Finite Group Databasebeta

← #11921667

Subgroups — #11921667

Order 1152.

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

normal subgroups only

This group has 497 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
1#5570⋊C3C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
3C3×C4×GL(2,𝔽3)C211yesmaximal, characteristic, complemented
4#10593215C212yesmaximal, complemented
5#10593207C212yesmaximal, complemented
6#85422—31nomaximal
7#79625—31nomaximal
8C2×C3⋊GL(2,𝔽3)C2212yes—
9C2×C3×GL(2,𝔽3)C2211yescharacteristic
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11#63909—41nomaximal
12Dic3×SL(2,𝔽3)C2212yes—
13C3×#859C2211yescharacteristic
14#5570D311yescharacteristic, complemented
15#5553—31no—
16#557⋊C3—31no—
17#5394—32no—
18C4⋊GL(2,𝔽3)—32no—
19C4⋊GL(2,𝔽3)—31no—
20C4⋊GL(2,𝔽3)—61no—
21C4×GL(2,𝔽3)D311yescharacteristic, complemented
22C3×C4×SD16—31no—
23#5163—32no—
24#5027—31no—
25#5007—31no—
26#4871—32no—
27#4817—32no—
28#4764—32no—
29#4694—31no—
30#4679—31no—
31C2×C4⋊(C3⋊D3)—41no—
32C2×C3×C4×D3—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C2×#626—42no—
35C3⋊GL(2,𝔽3)—22no—
36C3×GL(2,𝔽3)—21no—
37C3×C4⋊Dic3—41no—
38#3722—42no—
39#3368—91noSylow
40C2×#235—61no—
41C2×GL(2,𝔽3)—32no—
42C2×GL(2,𝔽3)—62no—
43C2×GL(2,𝔽3)C2×D311yescharacteristic
44C2×C3×SD16—31no—
45C3×C4×Q8C2×D311yescharacteristic
46C3×C4×D4—31no—
47Q8×Dic3C2×D312yes—
48C2×#215—32no—
49#940—32no—
50#930—61no—
51#895—121no—
52#893—32no—
53#890—32no—
54#884—62no—
55#874—31no—
56#869—31no—
57C4×Dic6—31no—
58C4×SL(2,𝔽3)—21no—
59C4×SL(2,𝔽3)—32no—
60C4×SL(2,𝔽3)C2×D311yescharacteristic
61#859C2×D311yescharacteristic
62C3×#106—31no—
63C3×#103—31no—
64C3×#102—31no—
65C3×C4×C8—31no—
66#832—32no—
67#821—32no—
68#821—32no—
69#815—32no—
70Dic3⋊C8—32no—
71#809—32no—
72#807—32no—
73C4⋊(C3⋊C8)—31no—
74C22×C3⋊D3—42no—
75C22×C3×D3—41no—
76C2×C32×C4—41no—
77C4⋊(C3⋊D3)—82no—
78C2×C3×Dic3—41no—
79C2×C3×Dic3—42no—
80C3×C4×D3—82no—
81C3×SL(2,𝔽3)#4111yescharacteristic, complemented
82#626—84no—
83#557—31no—
84#550—91no—
85#490—92no—
86#486—91no—
87#478—92no—
88#474—91no—
89#460—92no—
90#460—91no—
91#446—91no—
92#440—92no—
93C4×SD16—31no—
94C2×C3×Q8C22×D311yescharacteristic
95C2×C3×D4—31no—
96C22×C3×C4—61no—
97C2×#67—62no—
98C2×#67—62no—
99#235—124no—
100C2×D12—32no—
101C2×D12—61no—
102C2×D12—61no—
103C2×D12—121no—
104C2×D12—122no—
105C2×D12—241no—
106C2×C4×D3—41no—
107C2×C4×D3—62no—
108C2×Dic6—31no—
109C2×SL(2,𝔽3)—21no—
110C2×SL(2,𝔽3)C22×D311yescharacteristic
111GL(2,𝔽3)—21no—
112GL(2,𝔽3)—62no—
113GL(2,𝔽3)—122no—
114C3×SD16—61no—
115C3×SD16—61no—
116C2×C3×C8—31no—
117C2×C3×C8—31no—
118C3×C4⋊C4—31no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—121no—
121C3×C22⋊C4—61no—
122C3×C42—31no—
123#215—62no—
124#215—62no—
125(C2×D3)⋊C4—62no—
126(C2×D3)⋊C4—122no—
127C4⋊Dic3—32no—
128C4⋊Dic3—61no—
129C4⋊Dic3S411yescharacteristic
130Dic3⋊C4—32no—
131C4×Dic3—32no—
132C2×C3⋊C8—32no—
133C22×C32—41no—
134C2×C3⋊D3—82no—
135C2×C3⋊D3—82no—
136C2×C3⋊D3—82no—
137C2×C3×D3—42no—
138C2×C3×D3—81no—
139C2×C3×D3—81no—
140C32×C4—42no—
141C3×Dic3—82no—
142C3×Dic3—81no—
143C2×#41—181no—
144C2×SD16—31no—
145C2×SD16—92no—
146C4⋊Q8—91no—
147#127—91no—
148#127—92no—
149#125—92no—
150#121—182no—
151#121—181no—
152C4×Q8—32no—
153C4×Q8—91no—
154C4×Q8D3211yescharacteristic
155C4×D4—31no—
156C4×D4—92no—
