Finite Group Databasebeta

← #11921747

Subgroups — #11921747

Order 1152.

469 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 469 conjugacy classes of subgroups — too many to draw. Only its 51 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
1SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
2#10593625C211yesmaximal, characteristic, complemented
3C3×C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593217C212yesmaximal, complemented
5#10593209C212yesmaximal, complemented
6#85400—31nomaximal
7#79685—31nomaximal
8C2×D3×SL(2,𝔽3)C2212yes—
9C2×C3×GL(2,𝔽3)C2211yescharacteristic
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11#63908—41nomaximal
12C3⋊#859C2212yes—
13C3×#859C2211yescharacteristic
14Q8×D12D311yescharacteristic, complemented
15#5563—31no—
16D4×SL(2,𝔽3)—31no—
17#5421—32no—
18#5420—32no—
19C4×GL(2,𝔽3)D311yescharacteristic, complemented
20#5391—31no—
21#5391—61no—
22C3×C4×SD16—31no—
23#5170—32no—
24#5021—31no—
25#5019—31no—
26#4868—32no—
27#4802—32no—
28#4780—32no—
29#4688—31no—
30#4681—31no—
31C2×C3×D12—41no—
32C2×C3×C4×D3—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C2×#625—42no—
35D3×SL(2,𝔽3)—22no—
36C3×GL(2,𝔽3)—21no—
37C4⋊(C3⋊Dic3)—41no—
38(C2×D3)⋊Dic3—42no—
39#3329—91noSylow
40C2×D3×Q8C2×D312yes—
41C22×SL(2,𝔽3)—32no—
42C2×GL(2,𝔽3)C2×D311yescharacteristic
43C2×C3×SD16—31no—
44C3×C4×Q8C2×D311yescharacteristic
45C3×C4×D4—31no—
46#938—32no—
47#938—121no—
48C2×#214—32no—
49#930—61no—
50#926—61no—
51C2×ΓL(1,𝔽25)—32no—
52#897—32no—
53#892—32no—
54#885—62no—
55C4×D12—31no—
56#869—31no—
57#869—31no—
58C4×SL(2,𝔽3)—21no—
59C4×SL(2,𝔽3)C2×D311yescharacteristic
60#859—32no—
61#859—62no—
62#859C2×D311yescharacteristic
63C3×#106—31no—
64C3×#103—31no—
65C3×#102—31no—
66C3×C4×C8—31no—
67#835—32no—
68(C2×D3)⋊C8—32no—
69#816—32no—
70#816—32no—
71#810—32no—
72#807—32no—
73C4⋊(C3⋊C8)—31no—
74C22×C3×D3—41no—
75C22×C3×D3—42no—
76C2×C32×C4—41no—
77C2×C3⋊Dic3—42no—
78C2×C3×Dic3—41no—
79C3×D12—82no—
80C3×C4×D3—82no—
81C3×SL(2,𝔽3)#4111yescharacteristic
82#625—84no—
83#555—91no—
84D4×Q8—31no—
85#487—91no—
86#485—92no—
87#477—91no—
88#468—92no—
89#461—91no—
90#448—92no—
91C4×SD16—31no—
92#436—91no—
93#419—92no—
94C2×C3×Q8C22×D311yescharacteristic
95C2×C3×D4—31no—
96C2×C3×D4—121no—
97C22×C3×C4—61no—
98C2×#67—62no—
99C2×#67—122no—
100D3×Q8—22no—
101D3×Q8—62no—
102C2×D12S411yescharacteristic
103C2×C4×D3—32no—
104C2×C4×D3—41no—
105C2×Dic6—32no—
106C2×Dic6—61no—
107C2×SL(2,𝔽3)—21no—
108C2×SL(2,𝔽3)—62no—
109C2×SL(2,𝔽3)C22×D311yescharacteristic
110GL(2,𝔽3)—21no—
111C3×SD16—61no—
112C3×SD16—61no—
113C2×C3×C8—31no—
114C2×C3×C8—31no—
115C3×C4⋊C4—31no—
116C3×C4⋊C4—31no—
117C3×C22⋊C4—61no—
118C3×C42—31no—
119#217—62no—
120#217—122no—
121#214—122no—
122(C2×D3)⋊C4—32no—
123(C2×D3)⋊C4—62no—
124C4⋊Dic3—31no—
125C4⋊Dic3—32no—
126C4⋊Dic3—61no—
127C4⋊Dic3—61no—
128C4⋊Dic3—121no—
129C4⋊Dic3—241no—
130C4×Dic3—62no—
131C2×C3⋊C8—32no—
132ΓL(1,𝔽25)—122no—
133C22×C32—41no—
134C2×C3×D3—42no—
135C2×C3×D3—82no—
136C2×C3×D3—82no—
137C2×C3×D3—82no—
138C2×C3×D3—81no—
139C2×C3×D3—81no—
140C32×C4—42no—
141C3⋊Dic3—82no—
142C3×Dic3—81no—
143C22×Q8—32no—
144C2×SD16—31no—
145C2×SD16—92no—
146C2×M16—92no—
147C4⋊Q8—91no—
148C4⋊Q8—91no—
149#124—182no—
150C22⋊Q8—92no—
151#121—92no—
152#121—181no—
153C4×Q8D3211yescharacteristic
154C4×D4—31no—
155C4×D4—91no—
156#117—92no—
157#117—181no—
