Finite Group Databasebeta

← #11921704

Subgroups — #11921704

Order 1152.

591 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 591 conjugacy classes of subgroups — too many to draw. Only its 59 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
1C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C3×#5386C211yesmaximal, characteristic, complemented
4#10593218C212yesmaximal, complemented
5#10593214C212yesmaximal, complemented
6#85325—31nomaximal
7#80996—31nomaximal
8C2×#3799—41nomaximal
9C2×D3×SL(2,𝔽3)C2211yescharacteristic
10C2×C3⋊GL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2211yescharacteristic
13C3×#859C2212yes—
14#5725—31no—
15C4×D3×Q8D311yescharacteristic, complemented
16C2×C4×SL(2,𝔽3)—31no—
17#5421—32no—
18C4⋊GL(2,𝔽3)—31no—
19C4⋊GL(2,𝔽3)—61no—
20C4×GL(2,𝔽3)—32no—
21#5386D311yescharacteristic, complemented
22C3×#475—31no—
23#5094—31no—
24#5027—31no—
25#4863—32no—
26#4852—32no—
27#4848—31no—
28#4833—31no—
29#4806—32no—
30#4788—32no—
31C2×C4⋊(C3⋊D3)—41no—
32C2×C3×C4×D3—41no—
33C2×C3×Dic6—41no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35C2×#626—42no—
36C2×#624—42no—
37#3799—48no—
38D3×SL(2,𝔽3)—21no—
39C3⋊GL(2,𝔽3)—22no—
40#3103—91noSylow
41C2×#239—61no—
42C2×D3×Q8C2×D311yescharacteristic
43C2×#235—121no—
44C22×SL(2,𝔽3)—31no—
45C2×GL(2,𝔽3)—32no—
46C2×GL(2,𝔽3)—62no—
47C3×C4⋊Q8—31no—
48C3×C4×Q8C2×D311yescharacteristic
49#947—61no—
50Q8×Dic3C2×D311yescharacteristic
51C2×#215—32no—
52C2×#204—32no—
53#895—32no—
54#895—61no—
55#893—32no—
56#893—61no—
57D3×C4⋊C4—31no—
58#887—31no—
59#874—31no—
60C42×D3—31no—
61C4×Dic6—31no—
62C4×SL(2,𝔽3)—21no—
63C4×SL(2,𝔽3)—31no—
64C4×SL(2,𝔽3)C2×D311yescharacteristic
65#859C2×D312yes—
66C3×#106—32no—
67C3×C4⋊C8—31no—
68C3×#103—32no—
69#821—32no—
70(C2×D3)⋊C8—32no—
71C8×Dic3—32no—
72#810—32no—
73#809—32no—
74#808—32no—
75C4⋊(C3⋊C8)—31no—
76C22×C3⋊D3—42no—
77C22×C3×D3—41no—
78C2×C32×C4—41no—
79C4⋊(C3⋊D3)—44no—
80C2×C3×Dic3—42no—
81C2×C3×Dic3—41no—
82C3×C4×D3—44no—
83C3×Dic6—44no—
84C3×SL(2,𝔽3)#4111yescharacteristic, complemented
85#626—48no—
86#624—48no—
87#532—91no—
88C2×C4×Q8—31no—
89#481—92no—
90#475—31no—
91#475—91no—
92#460—91no—
93#460—91no—
94#448—92no—
95C4×SD16—92no—
96C4×SD16—92no—
97#431—91no—
98C2×C3×Q8—61no—
99C2×C3×Q8C22×D311yescharacteristic
100C22×C3×C4—121no—
101C2×#67—122no—
102#239—124no—
103D3×Q8—21no—
104D3×Q8—61no—
105#235—128no—
106C2×D12—32no—
107C2×D12—61no—
108C2×D12—62no—
109C2×D12—61no—
110C2×D12—121no—
111C2×D12—241no—
112C2×C4×D3—31no—
113C2×C4×D3—31no—
114C2×C4×D3—62no—
115C2×C4×D3—122no—
116C2×C4×D3S411yescharacteristic
117C2×Dic6—31no—
118C2×Dic6—41no—
119C2×SL(2,𝔽3)—21no—
120C2×SL(2,𝔽3)—61no—
121C2×SL(2,𝔽3)C22×D311yescharacteristic
122GL(2,𝔽3)—62no—
123GL(2,𝔽3)—122no—
124C2×C3×C8—32no—
125C3×C4⋊C4—31no—
126C3×C4⋊C4—32no—
127C3×C4⋊C4—61no—
128C3×C42—31no—
129#215—62no—
130#215—62no—
131(C2×D3)⋊C4—62no—
132(C2×D3)⋊C4—62no—
133C4⋊Dic3—31no—
134Dic3⋊C4—61no—
135C4×Dic3—31no—
136C4×Dic3—31no—
137C4×Dic3—31no—
138C2×C3⋊C8—32no—
139#204—64no—
140C22×C32—41no—
141C2×C3⋊D3—44no—
142C2×C3⋊D3—82no—
143C2×C3⋊D3—82no—
144C2×C3×D3—42no—
145C2×C3×D3—81no—
146C2×C3×D3—81no—
147C32×C4—42no—
148C3×Dic3—44no—
149C3×Dic3—42no—
150C2×#41—181no—
151C22×Q8—31no—
152C2×SD16—92no—
153C2×SD16—92no—
154C4⋊Q8—31no—
155#127—91no—
156#124—181no—
157#121—92no—
158#121—181no—
159C4×Q8—31no—
160C4×Q8—91no—
161C4×Q8—91no—
162C4×Q8D3211yescharacteristic
163C4×D4—92no—
164C4×D4—181no—
165C2×C4⋊C4—91no—
166C2×C42—91no—
167#106—32no—
168#106—92no—
169C4⋊C8—31no—
170C4⋊C8—91no—
171#103—32no—
172#103—92no—
173#102—92no—
174#102—92no—
175C22⋊C8—92no—
176C4×C8—92no—
177C23×C3—121no—