157#107—92no—
158#106—31no—
159C4⋊C8—91no—
160C4⋊C8—92no—
161#103—31no—
162#102—31no—
163#102—92no—
164#102—92no—
165#102—92no—
166#102—92no—
167#97—92no—
168C4×C8—31no—
169C23×C3—61no—
170C22×D3—41no—
171C22×D3—62no—
172C22×D3—122no—
173C22×D3—242no—
174C3×Q8—31no—
175C3×Q8#23911yescharacteristic
176C3×D4—61no—
177C3×D4—61no—
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×C4—122no—
185C2×C3×C4C2×S411yescharacteristic
186#67—124no—
187#67—242no—
188C2×Dic3—32no—
189C2×Dic3—41no—
190C2×Dic3C2×S412yes—
191D12—62no—
192D12—62no—
193D12—122no—
194D12—242no—
195D12—244no—
196D12—241no—
197D12—244no—
198C4×D3—82no—
199C4×D3—124no—
200Dic6—62no—
201SL(2,𝔽3)—21no—
202SL(2,𝔽3)#23511yescharacteristic, complemented
203C3×C8—61no—
204C3×C8—61no—
205C3⋊C8—62no—
206C2×C32—41no—
207C2×C32—41no—
208C2×C32—41no—
209C3⋊D3—162no—
210C3×D3—82no—
211#41—364no—
212C2×Q8—91no—
213C2×Q8C2×D3211yescharacteristic
214C2×D4—31no—
215C2×D4—92no—
216C2×D4—182no—
217C2×D4—181no—
218C2×D4—182no—
219C2×D4—181no—
220C22×C4—61no—
221C22×C4—182no—
222SD16—61no—
223SD16—61no—
224SD16—182no—
225SD16—182no—
226C2×C8—31no—
227C2×C8—31no—
228C2×C8—92no—
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—92no—
233C4⋊C4—92no—
234C4⋊C4—181no—
235C22⋊C4—61no—
236C22⋊C4—182no—
237C42—31no—
238C42—92no—
239C22×C3—41no—
240C22×C3—81no—
241C22×C3—121no—
242C22×C3—121no—
243C22×C3—121no—
244C22×C3C22×S411yescharacteristic
245C2×D3—42no—
246C2×D3—81no—
247C2×D3—81no—
248C2×D3—122no—
249C2×D3—122no—
250C2×D3—122no—
251C2×D3—242no—
252C2×D3—242no—
253C2×D3—242no—
254C2×D3—244no—
255C2×D3—482no—
256C2×D3—482no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—42no—
260C3×C4—61no—
261C3×C4—82no—
262C3×C4—121no—
263C3×C4—242no—
264C3×C4#98612yes—
265Dic3—22no—
266Dic3—62no—
267Dic3—81no—
268C32—41noSylow
269C23—61no—
270C23—182no—
271Q8—31no—
272Q8—182no—
273Q8#379911yescharacteristic
274D4—61no—
275D4—61no—
276D4—182no—
277D4—182no—
278D4—364no—
279D4—362no—
280D4—722no—
281D4—721no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—32no—
285C2×C4—61no—
286C2×C4—92no—
287C2×C4—122no—
288C2×C4—364no—
289C2×C4D3×S411yescharacteristic
290C8—61no—
291C8—61no—
292C8—182no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—81no—
297C2×C3—81no—
298C2×C3—81no—
299C2×C3—241no—
300C2×C3#591911yescharacteristic
301C2×C3C2×#98611yescharacteristic
302C2×C3#592611yescharacteristic
303D3—82no—
304D3—242no—
305D3—482no—
306D3—484no—
307C22—121no—
308C22—121no—
309C22—121no—
310C22—362no—
311C22—362no—
312C22—362no—
313C22C2×D3×S411yescharacteristic
314C4—31no—
315C4—31no—
316C4—62no—
317C4—61no—
318C4—121no—
319C4—182no—
320C4#6419912yes—
321C3—41no—
322C3—81no—
323C3#8542211yescharacteristic, complemented
324C2—241no—
325C2—722no—
326C2#1059642111yescharacteristic
327C2C2×#6419911yescharacteristic
328C2#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. #5570⋊C3C2semidirect
2. C4⋊(C3⋊GL(2,𝔽3))C2semidirect
3. C3×C4×GL(2,𝔽3)C2semidirect
4. #10593215C2semidirect
5. #10593207C2semidirect
14. #5570D3semidirect
21. C4×GL(2,𝔽3)D3semidirect
81. C3×SL(2,𝔽3)#41semidirect
202. SL(2,𝔽3)#235semidirect
323. C3#85422semidirect

Maximal subgroups

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

subgroupindex
#5570⋊C32
C4⋊(C3⋊GL(2,𝔽3))2
C3×C4×GL(2,𝔽3)2
#105932152
#105932072
#854223
#796253
#639094