158#107—92no—
159#106—31no—
160C4⋊C8—91no—
161#103—31no—
162#103—92no—
163#103—92no—
164#103—92no—
165#103—92no—
166#102—31no—
167C22⋊C8—92no—
168C4×C8—31no—
169C23×C3—61no—
170C23×C3—122no—
171C22×D3—41no—
172C22×D3C2×S412yes—
173C3×Q8—31no—
174C3×Q8#23711yescharacteristic
175C3×D4—61no—
176C3×D4—61no—
177C3×D4—242no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—41no—
181C2×C3×C4—61no—
182C2×C3×C4—81no—
183C2×C3×C4—122no—
184C2×C3×C4C2×S411yescharacteristic
185#67—242no—
186#67—244no—
187C2×Dic3—32no—
188C2×Dic3—41no—
189C2×Dic3—62no—
190C2×Dic3—122no—
191C2×Dic3—242no—
192D12—22no—
193C4×D3—62no—
194C4×D3—62no—
195C4×D3—82no—
196Dic6—62no—
197Dic6—62no—
198Dic6—241no—
199SL(2,𝔽3)—21no—
200SL(2,𝔽3)#23511yescharacteristic
201C3×C8—61no—
202C3×C8—61no—
203C3⋊C8—62no—
204C2×C32—41no—
205C2×C32—41no—
206C2×C32—41no—
207C3×D3—82no—
208C3×D3—162no—
209C2×Q8—62no—
210C2×Q8—92no—
211C2×Q8—181no—
212C2×Q8—182no—
213C2×Q8C2×D3211yescharacteristic
214C2×D4—31no—
215C2×D4—31no—
216C2×D4—182no—
217C22×C4—61no—
218C22×C4—92no—
219SD16—61no—
220SD16—61no—
221SD16—362no—
222M16—362no—
223C2×C8—31no—
224C2×C8—31no—
225C2×C8—92no—
226C4⋊C4—31no—
227C4⋊C4—31no—
228C4⋊C4—92no—
229C4⋊C4—91no—
230C4⋊C4—181no—
231C4⋊C4—181no—
232C22⋊C4—61no—
233C22⋊C4—92no—
234C22⋊C4—182no—
235C22⋊C4—182no—
236C42—31no—
237C42—182no—
238C22×C3—41no—
239C22×C3—81no—
240C22×C3—121no—
241C22×C3—121no—
242C22×C3—121no—
243C22×C3—242no—
244C22×C3—242no—
245C22×C3—242no—
246C22×C3C22×S411yescharacteristic
247C2×D3—22no—
248C2×D3—22no—
249C2×D3—22no—
250C2×D3—42no—
251C2×D3—81no—
252C2×D3—81no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—42no—
256C3×C4—61no—
257C3×C4—82no—
258C3×C4—121no—
259C3×C4#98312yes—
260Dic3—62no—
261Dic3—81no—
262Dic3—122no—
263Dic3—242no—
264Dic3—482no—
265C32—41noSylow
266C23—32no—
267C23—61no—
268Q8—31no—
269Q8—182no—
270Q8—182no—
271Q8—721no—
272Q8#379511yescharacteristic
273D4—61no—
274D4—61no—
275D4—62no—
276D4—722no—
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—61no—
280C2×C4—92no—
281C2×C4—122no—
282C2×C4—182no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4D3×S411yescharacteristic
286C8—61no—
287C8—61no—
288C8—182no—
289C2×C3—41no—
290C2×C3—41no—
291C2×C3—41no—
292C2×C3—81no—
293C2×C3—81no—
294C2×C3—81no—
295C2×C3—241no—
296C2×C3—482no—
297C2×C3#591411yescharacteristic
298C2×C3C2×#98311yescharacteristic
299C2×C3#592611yescharacteristic
300D3—42no—
301D3—82no—
302C22—62no—
303C22—62no—
304C22—62no—
305C22—121no—
306C22—121no—
307C22—121no—
308C22C2×D3×S411yescharacteristic
309C4—31no—
310C4—31no—
311C4—61no—
312C4—121no—
313C4—182no—
314C4—362no—
315C4#6420112yes—
316C3—41no—
317C3—81no—
318C3#8540011yescharacteristic, complemented
319C2—122no—
320C2—241no—
321C2#1059642411yescharacteristic
322C2C2×#6420111yescharacteristic
323C2#1059645911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FittingC3×C4×Q8C2×D3
socleC22×C3C22×S4
hypercenterC2×C4D3×S4
hypocenterC3×SL(2,𝔽3)#41
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. SL(2,𝔽3)×D12C2semidirect
2. #10593625C2semidirect
3. C3×C4×GL(2,𝔽3)C2semidirect
4. #10593217C2semidirect
5. #10593209C2semidirect
14. Q8×D12D3semidirect
19. C4×GL(2,𝔽3)D3semidirect
318. C3#85400semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D122
#105936252
C3×C4×GL(2,𝔽3)2
#105932172
#105932092
#854003
#796853
#639084