Finite Group Databasebeta

← #87046⋊C3 #11923293

Subgroups — #87046⋊C3 #11923293

Order 1152.

533 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 533 conjugacy classes of subgroups — too many to draw. Only its 85 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
1C3×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5572⋊C3C212yesmaximal
3C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic
4#5570⋊C3C211yesmaximal, characteristic, complemented
5#5569⋊C3C212yesmaximal, complemented
6#87046C311yesmaximal, characteristic, complemented
7#3502⋊C3—31nomaximal
8C2×D3×SL(2,𝔽3)C2211yescharacteristic
9C3×#946—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C3×Q82C2×C311yescharacteristic, complemented
15#5729—32no—
16#5728—31no—
17#5722—31no—
18#5721—32no—
19#5578—31no—
20#5572C2×C312yes—
21C4×D3×Q8C2×C311yescharacteristic
22#5570C2×C311yescharacteristic, complemented
23#5569C2×C312yescomplemented
24#5567—31no—
25Q8×SL(2,𝔽3)—21no—
26Q8×SL(2,𝔽3)D311yescharacteristic, complemented
27#557⋊C3—32no—
28#557⋊C3—31no—
29#519⋊C3—32no—
30C2×C4×SL(2,𝔽3)—31no—
31C2×C32×Q8—41no—
32C2×C3×C4×D3—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34D3×SL(2,𝔽3)Q812yescomplemented
35C3×(C2×D3)⋊C4—42no—
36C3×C4⋊Dic3—41no—
37C3×Dic3⋊C4—42no—
38#3502—31noSylow
39C2×D3×Q8C22×C311yescharacteristic
40C22×SL(2,𝔽3)—31no—
41C3×C4⋊Q8—32no—
42C3×C4⋊Q8—31no—
43C3×C22⋊Q8—121no—
44C3×C4×Q8—31no—
45C3×C4×Q8C22×C312yes—
46C3×C4×Q8C22×C311yescharacteristic
47#946—32no—
48#946A411yescharacteristic
49Q8×Dic3—32no—
50Q8×Dic3C22×C311yescharacteristic
51Q8×Dic3C22×C312yes—
52#944—34no—
53#944—31no—
54#897—31no—
55#896—32no—
56#892—32no—
57D3×C4⋊C4—31no—
58#890—32no—
59#890—32no—
60#889—34no—
61#889—32no—
62#888—32no—
63#888—31no—
64#887—32no—
65#887—32no—
66#887—31no—
67#872—32no—
68C42×D3—31no—
69#870—32no—
70#869—31no—
71C4×Dic6—31no—
72C4×Dic6—32no—
73C4×Dic6—32no—
74C4×Dic6—31no—
75C4×SL(2,𝔽3)—21no—
76C4×SL(2,𝔽3)—22no—
77C4×SL(2,𝔽3)—31no—
78C4×SL(2,𝔽3)—32no—
79C4×SL(2,𝔽3)C2×D311yescharacteristic
80C4×SL(2,𝔽3)C2×D312yes—
81C22×C3×D3—41no—
82C32×Q8—82no—
83C2×C32×C4—41no—
84C2×C32×C4—42no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—42no—
87C3×C4×D3—82no—
88C3×SL(2,𝔽3)C2×Q811yescharacteristic
89Q82—91no—
90Q82C3×D311yescharacteristic, complemented
91#557—32no—
92#557—31no—
93#557—91no—
94#557—92no—
95#544—92no—
96#533—91no—
97#533—91no—
98#519—32no—
99C2×C4×Q8—31no—
100C2×C3×Q8—32no—
101C2×C3×Q8—41no—
102C2×C3×Q8—81no—
