Finite Group Databasebeta

← #11922162

Subgroups — #11922162

Order 1152.

631 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 75 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
1C22×C3×GL(2,𝔽3)C211yesmaximal, characteristic
2#5812⋊C3C211yesmaximal, characteristic, complemented
3#10593654C211yesmaximal, characteristic, complemented
4Dic3×GL(2,𝔽3)C212yesmaximal, complemented
5#10593204C212yesmaximal, complemented
6#85353—31nomaximal
7#83843—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3×GL(2,𝔽3)C414yescomplemented
10C2×C3×GL(2,𝔽3)C2211yescharacteristic
11C2×C3×GL(2,𝔽3)C2211yescharacteristic
12D3×#217—41nomaximal
13Dic3×SL(2,𝔽3)C2212yescomplemented
14C3⋊#859C2212yescomplemented
15C22×GL(2,𝔽3)D311yescharacteristic
16C22×C3×SD16—31no—
17#5812D311yescharacteristic, complemented
18#5798—31no—
19#519⋊C3—31no—
20#5420—31no—
21#5420—61no—
22#5394—32no—
23C4×GL(2,𝔽3)—32no—
24#5225—31no—
25#5215—31no—
26#5164—32no—
27#5164—32no—
28Dic3×SD16—32no—
29Dic3×SD16—32no—
30#5106—31no—
31#5101—31no—
32C23×C3×D3—41no—
33C2×C3×SL(2,𝔽3)C2×C412yescomplemented
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×D3×Dic3—42no—
36C3×GL(2,𝔽3)C2×C412yescomplemented
37C3×GL(2,𝔽3)C2×C412yescomplemented
38#3757—41no—
39C3×#217—41no—
40(C2×D3)⋊Dic3—42no—
41#2949—91noSylow
42C22×C3×Q8C2×D311yescharacteristic
43C22×C3×D4—31no—
44C22×SL(2,𝔽3)—21no—
45C22×SL(2,𝔽3)C2×D311yescharacteristic
46C2×GL(2,𝔽3)Dic314yescomplemented
47C2×GL(2,𝔽3)C2×D311yescharacteristic
48C2×GL(2,𝔽3)C2×D311yescharacteristic
49C2×C3×SD16—34no—
50C2×C3×SD16—31no—
51C2×C3×SD16—31no—
52C2×C3×SD16—34no—
53C2×C3×SD16—31no—
54C2×C3×SD16—31no—
55C22×C3×C8—31no—
56C2×#217—61no—
57C2×#217—61no—
58Q8×Dic3—32no—
59Q8×Dic3C2×D312yescomplemented
60D4×Dic3—32no—
61D4×Dic3—32no—
62D4×Dic3—62no—
63#926—31no—
64#926—31no—
65#926—31no—
66C2×#208—31no—
67D3×C22⋊C4—121no—
68C4×SL(2,𝔽3)—32no—
69#859—32no—
70#859—62no—
71#835—32no—
72#835—32no—
73#832—32no—
74#832—32no—
75#818—32no—
76#817—32no—
77#815—32no—
78C8×Dic3—32no—
79C23×C32—41no—
80C22×C3×D3—41no—
81C22×C3×D3—41no—
82C22×C3×D3—44no—
83C22×C3×D3—84no—
84C2×C3⋊Dic3—42no—
85C2×C3×Dic3—42no—
86C3×SL(2,𝔽3)C22×C411yescharacteristic
87D3×Dic3—84no—
88C22×SD16—31no—
89#519—31no—
90#518—91no—
91#440—92no—
92#440—92no—
93C4×SD16—92no—
94C4×SD16—92no—
95#427—91no—
96#419—91no—
97#418—91no—
98#405—91no—
99C24×C3—61no—
100C23×D3—41no—
101C2×C3×Q8—31no—
102C2×C3×Q8—32no—
103C2×C3×Q8C4×D312yescomplemented
104C2×C3×Q8C22×D311yescharacteristic
105C2×C3×D4—31no—
106C2×C3×D4—31no—
107C2×C3×D4—34no—
108C2×C3×D4—61no—
