2×#135—31no—
44#568—32no—
45#568C2×D312yes—
46C2×#131—31no—
47C2×#131C2×D311yescharacteristic
48#444—32no—
49#444—32no—
50#444—32no—
51#444—32no—
52#444—62no—
53#444—62no—
54#443—62no—
55#443—62no—
56#430—32no—
57#430—32no—
58#430—61no—
59#430—61no—
60C2×#108—31no—
61C2×#108—31no—
62C2×#108—31no—
63C2×#108—31no—
64C2×#108—32no—
65C2×#108—62no—
66#427—31no—
67#423—32no—
68#421—32no—
69#421—32no—
70C2×#104—31no—
71C2×#104—31no—
72#405—32no—
73C2×#97—31no—
74#344—32no—
75#344—32no—
76#339—32no—
77C2×C4×D3—41no—
78#41⋊C3C22×C411yescharacteristic
79#41⋊C3C2×D411yescharacteristic
80C2×SL(2,𝔽3)C2×Q811yescharacteristic
81GL(2,𝔽3)C4⋊C412yes—
82#226C4⋊C412yes—
83C3×M16—42no—
84C2×C3×C8—41no—
85#208—42no—
86C2×C3⋊C8—41no—
87ΓL(1,𝔽25)—42no—
88ΓL(1,𝔽25)—42no—
89ΓL(1,𝔽25)—82no—
90D3×C8—82no—
91C2×#41—31no—
92C2×#41C22×D311yescharacteristic
93#135—32no—
94#135—32no—
95#135—32no—
96#135—32no—
97C2×Q16—31no—
98C2×SD16—31no—
99C2×SD16—31no—
100C2×D8—31no—
101#131—22no—
102#131—31no—
103#131—31no—
104#131—32no—
105#131—32no—
106#131—61no—
107#131—61no—
108#131—62no—
109#131—62no—
110#131—62no—
111#131—122no—
112#131C4×D311yescharacteristic
113#131C4×D311yescharacteristic
114C2×M16—31no—
115C2×M16—31no—
116C2×M16—32no—
117C2×M16—32no—
118C2×M16—32no—
119C2×M16—61no—
120C2×M16—62no—
121C2×M16—62no—
122C22×C8—31no—
123C22×C8—31no—
124C22×C8—61no—
125#117—32no—
126C2×C42—31no—
127#108—32no—
128#108—32no—
129#108—32no—
130#108—31no—
131#108—61no—
132#108—61no—
133#108—62no—
134#108—62no—
135#108—62no—
136#108—62no—
137#108—61no—
138#108S411yescharacteristic
139#107—61no—
140#106—61no—
141C4⋊C8—122no—
142#104—32no—
143#104—32no—
144#104—32no—
145#104—32no—
146#104—62no—
147#104—62no—
148#104—62no—
149#104—62no—
150#97—32no—
151#97—32no—
152#97—62no—
153C4×C8—62no—
154C22×D3—41no—
155C2×C3×C4—41no—
156C2×Dic3—41no—
157C4×D3—42no—
158C4×D3—42no—
159SL(2,𝔽3)C2×C4⋊C411yescharacteristic
160C3×C8—41no—
161C3×C8—41no—
162C3×C8—82no—
163C3⋊C8—42no—
164C3⋊C8—82no—
165#41—31no—
166#41—31no—
167#41—32no—
168#41—32no—
169#41—61no—
170#41—61no—
171#41C2×C4×D311yescharacteristic
172#41D3×D411yescharacteristic
173C2×Q8—31no—
174C2×Q8D3×Q811yescharacteristic
175C2×D4—31no—
176C2×D4—31no—
177C2×D4—61no—
178C22×C4—31no—
179C22×C4—61no—
180C22×C4—61no—
181Q16—32no—
182Q16—32no—
183SD16—32no—
184SD16—32no—
185SD16—32no—
186SD16—32no—
187D8—32no—
188D8—32no—
189M16—31no—
190M16—31no—
191M16—32no—
192M16—32no—
193M16—62no—
194M16—61no—
195M16—61no—
196M16—62no—
197M16—62no—
198M16—62no—
199M16—62no—
200M16—62no—
201M16—122no—
202M16—122no—
203M16C2×S412yes—
204C2×C8—32no—
205C2×C8—32no—
206C2×C8—31no—
207C2×C8—31no—
208C2×C8—31no—
209C2×C8—31no—
210C2×C8—31no—
211C2×C8—31no—
212C2×C8—31no—
213C2×C8—62no—
214C2×C8—62no—
215C2×C8—62no—
216C2×C8—61no—
217C2×C8—62no—
218C2×C8—62no—
219C2×C8—122no—
220C2×C8—122no—
221C2×C8C2×S411yescharacteristic
222C4⋊C4—62no—
223C22⋊C4—62no—
224C42—32no—
225C42—61no—
226C42—62no—
227C22×C3—41no—
228C2×D3—42no—
229C2×D3—161no—
230C3×C4—41no—
231C3×C4—41no—
232Dic3—42no—
233C23—31no—
234C23—61no—
235Q8—31no—
236Q8—32no—
237Q8—61no—
238Q8D3×C4⋊C411yescharacteristic
239D4—31no—
240D4—31no—
241D4—32no—
242D4—61no—
243D4—61no—
244D4—62no—
245D4—62no—
246C2×C4—31no—
247C2×C4—31no—
248C2×C4—31no—
249C2×C4—31no—
250C2×C4—31no—
251C2×C4—61no—
252C2×C4—62no—
253C2×C4—62no—
254C2×C4—62no—
255C2×C4—122no—
256C2×C4—241no—
257C2×C4C22×S411yescharacteristic
258C8—22no—
259C8—32no—
260C8—32no—
261C8—31no—
262C8—31no—
263C8—62no—
264C8—62no—
265C8—122no—
266C8C4×S411yescharacteristic
267C8C4×S411yescharacteristic
268C2×C3—41no—
269C2×C3—81no—
270D3—82no—
271C22—31no—
272C22—31no—
273C22—62no—
274C22—121no—
275C22—241no—
276C22Q8×S411yescharacteristic
277C4—31no—
278C4—31no—
279C4—62no—
280C4—122no—
281C4—242no—
282C4C2×C4×S411yescharacteristic
283C4D4×S411yescharacteristic
284C3—41noSylow
285C2—21no—
286C2—61no—
287C2—61no—
288C2—122no—
289C2C4⋊C4×S411yescharacteristic
subgroupG/H
centerC4D4×S4
commutator#41⋊C3C22×C4
FrattiniC2×C4C22×S4
Fitting#2916D3
socleC2C4⋊C4×S4
prime radicalC2×#985C22
cosocleC2×#41⋊C3C23
hypercenter#108S4
hypocenterSL(2,𝔽3)C2×C4⋊C4
LCSG ▷ #41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C4 ◁ C2×C4 ◁ #108

Complements

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

NH ≅ G/Ntype
3. #2916⋊C3C2semidirect
4. #73129C2semidirect
5. #73120C2semidirect
13. #5405C4semidirect
14. #5405C4semidirect
20. #2916D3semidirect

Maximal subgroups

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

subgroupindex
#854522
C2×#54052
#2916⋊C32
#731292
#731202
#226953
#48934