Finite Group Databasebeta

← #11921671

Subgroups — #11921671

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#10593204C212yesmaximal, complemented
6#85422—31nomaximal
7#80371—31nomaximal
8C2×C3⋊GL(2,𝔽3)C2211yescharacteristic
9C2×C3×GL(2,𝔽3)C2212yes—
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11#63916—41nomaximal
12Dic3×SL(2,𝔽3)C2212yes—
13C3⋊#859C2211yescharacteristic
14#5718—31no—
15#5570D311yescharacteristic, complemented
16#557⋊C3—31no—
17#5394—32no—
18C4⋊GL(2,𝔽3)—32no—
19C4⋊GL(2,𝔽3)D311yescharacteristic, complemented
20C4×GL(2,𝔽3)—31no—
21C4×GL(2,𝔽3)—61no—
22C3×#460—31no—
23#5164—32no—
24#5163—32no—
25#5070—31no—
26C4×#215—31no—
27#4846—31no—
28#4831—31no—
29#4766—32no—
30#4764—32no—
31C2×C4×C3⋊D3—41no—
32C2×C3×D12—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C2×#626—42no—
35C3⋊GL(2,𝔽3)—21no—
36C3×GL(2,𝔽3)—22no—
37C3×C4⋊Dic3—41no—
38(C2×D3)⋊Dic3—42no—
39#3368—91noSylow
40C2×#237—61no—
41C2×GL(2,𝔽3)—31no—
42C2×GL(2,𝔽3)—61no—
43C2×GL(2,𝔽3)C2×D312yes—
44C2×C3×SD16—32no—
45C3×#127—31no—
46C3×C4×Q8C2×D311yescharacteristic
47Q8×Dic3C2×D312yes—
48C2×#215—31no—
49#940—32no—
50#939—62no—
51#938—61no—
52D4×Dic3—32no—
53#895—121no—
54#890—32no—
55C4×D12—31no—
56#869—31no—
57C4×Dic6—31no—
58C4×SL(2,𝔽3)—21no—
59C4×SL(2,𝔽3)—32no—
60C4×SL(2,𝔽3)C2×D311yescharacteristic
61#859—31no—
62#859—61no—
63C3×C4⋊C8—31no—
64C3×#102—32no—
65#835—31no—
66#832—32no—
67#832—32no—
68#821—32no—
69#818—32no—
70#815—32no—
71Dic3⋊C8—32no—
72#809—31no—
73#808—31no—
74C4×C3⋊C8—31no—
75C22×C3⋊D3—41no—
76C22×C3×D3—42no—
77C2×C32×C4—41no—
78C2×C3⋊Dic3—41no—
79C4×C3⋊D3—82no—
80C2×C3×Dic3—42no—
81C3×D12—82no—
82C3×SL(2,𝔽3)#4111yescharacteristic, complemented
83#626—84no—
84#557—31no—
85#550—91no—
86#490—92no—
87#486—91no—
88#478—92no—
89#474—91no—
90#460—31no—
91#460—92no—
92#446—91no—
93#440—92no—
94C4×SD16—91no—
95C2×C3×Q8C22×D311yescharacteristic
96C2×C3×D4—32no—
97C2×C3×D4—61no—
98C2×C3×D4—61no—
99C2×#67—62no—
100C2×#67—62no—
101C22×Dic3—62no—
102#237—124no—
103C2×D12—31no—
104C2×D12—41no—
105C2×D12—122no—
106C2×C4×D3—61no—
107C2×C4×D3—121no—
108C2×C4×D3—241no—
109C2×Dic6—31no—
110C2×SL(2,𝔽3)—21no—
111C2×SL(2,𝔽3)C22×D311yescharacteristic
112GL(2,𝔽3)—22no—
113GL(2,𝔽3)—62no—
114GL(2,𝔽3)—61no—
115C3×SD16—62no—
116C3×SD16—62no—
117C2×C3×C8—32no—
118C3×C4⋊C4—31no—
119C3×C4⋊C4—121no—
120C3×C42—31no—
121#217—62no—
122#215—61no—
123#215—61no—
124(C2×D3)⋊C4—61no—
125(C2×D3)⋊C4—122no—
126C4⋊Dic3—32no—
127C4⋊Dic3—31no—
128C4⋊Dic3—61no—
129C4⋊Dic3S411yescharacteristic
130Dic3⋊C4—32no—
131C4×Dic3—32no—
132C2×C3⋊C8—31no—
133C2×C3⋊C8—31no—
134C22×C32—41no—
135C2×C3⋊D3—42no—
136C2×C3⋊D3—81no—
137C2×C3⋊D3—81no—
138C2×C3×D3—82no—
139C2×C3×D3—82no—
140C2×C3×D3—82no—
141C32×C4—42no—
142C3⋊Dic3—81no—
143C3×Dic3—82no—
144C2×#41—181no—
145C2×SD16—32no—
146C2×SD16—91no—
147C4⋊Q8—91no—
148#127—31no—
149#127—92no—
150#125—92no—
151#121—181no—
152#121—182no—
153C4×Q8—32no—
154C4×Q8—91no—
155C4×Q8D3211yescharacteristic
156C4×D4—91no—
157C4×D4—92no—
158#107—92no—
159#106—91no—
160C4⋊C8—31no—
161C4⋊C8—92no—
162#103—91no—
163#102—32no—
164#102—91no—
165#102—92no—
166#102—92no—
167#102—92no—