109C2×C3×D4—62no—
110C22×C3×C4—31no—
111C22×Dic3—62no—
112C22×Dic3—62no—
113C2×C4×D3—122no—
114C2×SL(2,𝔽3)—22no—
115C2×SL(2,𝔽3)—21no—
116C2×SL(2,𝔽3)C2×Dic312yescomplemented
117C2×SL(2,𝔽3)C22×D311yescharacteristic
118GL(2,𝔽3)C2×Dic312yescomplemented
119GL(2,𝔽3)C2×Dic312yescomplemented
120C3×SD16—34no—
121C3×SD16—34no—
122C3×SD16—32no—
123C3×SD16—32no—
124C3×SD16—32no—
125C3×SD16—32no—
126C2×C3×C8—31no—
127C2×C3×C8—31no—
128C2×C3×C8—34no—
129C3×C22⋊C4—121no—
130#217—31no—
131#217—62no—
132#217—61no—
133#217—62no—
134#217—121no—
135#217—241no—
136#217S411yescharacteristic
137(C2×D3)⋊C4—122no—
138C4⋊Dic3—32no—
139C4⋊Dic3—32no—
140C4⋊Dic3—32no—
141C4×Dic3—32no—
142C4×Dic3—61no—
143C4×Dic3—61no—
144#208—62no—
145C2×C3⋊C8—32no—
146C22×C32—42no—
147C22×C32—41no—
148C22×C32—82no—
149C2×C3×D3—42no—
150C2×C3×D3—42no—
151C2×C3×D3—44no—
152C2×C3×D3—82no—
153C2×C3×D3—88no—
154C3⋊Dic3—82no—
155C3×Dic3—82no—
156C22×Q8D3211yescharacteristic
157C22×D4—31no—
158C2×SD16—31no—
159C2×SD16—31no—
160C2×SD16—34no—
161C2×SD16—34no—
162C2×SD16—31no—
163C2×SD16—31no—
164C2×M16—91no—
165C22×C8—31no—
166C4×Q8—32no—
167C4×Q8—92no—
168C4×D4—92no—
169C4×D4—92no—
170C4×D4—182no—
171#117—91no—
172#117—91no—
173#117—91no—
174C2×C22⋊C4—181no—
175C2×C22⋊C4—181no—
176#107—92no—
177#106—92no—
178#103—92no—
179#103—92no—
180#102—92no—
181#102—92no—
182#97—92no—
183C4×C8—92no—
184C23×C3—41no—
185C23×C3—61no—
186C23×C3—61no—
187C23×C3—64no—
188C23×C3—81no—
189C23×C3—242no—
190C23×C3C2×S411yescharacteristic
191C22×D3—41no—
192C22×D3—41no—
193C22×D3—44no—
194C22×D3—84no—
195C3×Q8—32no—
196C3×Q8—31no—
197C3×Q8—61no—
198C3×Q8C2×C4×D311yescharacteristic
199C3×D4—32no—
200C3×D4—32no—
201C3×D4—61no—
202C3×D4—61no—
203C3×D4—64no—
204C2×C3×C4—32no—
205C2×C3×C4—32no—
206C2×C3×C4—31no—
207C2×C3×C4—31no—
208C2×C3×C4—122no—
209C2×Dic3—32no—
210C2×Dic3—62no—
211C2×Dic3—122no—
212C2×Dic3—242no—
213C2×Dic3—242no—
214C2×Dic3—242no—
215C2×Dic3C2×S412yes—
216C4×D3—244no—
217SL(2,𝔽3)—21no—
218SL(2,𝔽3)C22×Dic311yescharacteristic
219C3×C8—32no—
220C3×C8—32no—
221C3⋊C8—62no—
222C2×C32—41no—
223C2×C32—42no—
224C2×C32—82no—
225C3×D3—44no—
226C3×D3—82no—
227C24—61no—
228C2×Q8—31no—
229C2×Q8—32no—
230C2×Q8D3×Dic312yescomplemented
231C2×Q8C2×D3211yescharacteristic
232C2×D4—31no—
233C2×D4—31no—
234C2×D4—34no—
235C2×D4—61no—
236C2×D4—62no—
237C22×C4—31no—
238C22×C4—182no—
239C22×C4—182no—
240SD16—34no—
241SD16—34no—
242SD16—32no—
243SD16—32no—
244SD16—32no—
245SD16—32no—
246M16—182no—
247C2×C8—31no—
248C2×C8—31no—
249C2×C8—34no—
250C2×C8—92no—
251C4⋊C4—92no—
252C4⋊C4—92no—
