Finite Group Databasebeta

← #11921732

Subgroups — #11921732

Order 1152.

517 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 517 conjugacy classes of subgroups — too many to draw. Only its 53 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×#5391C211yesmaximal, characteristic, complemented
4#10593218C212yesmaximal, complemented
5#10593207C212yesmaximal, complemented
6#85332—31nomaximal
7#81094—31nomaximal
8C2×D3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊GL(2,𝔽3)C2212yes—
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11#63909—41nomaximal
12Dic3×SL(2,𝔽3)C2211yescharacteristic
13C3×#859C2212yes—
14#5725—31no—
15C4×D3×Q8D311yescharacteristic, complemented
16C2×C4×SL(2,𝔽3)—31no—
17#5421—32no—
18#5394—32no—
19C4⋊GL(2,𝔽3)—31no—
20C4⋊GL(2,𝔽3)—61no—
21#5391D311yescharacteristic, complemented
22C3×#477—31no—
23#5091—31no—
24#5027—31no—
25#4889—32no—
26#4882—32no—
27#4848—31no—
28#4835—31no—
29#4817—32no—
30#4806—32no—
31C2×C4⋊(C3⋊D3)—41no—
32C2×C3×C4×D3—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C2×#626—42no—
35D3×SL(2,𝔽3)—21no—
36C3⋊GL(2,𝔽3)—22no—
37C3×C4⋊Dic3—41no—
38#3722—42no—
39#3112—91noSylow
40C2×#239—61no—
41C2×D3×Q8C2×D311yescharacteristic
42C22×SL(2,𝔽3)—31no—
43C2×GL(2,𝔽3)—32no—
44C2×GL(2,𝔽3)—62no—
45C3×C4⋊Q8—31no—
46C3×C4×Q8C2×D311yescharacteristic
47#947—61no—
48Q8×Dic3C2×D311yescharacteristic
49C2×#215—32no—
50#930—121no—
51C2×#204—32no—
52#895—32no—
53#895—61no—
54#893—32no—
55#893—61no—
56D3×C4⋊C4—31no—
57#887—31no—
58#874—31no—
59C42×D3—31no—
60C4×Dic6—31no—
61C4×SL(2,𝔽3)—21no—
62C4×SL(2,𝔽3)—31no—
63C4×SL(2,𝔽3)C2×D311yescharacteristic
64#859C2×D312yes—
65C3×#107—32no—
66C3×C4⋊C8—31no—
67C3×#103—32no—
68#821—32no—
69(C2×D3)⋊C8—32no—
70#815—32no—
71#810—32no—
72#809—32no—
73#807—32no—
74C4⋊(C3⋊C8)—31no—
75C22×C3⋊D3—42no—
76C22×C3×D3—41no—
77C2×C32×C4—41no—
78C4⋊(C3⋊D3)—82no—
79C2×C3×Dic3—42no—
80C2×C3×Dic3—41no—
81C3×C4×D3—82no—
82C3×SL(2,𝔽3)#4111yescharacteristic, complemented
83#626—84no—
84#532—91no—
85C2×C4×Q8—31no—
86#480—92no—
87#477—31no—
88#477—91no—
89#460—91no—
90#460—91no—
91#448—92no—
92#440—92no—
93#440—92no—
94#432—91no—
95C2×C3×Q8—61no—
96C2×C3×Q8C22×D311yescharacteristic
97C22×C3×C4—121no—
98C2×#67—122no—
99#239—124no—
100D3×Q8—21no—
101D3×Q8—61no—
102C2×D12—32no—
103C2×D12—61no—
104C2×D12—62no—
105C2×D12—61no—
106C2×D12—121no—
107C2×D12—241no—
108C2×C4×D3—31no—
109C2×C4×D3—31no—
110C2×C4×D3—62no—
111C2×C4×D3S411yescharacteristic
112C2×Dic6—31no—
113C2×SL(2,𝔽3)—21no—
114C2×SL(2,𝔽3)—61no—
115C2×SL(2,𝔽3)C22×D311yescharacteristic
116GL(2,𝔽3)—62no—
117GL(2,𝔽3)—122no—
118C2×C3×C8—32no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—61no—
122C3×C42—31no—
123#215—62no—
124#215—62no—
125(C2×D3)⋊C4—62no—
126(C2×D3)⋊C4—62no—
127(C2×D3)⋊C4—122no—
128C4⋊Dic3—31no—
129C4⋊Dic3—41no—
130Dic3⋊C4—61no—
131C4×Dic3—31no—
132C4×Dic3—31no—
133C4×Dic3—31no—
134C2×C3⋊C8—32no—
135#204—64no—
136C22×C32—41no—
137C2×C3⋊D3—82no—
138C2×C3⋊D3—82no—
139C2×C3⋊D3—82no—
140C2×C3×D3—42no—
141C2×C3×D3—81no—
142C2×C3×D3—81no—
143C32×C4—42no—
144C3×Dic3—81no—
145C3×Dic3—82no—
146C2×#41—181no—
147C22×Q8—31no—
148C2×SD16—92no—
149C2×SD16—92no—
150C4⋊Q8—31no—
151#127—91no—
152#124—181no—
153#121—92no—
154#121—181no—
155C4×Q8—31no—
156C4×Q8—91no—
157C4×Q8—91no—
158C4×Q8D3211yescharacteristic
159C4×D4—92no—
160C4×D4—181no—
161C2×C4⋊C4—91no—
162C2×C42—91no—
163#107—32no—