168#97—92no—
169C4×C8—91no—
170C23×C3—62no—
171C22×D3—42no—
172C22×D3—61no—
173C22×D3—121no—
174C22×D3—241no—
175C3×Q8—31no—
176C3×Q8#23911yescharacteristic
177C3×D4—62no—
178C3×D4—62no—
179C3×D4—122no—
180C3×D4—241no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—41no—
184C2×C3×C4—81no—
185C2×C3×C4—122no—
186C2×C3×C4C2×S411yescharacteristic
187#67—124no—
188#67—242no—
189C2×Dic3—32no—
190C2×Dic3—61no—
191C2×Dic3—121no—
192C2×Dic3—124no—
193C2×Dic3—241no—
194C2×Dic3C2×S412yes—
195D12—61no—
196D12—61no—
197D12—82no—
198D12—244no—
199C4×D3—122no—
200C4×D3—242no—
201C4×D3—244no—
202Dic6—62no—
203SL(2,𝔽3)—21no—
204SL(2,𝔽3)#23711yescharacteristic, complemented
205C3×C8—62no—
206C3⋊C8—61no—
207C3⋊C8—61no—
208C2×C32—41no—
209C2×C32—41no—
210C2×C32—41no—
211C3⋊D3—82no—
212C3×D3—162no—
213#41—364no—
214C2×Q8—91no—
215C2×Q8C2×D3211yescharacteristic
216C2×D4—32no—
217C2×D4—61no—
218C2×D4—61no—
219C2×D4—91no—
220C2×D4—182no—
221C2×D4—182no—
222C22×C4—182no—
223C22×C4—181no—
224SD16—62no—
225SD16—62no—
226SD16—181no—
227SD16—181no—
228C2×C8—32no—
229C2×C8—91no—
230C2×C8—91no—
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—92no—
234C4⋊C4—92no—
235C4⋊C4—91no—
236C4⋊C4—181no—
237C22⋊C4—182no—
238C22⋊C4—181no—
239C42—31no—
240C42—92no—
241C22×C3—41no—
242C22×C3—81no—
243C22×C3—122no—
244C22×C3—122no—
245C22×C3—122no—
246C22×C3C22×S411yescharacteristic
247C2×D3—82no—
248C2×D3—82no—
249C2×D3—82no—
250C2×D3—122no—
251C2×D3—121no—
252C2×D3—121no—
253C2×D3—121no—
254C2×D3—241no—
255C2×D3—241no—
256C2×D3—242no—
257C2×D3—242no—
258C2×D3—242no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—42no—
262C3×C4—61no—
263C3×C4—82no—
264C3×C4—242no—
265C3×C4#98612yes—
266Dic3—22no—
267Dic3—62no—
268Dic3—121no—
269Dic3—241no—
270Dic3—242no—
271C32—41noSylow
272C23—62no—
273C23—181no—
274Q8—31no—
275Q8—182no—
276Q8#379611yescharacteristic
277D4—62no—
278D4—62no—
279D4—122no—
280D4—181no—
281D4—181no—
282D4—241no—
283D4—364no—
284D4—722no—
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—32no—
288C2×C4—92no—
289C2×C4—181no—
290C2×C4—364no—
291C2×C4—362no—
292C2×C4D3×S411yescharacteristic
293C8—62no—
294C8—181no—
295C8—181no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—41no—
299C2×C3—81no—
300C2×C3—81no—
301C2×C3—81no—
302C2×C3—242no—
303C2×C3#591911yescharacteristic
304C2×C3C2×#98611yescharacteristic
305C2×C3#592611yescharacteristic
306D3—162no—
307D3—242no—
308D3—244no—
309D3—241no—
310C22—122no—
311C22—122no—
312C22—122no—
313C22—361no—
314C22—361no—
315C22—361no—
316C22C2×D3×S411yescharacteristic
317C4—31no—
318C4—31no—
319C4—61no—
320C4—62no—
321C4—182no—
322C4—361no—
323C4#6419912yes—
324C3—41no—
325C3—81no—
326C3#8542211yescharacteristic, complemented
327C2—242no—
328C2—721no—
329C2#1059642011yescharacteristic
330C2C2×#6419911yescharacteristic
331C2#1059646011yescharacteristic
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. #10593204C2semidirect
15. #5570D3semidirect
19. C4⋊GL(2,𝔽3)D3semidirect
82. C3×SL(2,𝔽3)#41semidirect
204. SL(2,𝔽3)#237semidirect
326. 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
#105932042
#854223
#803713
#639164