Finite Group Databasebeta

← #11920691

Subgroups — #11920691

Order 1152.

413 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 45 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×#64267C211yesmaximal, characteristic
2#5827⋊C3C211yesmaximal, characteristic, complemented
3C2×C3×GU(2,𝔽3)C211yesmaximal, characteristic, complemented
4#73078—31nomaximal
5#70366—31nomaximal
6C2×C3×#41⋊C3C2211yescharacteristic
7#64267—21no—
8#64267C412yescomplemented
9C2×C3⋊GL(2,𝔽3)—21no—
10Dic3×SL(2,𝔽3)—21no—
11C3×GU(2,𝔽3)—21no—
12#63564—41nomaximal
13C2×#986—31no—
14C2×#986—61no—
15#5827D311yescharacteristic, complemented
16C2×#949—31no—
17#520⋊C3—31no—
18C2×GU(2,𝔽3)D311yescharacteristic, complemented
19C2×C3×#104—31no—
20#5131—31no—
21#4939—31no—
22#4554—31no—
23#4523—31no—
24C2×C4⋊(C3⋊D3)—41no—
25C3×#41⋊C3C2×C411yescharacteristic
26C3×#41⋊C3D411yescharacteristic
27C2×C3×SL(2,𝔽3)D411yescharacteristic
28C3⋊GL(2,𝔽3)—22no—
29C3×C4⋊Dic3—41no—
30C2×C3×C3⋊C8—41no—
31#1883—91noSylow
32C2×C3×#41C2×D311yescharacteristic
33C22×D12—31no—
34C2×#41⋊C3—21no—
35C2×#41⋊C3C2×D311yescharacteristic
36#986—32no—
37#986—62no—
38#986—62no—
39#986—61no—
40C2×GL(2,𝔽3)—62no—
41C2×GL(2,𝔽3)—61no—
42C2×C3×M16—31no—
43C2×C3×C42—31no—
44#949—32no—
45#949—32no—
46#949—61no—
47#949—61no—
48Q8×Dic3—21no—
49C2×#215—61no—
50D4×Dic3—61no—
51C2×#213—61no—
52C2×(C2×D3)⋊C4—62no—
53#926—31no—
54C2×C4⋊Dic3—31no—
55C2×#208—31no—
56C4×SL(2,𝔽3)—61no—
57GU(2,𝔽3)—21no—
58C3×#104—61no—
59C3×#104—61no—
60#821—61no—
61#821—61no—
62#809—121no—
63C22×C3⋊D3—81no—
64C2×C32×C4—41no—
65C4⋊(C3⋊D3)—42no—
66C4⋊(C3⋊D3)—81no—
67C2×C3×Dic3—81no—
68C3×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
69C3×C3⋊C8—81no—
70C2×AGL(1,ℤ/8)—91no—
71#520—31no—
72C2×#104—31no—
73#418—91no—
74#390—91no—
75#343—91no—
76#339—91no—
77C23×D3—61no—
78C3×#41—31no—
79C3×#41—31no—
80C3×#41C4×D311yescharacteristic
81C3×#41D1211yescharacteristic
82C2×C3×Q8#6711yescharacteristic
83C2×C3×D4—31no—
84C22×C3×C4—31no—
85C22×C3×C4—61no—
86C22×Dic3—32no—
87C2×D12—32no—
88C2×D12—61no—
89C2×D12—61no—
90C2×D12—61no—
91C2×D12—61no—
92C2×D12—61no—
93C2×D12—121no—
94C2×D12—241no—
95#41⋊C3—21no—
96#41⋊C3—21no—
97#41⋊C3C4×D311yescharacteristic
98#41⋊C3#6711yescharacteristic
99C2×SL(2,𝔽3)—21no—
100C2×SL(2,𝔽3)D1211yescharacteristic
101GL(2,𝔽3)—64no—
102GL(2,𝔽3)—62no—
103C3×M16—61no—
104C3×M16—61no—
105C2×C3×C8—61no—
106C3×C4⋊C4—121no—
107C3×C42—61no—
108C3×C42—61no—
109#217—61no—
110#215—62no—
111#215—62no—
112#213—62no—
113#213—62no—
114(C2×D3)⋊C4—124no—
115C4⋊Dic3—31no—
116C4⋊Dic3—61no—
117C4⋊Dic3S411yescharacteristic
118C4×Dic3—61no—
119#208—32no—
120#208—61no—
121C2×C3⋊C8—41no—
122C2×C3⋊C8—61no—
123C22×C32—41no—
124C2×C3⋊D3—82no—
125C2×C3⋊D3—161no—
126C2×C3⋊D3—161no—
127C32×C4—41no—
128C32×C4—41no—
129C3×Dic3—161no—
130C2×#41D3211yescharacteristic
131C22×D4—91no—
132AGL(1,ℤ/8)—92no—
133AGL(1,ℤ/8)—92no—
134AGL(1,ℤ/8)—181no—
135AGL(1,ℤ/8)—181no—
136C2×SD16—181no—
137C2×D8—181no—
138C2×M16—31no—
139C2×M16—91no—
140C4×Q8—61no—
141C4×D4—181no—
142#117—91no—
143C2×C4⋊C4—91no—
144C2×C22⋊C4—182no—
145C2×C42—31no—
146#104—61no—
147#104—61no—
148#102—181no—
149#102—181no—
150C23×C3—31no—
151C22×D3—62no—
152C22×D3—121no—
153C22×D3—122no—
154C22×D3—122no—
155C22×D3—121no—
156C22×D3—241no—
157C22×D3—242no—
158C3×Q8—31no—
159C3×Q8(C2×D3)⋊C411yescharacteristic
160C3×D4—31no—
161C3×D4—31no—
162C3×D4—61no—
163C2×C3×C4—31no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—41no—