253C4⋊C4—92no—
254C22⋊C4—31no—
255C22⋊C4—91no—
256C22⋊C4—182no—
257C22⋊C4—182no—
258C22⋊C4—181no—
259C42—92no—
260C42—181no—
261C42—181no—
262C22×C3—22no—
263C22×C3—41no—
264C22×C3—42no—
265C22×C3—62no—
266C22×C3—62no—
267C22×C3—82no—
268C22×C3—81no—
269C22×C3—82no—
270C22×C3—122no—
271C22×C3—122no—
272C22×C3—162no—
273C22×C3—244no—
274C22×C3C4×S412yes—
275C22×C3C22×S411yescharacteristic
276C2×D3—42no—
277C2×D3—42no—
278C2×D3—44no—
279C2×D3—82no—
280C2×D3—88no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—32no—
284C3×C4—242no—
285Dic3—22no—
286Dic3—62no—
287Dic3—122no—
288Dic3—242no—
289Dic3—482no—
290C32—41noSylow
291C23—61no—
292C23—61no—
293C23—64no—
294C23—242no—
295C23D3×S411yescharacteristic
296Q8—32no—
297Q8—31no—
298Q8—61no—
299Q8C2×D3×Dic311yescharacteristic
300D4—32no—
301D4—32no—
302D4—61no—
303D4—61no—
304D4—64no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—32no—
308C2×C4—32no—
309C2×C4—32no—
310C2×C4—92no—
311C2×C4—182no—
312C2×C4—722no—
313C2×C4—722no—
314C8—32no—
315C8—32no—
316C8—182no—
317C2×C3—22no—
318C2×C3—42no—
319C2×C3—41no—
320C2×C3—82no—
321C2×C3—82no—
322C2×C3—81no—
323C2×C3—122no—
324C2×C3—122no—
325C2×C3—162no—
326C2×C3C2×C4×S411yescharacteristic
327C2×C3#592612yes—
328D3—44no—
329D3—82no—
330C22—22no—
331C22—62no—
332C22—62no—
333C22—122no—
334C22—122no—
335C22—244no—
336C22Dic3×S412yes—
337C22C2×D3×S411yescharacteristic
338C4—31no—
339C4—31no—
340C4—32no—
341C4—62no—
342C4—182no—
343C4—362no—
344C3—41no—
345C3—81no—
346C3#8535311yescharacteristic, complemented
347C2—22no—
348C2—122no—
349C2—122no—
350C2#1059644512yes—
351C2C2×Dic3×S411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC22×C3×Q8C2×D3
socleC22×C3C22×S4
prime radicalC22×C3×GL(2,𝔽3)C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC23D3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. #5812⋊C3C2semidirect
3. #10593654C2semidirect
4. Dic3×GL(2,𝔽3)C2semidirect
5. #10593204C2semidirect
9. C2×C3×GL(2,𝔽3)C4semidirect
13. Dic3×SL(2,𝔽3)C22semidirect
14. C3⋊#859C22semidirect
17. #5812D3semidirect
33. C2×C3×SL(2,𝔽3)C2×C4semidirect
36. C3×GL(2,𝔽3)C2×C4semidirect
37. C3×GL(2,𝔽3)C2×C4semidirect
46. C2×GL(2,𝔽3)Dic3semidirect
59. Q8×Dic3C2×D3semidirect
103. C2×C3×Q8C4×D3semidirect
116. C2×SL(2,𝔽3)C2×Dic3semidirect
118. GL(2,𝔽3)C2×Dic3semidirect
119. GL(2,𝔽3)C2×Dic3semidirect
230. C2×Q8D3×Dic3semidirect
346. C3#85353semidirect

Maximal subgroups

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

subgroupindex
C22×C3×GL(2,𝔽3)2
#5812⋊C32
#105936542
Dic3×GL(2,𝔽3)2
#105932042
#853533
#838433
D3×#2174