178C22×D3—62no—
179C22×D3—122no—
180C22×D3—242no—
181C22×D3C2×S411yescharacteristic
182C3×Q8—31no—
183C3×Q8—122no—
184C3×Q8#23511yescharacteristic, complemented
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—41no—
188C2×C3×C4—62no—
189C2×C3×C4—81no—
190C2×C3×C4—124no—
191C2×C3×C4—121no—
192C2×C3×C4C2×S411yescharacteristic
193#67—128no—
194C2×Dic3—31no—
195C2×Dic3—31no—
196C2×Dic3—42no—
197C2×Dic3C2×S411yescharacteristic
198D12—62no—
199D12—62no—
200D12—124no—
201D12—122no—
202D12—124no—
203D12—122no—
204D12—244no—
205C4×D3—61no—
206C4×D3—61no—
207C4×D3—128no—
208C4×D3—121no—
209C4×D3—124no—
210C4×D3GL(2,𝔽3)14yescomplemented
211Dic6—32no—
212Dic6—44no—
213Dic6—61no—
214SL(2,𝔽3)—21no—
215SL(2,𝔽3)#23911yescharacteristic, complemented
216C3×C8—62no—
217C3⋊C8—62no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—41no—
221C3⋊D3—84no—
222C3×D3—82no—
223#41—364no—
224C2×Q8—61no—
225C2×Q8—61no—
226C2×Q8—91no—
227C2×Q8—181no—
228C2×Q8C2×D3211yescharacteristic
229C2×D4—92no—
230C2×D4—181no—
231C2×D4—182no—
232C2×D4—181no—
233C22×C4—31no—
234C22×C4—91no—
235C22×C4—91no—
236C22×C4—182no—
237SD16—184no—
238SD16—182no—
239SD16—182no—
240C2×C8—32no—
241C2×C8—92no—
242C4⋊C4—31no—
243C4⋊C4—32no—
244C4⋊C4—61no—
245C4⋊C4—91no—
246C4⋊C4—181no—
247C22⋊C4—182no—
248C22⋊C4—182no—
249C42—31no—
250C42—91no—
251C42—91no—
252C42—91no—
253C22×C3—41no—
254C22×C3—81no—
255C22×C3—122no—
256C22×C3—241no—
257C22×C3—241no—
258C22×C3C22×S411yescharacteristic
259C2×D3—21no—
260C2×D3—21no—
261C2×D3—124no—
262C2×D3—122no—
263C2×D3—122no—
264C2×D3—122no—
265C2×D3—242no—
266C2×D3—242no—
267C2×D3—244no—
268C2×D3—482no—
269C2×D3—482no—
270C2×D3C2×GL(2,𝔽3)12yes—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—42no—
274C3×C4—61no—
275C3×C4—82no—
276C3×C4—122no—
277C3×C4—122no—
278C3×C4C2×GL(2,𝔽3)12yes—
279Dic3—44no—
280Dic3—61no—
281Dic3—61no—
282Dic3C2×GL(2,𝔽3)12yes—
283C32—41noSylow
284C23—31no—
285C23—182no—
286Q8—31no—
287Q8—92no—
288Q8—122no—
289Q8—181no—
290Q8#379911yescharacteristic, complemented
291D4—182no—
292D4—182no—
293D4—362no—
294D4—364no—
295D4—362no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—34no—
299C2×C4—31no—
300C2×C4—62no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—181no—
304C2×C4—181no—
305C2×C4—364no—
306C2×C4—361no—
307C2×C4D3×S411yescharacteristic
308C8—62no—
309C8—182no—
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—81no—
314C2×C3—81no—
315C2×C3—81no—
316C2×C3—242no—
317C2×C3#591211yescharacteristic
318C2×C3C22×GL(2,𝔽3)11yescharacteristic
319C2×C3#592311yescharacteristic
320D3—22no—
321D3—244no—
322D3—242no—
323D3—484no—
324C22—32no—
325C22—61no—
326C22—61no—
327C22—362no—
328C22—362no—
329C22—362no—
330C22C2×D3×S411yescharacteristic
331C4—31no—
332C4—31no—
333C4—32no—
334C4—61no—
335C4—122no—
336C4—181no—
337C4—181no—
338C4D3×GL(2,𝔽3)12yes—
339C3—41no—
340C3—81no—
341C3#8532511yescharacteristic, complemented
342C2—62no—
343C2—722no—
344C2#1059642811yescharacteristic
345C2C2×D3×GL(2,𝔽3)11yescharacteristic
346C2#1059645611yescharacteristic
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. C4×D3×SL(2,𝔽3)C2semidirect
2. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C3×#5386C2semidirect
4. #10593218C2semidirect
5. #10593214C2semidirect
15. C4×D3×Q8D3semidirect
21. #5386D3semidirect
84. C3×SL(2,𝔽3)#41semidirect
184. C3×Q8#235semidirect
210. C4×D3GL(2,𝔽3)semidirect
215. SL(2,𝔽3)#239semidirect
290. Q8#3799semidirect
341. C3#85325semidirect

Maximal subgroups

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

subgroupindex
C4×D3×SL(2,𝔽3)2
C4⋊(C3⋊GL(2,𝔽3))2
C3×#53862
#105932182
#105932142
#853253
#809963
C2×#37994