164#107—92no—
165C4⋊C8—31no—
166C4⋊C8—91no—
167#103—32no—
168#103—92no—
169#102—92no—
170#102—92no—
171C22⋊C8—92no—
172#97—92no—
173C23×C3—121no—
174C22×D3—62no—
175C22×D3—122no—
176C22×D3—242no—
177C22×D3C2×S411yescharacteristic
178C3×Q8—31no—
179C3×Q8—122no—
180C3×Q8#23511yescharacteristic
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—41no—
184C2×C3×C4—62no—
185C2×C3×C4—81no—
186C2×C3×C4—121no—
187C2×C3×C4—242no—
188C2×C3×C4C2×S411yescharacteristic
189#67—244no—
190C2×Dic3—31no—
191C2×Dic3—31no—
192C2×Dic3—42no—
193C2×Dic3C2×S411yescharacteristic
194D12—62no—
195D12—62no—
196D12—122no—
197D12—124no—
198D12—122no—
199D12—242no—
200D12—244no—
201C4×D3—22no—
202C4×D3—61no—
203C4×D3—61no—
204C4×D3—62no—
205C4×D3—124no—
206Dic6—32no—
207Dic6—61no—
208SL(2,𝔽3)—21no—
209SL(2,𝔽3)#23911yescharacteristic, complemented
210C3×C8—62no—
211C3⋊C8—62no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C3⋊D3—162no—
216C3×D3—82no—
217#41—364no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—91no—
221C2×Q8—181no—
222C2×Q8C2×D3211yescharacteristic
223C2×D4—92no—
224C2×D4—181no—
225C2×D4—182no—
226C2×D4—181no—
227C22×C4—31no—
228C22×C4—91no—
229C22×C4—91no—
230C22×C4—182no—
231SD16—184no—
232SD16—182no—
233SD16—182no—
234C2×C8—32no—
235C2×C8—92no—
236C4⋊C4—31no—
237C4⋊C4—32no—
238C4⋊C4—61no—
239C4⋊C4—91no—
240C4⋊C4—181no—
241C22⋊C4—182no—
242C22⋊C4—182no—
243C42—31no—
244C42—91no—
245C42—91no—
246C42—91no—
247C22×C3—41no—
248C22×C3—81no—
249C22×C3—122no—
250C22×C3—241no—
251C22×C3—241no—
252C22×C3C22×S411yescharacteristic
253C2×D3—21no—
254C2×D3—21no—
255C2×D3—122no—
256C2×D3—122no—
257C2×D3—122no—
258C2×D3—242no—
259C2×D3—242no—
260C2×D3—242no—
261C2×D3—244no—
262C2×D3—482no—
263C2×D3—482no—
264C2×D3#98612yes—
265C3×C4—31no—
266C3×C4—31no—
267C3×C4—42no—
268C3×C4—61no—
269C3×C4—82no—
270C3×C4—122no—
271C3×C4—241no—
272C3×C4#98312yes—
273Dic3—21no—
274Dic3—61no—
275Dic3—61no—
276Dic3—82no—
277C32—41noSylow
278C23—31no—
279C23—182no—
280Q8—31no—
281Q8—92no—
282Q8—122no—
283Q8—181no—
284Q8#379911yescharacteristic
285D4—182no—
286D4—182no—
287D4—362no—
288D4—364no—
289D4—362no—
290C2×C4—31no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—91no—
296C2×C4—91no—
297C2×C4—181no—
298C2×C4—181no—
299C2×C4—182no—
300C2×C4—364no—
301C2×C4D3×S411yescharacteristic
302C8—62no—
303C8—182no—
304C2×C3—41no—
305C2×C3—41no—
306C2×C3—41no—
307C2×C3—81no—
308C2×C3—81no—
309C2×C3—81no—
310C2×C3—242no—
311C2×C3#591211yescharacteristic
312C2×C3C2×#98311yescharacteristic
313C2×C3C2×#98611yescharacteristic
314D3—22no—
315D3—242no—
316D3—482no—
317D3—484no—
318C22—32no—
319C22—61no—
320C22—61no—
321C22—362no—
322C22—362no—
323C22—362no—
324C22C2×D3×S411yescharacteristic
325C4—31no—
326C4—31no—
327C4—61no—
328C4—61no—
329C4—122no—
330C4—181no—
331C4—181no—
332C4#6420212yes—
333C3—41no—
334C3—81no—
335C3#8533211yescharacteristic, complemented
336C2—62no—
337C2—722no—
338C2#1059642811yescharacteristic
339C2C2×#6420211yescharacteristic
340C2D3×#98611yescharacteristic
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×#5391C2semidirect
4. #10593218C2semidirect
5. #10593207C2semidirect
15. C4×D3×Q8D3semidirect
21. #5391D3semidirect
82. C3×SL(2,𝔽3)#41semidirect
209. SL(2,𝔽3)#239semidirect
335. C3#85332semidirect

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×#53912
#105932182
#105932072
#853323
#810943
#639094