103C2×C3×Q8C2×A411yescharacteristic
104C2×C3×Q8C23×C311yescharacteristic
105C22×C3×C4—121no—
106D3×Q8—32no—
107D3×Q8C3×Q812yescomplemented
108C2×C4×D3—31no—
109C2×C4×D3—31no—
110C2×C4×D3C2×A411yescharacteristic
111C2×Dic6—31no—
112C2×Dic6—31no—
113C2×Dic6—32no—
114C2×SL(2,𝔽3)—21no—
115C2×SL(2,𝔽3)—32no—
116C2×SL(2,𝔽3)C22×D311yescharacteristic
117C3×C4⋊C4—32no—
118C3×C4⋊C4—31no—
119C3×C4⋊C4—31no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—122no—
122C3×C4⋊C4—121no—
123C3×C22⋊C4—122no—
124C3×C42—31no—
125C3×C42—32no—
126(C2×D3)⋊C4—32no—
127(C2×D3)⋊C4C2×A412yes—
128C4⋊Dic3—32no—
129C4⋊Dic3—32no—
130C4⋊Dic3—32no—
131C4⋊Dic3—31no—
132C4⋊Dic3C2×A411yescharacteristic
133Dic3⋊C4—32no—
134Dic3⋊C4—34no—
135Dic3⋊C4—32no—
136Dic3⋊C4—32no—
137Dic3⋊C4—32no—
138Dic3⋊C4—32no—
139Dic3⋊C4C2×A412yes—
140C4×Dic3—34no—
141C4×Dic3—31no—
142C4×Dic3—31no—
143C4×Dic3—32no—
144C4×Dic3—31no—
145C22×C32—41no—
146C2×C3×D3—42no—
147C2×C3×D3—81no—
148C2×C3×D3—81no—
149C32×C4—42no—
150C32×C4—82no—
151C3×Dic3—81no—
152C3×Dic3—82no—
153C22×Q8—31no—
154C4⋊Q8—31no—
155C4⋊Q8—32no—
156C4⋊Q8—91no—
157C4⋊Q8—91no—
158C4⋊Q8—92no—
159C4⋊Q8—94no—
160C4⋊Q8—91no—
161#125—92no—
162#125—92no—
163#125—92no—
164#125—92no—
165#125—94no—
166C22⋊Q8—31no—
167C22⋊Q8—92no—
168C22⋊Q8—91no—
169C22⋊Q8—92no—
170C4×Q8—31no—
171C4×Q8—31no—
172C4×Q8—32no—
173C4×Q8—92no—
174C4×Q8—91no—
175C4×Q8—92no—
176C4×Q8—91no—
177C4×Q8—92no—
178C4×Q8—92no—
179C4×Q8—92no—
180C4×Q8—91no—
181C4×Q8C2×C3×D311yescharacteristic
182C4×Q8C2×C3×D312yes—
183#117—92no—
184#117—92no—
185C2×C4⋊C4—91no—
186C2×C42—91no—
187C23×C3—121no—
188C22×D3C22×A411yescharacteristic
189C3×Q8—22no—
190C3×Q8—31no—
191C3×Q8—82no—
192C3×Q8—84no—
193C3×Q8—122no—
194C3×Q8C2×C3×Q811yescharacteristic
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—31no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—82no—
201C2×C3×C4—81no—
202C2×C3×C4—122no—
203C2×C3×C4—121no—
204C2×C3×C4—242no—
205C2×C3×C4C22×A412yes—
206C2×C3×C4C22×A411yescharacteristic
207C2×Dic3—31no—
208C2×Dic3—32no—
209C2×Dic3—31no—
210C2×Dic3C22×A411yescharacteristic
211C2×Dic3C22×A412yes—
212C4×D3—22no—
213C4×D3—32no—
214C4×D3—32no—
215C4×D3—121no—
216Dic6—32no—
217Dic6—32no—
218Dic6—62no—
219Dic6—122no—
220SL(2,𝔽3)—21no—
221SL(2,𝔽3)D3×Q811yescharacteristic
222C2×C32—41no—
223C2×C32—41no—
224C2×C32—41no—
225C3×D3—82no—
226C2×Q8—32no—
227C2×Q8—32no—
228C2×Q8—92no—
229C2×Q8—91no—
230C2×Q8—91no—
231C2×Q8—92no—
232C2×Q8D3×A411yescharacteristic