169C2×C3×C4—62no—
170C2×C3×C4—81no—
171C2×C3×C4—121no—
172C2×C3×C4—121no—
173C2×C3×C4—241no—
174C2×C3×C4C2×S411yescharacteristic
175C2×Dic3—21no—
176C2×Dic3—61no—
177C2×Dic3—122no—
178D12—62no—
179D12—62no—
180D12—122no—
181D12—121no—
182D12—121no—
183D12—242no—
184D12—241no—
185D12—242no—
186SL(2,𝔽3)—21no—
187SL(2,𝔽3)(C2×D3)⋊C411yescharacteristic, complemented
188C3×C8—121no—
189C3⋊C8—62no—
190C3⋊C8—81no—
191C2×C32—41no—
192C2×C32—41no—
193C2×C32—41no—
194C3⋊D3—162no—
195C24—181no—
196#41—31no—
197#41—31no—
198#41#62411yescharacteristic
199#41#62611yescharacteristic
200C2×Q8#62611yescharacteristic
201C2×D4—31no—
202C2×D4—92no—
203C2×D4—181no—
204C2×D4—181no—
205C2×D4—181no—
206C2×D4—181no—
207C2×D4—181no—
208C22×C4—31no—
209C22×C4—61no—
210C22×C4—92no—
211SD16—182no—
212SD16—182no—
213D8—182no—
214D8—182no—
215M16—61no—
216M16—61no—
217M16—92no—
218M16—181no—
219C2×C8—61no—
220C2×C8—181no—
221C4⋊C4—31no—
222C4⋊C4—91no—
223C4⋊C4—181no—
224C22⋊C4—181no—
225C22⋊C4—364no—
226C42—61no—
227C42—61no—
228C42—181no—
229C22×C3—31no—
230C22×C3—31no—
231C22×C3—41no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3—81no—
235C22×C3GL(2,ℤ/4)11yescharacteristic
236C2×D3—124no—
237C2×D3—122no—
238C2×D3—242no—
239C2×D3—244no—
240C2×D3—242no—
241C2×D3—241no—
242C2×D3—241no—
243C2×D3—481no—
244C2×D3—481no—
245C2×D3—482no—
246C2×D3—482no—
247C3×C4—31no—
248C3×C4—31no—
249C3×C4—41no—
250C3×C4—41no—
251C3×C4—81no—
252C3×C4—81no—
253C3×C4—122no—
254C3×C4—481no—
255C3×C4C4×S411yescharacteristic
256C3×C4C4⋊S411yescharacteristic
257Dic3—41no—
258Dic3—121no—
259C32—41noSylow
260C23—31no—
261C23—182no—
262C23—361no—
263C23—362no—
264C23—362no—
265C23—361no—
266Q8—31no—
267Q8#372211yescharacteristic
268D4—31no—
269D4—31no—
270D4—61no—
271D4—182no—
272D4—182no—
273D4—361no—
274D4—361no—
275C2×C4—31no—
276C2×C4—31no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—31no—
280C2×C4—62no—
281C2×C4—61no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—181no—
285C2×C4—362no—
286C2×C4D3×S411yescharacteristic
287C8—121no—
288C8—182no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3—61no—
293C2×C3—61no—
294C2×C3—81no—
295C2×C3—81no—
296C2×C3—81no—
297C2×C3(C2×S4)⋊C411yescharacteristic
298C2×C3#542911yescharacteristic
299C2×C3#543111yescharacteristic
300D3—242no—
301D3—482no—
302D3—484no—
303C22—31no—
304C22—31no—
305C22—61no—
306C22—61no—
307C22—364no—
308C22—362no—
309C22—722no—
310C22—721no—
311C22—721no—
312C22#6421011yescharacteristic
313C4—31no—
314C4—31no—
315C4—122no—
316C4—121no—
317C4—361no—
318C4(C3⋊S4)⋊C411yescharacteristic
319C4#6420711yescharacteristic
320C3—41no—
321C3—81no—
322C3#7307811yescharacteristic, complemented
323C2—61no—
324C2—61no—
325C2—722no—
326C2#1059323011yescharacteristic
327C2#1059323411yescharacteristic
328C2#1059324211yescharacteristic
subgroupG/H
centerC22#64210
commutatorC3×#41⋊C3C2×C4
Frattini, hypercenterC2×C4D3×S4
FittingC2×C3×#41C2×D3
socleC22×C3GL(2,ℤ/4)
prime radicalC2×#64267C2
cosocleC2×C3×#41⋊C3C22
hypocenterC3×SL(2,𝔽3)C22⋊C4
LCSG ▷ C3×#41⋊C3 ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #5827⋊C3C2semidirect
3. C2×C3×GU(2,𝔽3)C2semidirect
8. #64267C4semidirect
15. #5827D3semidirect
18. C2×GU(2,𝔽3)D3semidirect
68. C3×SL(2,𝔽3)C22⋊C4semidirect
187. SL(2,𝔽3)(C2×D3)⋊C4semidirect
322. C3#73078semidirect

Maximal subgroups

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

subgroupindex
C2×#642672
#5827⋊C32
C2×C3×GU(2,𝔽3)2
#730783
#703663
#635644