233C2×Q8C22×C3×D311yescharacteristic
234C22×C4—31no—
235C22×C4—91no—
236C22×C4—91no—
237C4⋊C4—31no—
238C4⋊C4—32no—
239C4⋊C4—31no—
240C4⋊C4—32no—
241C4⋊C4—32no—
242C4⋊C4—31no—
243C4⋊C4—92no—
244C4⋊C4—92no—
245C4⋊C4—92no—
246C4⋊C4—91no—
247C4⋊C4—92no—
248C4⋊C4—92no—
249C4⋊C4—92no—
250C4⋊C4—94no—
251C4⋊C4—92no—
252C4⋊C4—92no—
253C22⋊C4—32no—
254C22⋊C4—92no—
255C42—31no—
256C42—32no—
257C42—91no—
258C42—91no—
259C42—92no—
260C42—94no—
261C42—91no—
262C22×C3—41no—
263C22×C3—81no—
264C22×C3—122no—
265C22×C3—241no—
266C22×C3—241no—
267C22×C3C23×A411yescharacteristic
268C2×D3—21no—
269C2×D3—21no—
270C2×D3Q8×A412yes—
271C3×C4—22no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—42no—
275C3×C4—62no—
276C3×C4—61no—
277C3×C4—82no—
278C3×C4—84no—
279C3×C4—82no—
280C3×C4—241no—
281C3×C4—242no—
282C3×C42−1+4⋊C312yes—
283Dic3—21no—
284Dic3—22no—
285Dic3—32no—
286Dic3—62no—
287Dic3—61no—
288C32—41noSylow
289C23—31no—
290Q8—22no—
291Q8—31no—
292Q8—92no—
293Q8—92no—
294Q8—122no—
295Q8—182no—
296Q8—362no—
297Q8C3×D3×Q811yescharacteristic
298C2×C4—31no—
299C2×C4—32no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—32no—
303C2×C4—62no—
304C2×C4—92no—
305C2×C4—92no—
306C2×C4—91no—
307C2×C4—92no—
308C2×C4—91no—
309C2×C4—361no—
310C2×C4C2×D3×A412yes—
311C2×C4C2×D3×A411yescharacteristic
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—81no—
316C2×C3—81no—
317C2×C3—81no—
318C2×C3—242no—
319C2×C3C2×Q8×A411yescharacteristic
320C2×C3C2×2−1+4⋊C311yescharacteristic
321C2×C3#594511yescharacteristic
322D3—22no—
323C22—32no—
324C22—61no—
325C22—61no—
326C22C22×D3×A411yescharacteristic
327C4—22no—
328C4—31no—
329C4—31no—
330C4—61no—
331C4—61no—
332C4—62no—
333C4—62no—
334C4—92no—
335C4—182no—
336C4—181no—
337C4#1007⋊C312yes—
338C3—41no—
339C3—81no—
340C3#3502⋊C311yescharacteristic, complemented
341C2—62no—
342C2C2×#1007⋊C311yescharacteristic
343C2D3×Q8×A411yescharacteristic
344C2#1059662311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×Q82C2×C3
socleC22×C3C23×A4
prime radicalC2×D3×SL(2,𝔽3)C22
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
4. #5570⋊C3C2semidirect
5. #5569⋊C3C2semidirect
6. #87046C3semidirect
14. C3×Q82C2×C3semidirect
22. #5570C2×C3semidirect
23. #5569C2×C3semidirect
26. Q8×SL(2,𝔽3)D3semidirect
34. D3×SL(2,𝔽3)Q8semidirect
90. Q82C3×D3semidirect
107. D3×Q8C3×Q8semidirect
340. C3#3502⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
#5572⋊C32
C4×D3×SL(2,𝔽3)2
#5570⋊C32
#5569⋊C32
#870463
#3502⋊C33
C3×#9464