Finite Group Databasebeta

← #87031⋊C3 #11923286

Subgroups — #87031⋊C3 #11923286

Order 1152.

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

normal subgroups only

This group has 527 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×#557⋊C3C211yesmaximal, characteristic, complemented
2#5572⋊C3C212yesmaximal
3C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic
4#5570⋊C3C212yesmaximal, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#87031C311yesmaximal, characteristic, complemented
7#3502⋊C3—31nomaximal
8C2×D3×SL(2,𝔽3)C2211yescharacteristic
9C3×#897—41nomaximal
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2212yes—
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×#557C2×C311yescharacteristic, complemented
15#5728—31no—
16#5721—31no—
17#5689—32no—
18#5685—32no—
19#5578—31no—
20#5572C2×C312yes—
21C4×D3×Q8C2×C311yescharacteristic
22#5570C2×C312yescomplemented
23Q8×Dic6—31no—
24Q8×Dic6C2×C311yescharacteristic, complemented
25Q8×SL(2,𝔽3)—31no—
26#557⋊C3—21no—
27#557⋊C3—32no—
28#557⋊C3D311yescharacteristic, complemented
29#519⋊C3—32no—
30C2×C4×SL(2,𝔽3)—31no—
31C2×C3×C4×D3—41no—
32C2×C3×Dic6—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34D3×SL(2,𝔽3)Q812yescomplemented
35C32×C4⋊C4—41no—
36C3×(C2×D3)⋊C4—42no—
37C3×C4⋊Dic3—42no—
38#3502—31noSylow
39C2×D3×Q8C22×C311yescharacteristic
40C22×SL(2,𝔽3)—31no—
41C3×C4⋊Q8—31no—
42C3×#125—32no—
43C3×C22⋊Q8—121no—
44C3×C4×Q8—31no—
45C3×C4×Q8C22×C311yescharacteristic
46C3×C4×Q8C22×C312yes—
47#946—31no—
48Q8×Dic3—32no—
49Q8×Dic3C22×C312yes—
50Q8×Dic3C22×C311yescharacteristic
51#944—31no—
52#897—32no—
53#897A411yescharacteristic
54#896—32no—
55#892—32no—
56D3×C4⋊C4—31no—
57#890—32no—
58#890—32no—
59#889—34no—
60#889—32no—
61#888—34no—
62#888—31no—
63#888—32no—
64#887—32no—
65#887—32no—
66#887—31no—
67#872—32no—
68C42×D3—31no—
69#869—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—
82C2×C32×C4—41no—
83C2×C32×C4—42no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—42no—
86C3×C4×D3—82no—
87C3×Dic6—82no—
88C3×SL(2,𝔽3)C2×Q811yescharacteristic
89Q82—31no—
90Q82—91no—
91#557—32no—
92#557—92no—
93#557—91no—
94#557C3×D311yescharacteristic, complemented
95#544—92no—
96#533—91no—
97#533—91no—
98#519—32no—
99C2×C4×Q8—31no—
100C2×C3×Q8—31no—
101C2×C3×Q8—121no—
102C2×C3×Q8C23×C311yescharacteristic
103C22×C3×C4—121no—
104D3×Q8—32no—
105D3×Q8C3×Q812yescomplemented
106C2×C4×D3—31no—
107C2×C4×D3—31no—
108C2×C4×D3C2×A411yescharacteristic
109C2×Dic6—31no—
110C2×Dic6—32no—
111C2×Dic6—32no—
112C2×Dic6C2×A411yescharacteristic
113C2×SL(2,𝔽3)—21no—
114C2×SL(2,𝔽3)—32no—
115C2×SL(2,𝔽3)C22×D311yescharacteristic
116C3×C4⋊C4—31no—
117C3×C4⋊C4—32no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—32no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—81no—
122C3×C4⋊C4—122no—
123C3×C4⋊C4C2×A411yescharacteristic
124C3×C22⋊C4—122no—
125C3×C42—31no—
126C3×C42—32no—
127(C2×D3)⋊C4—32no—
128(C2×D3)⋊C4C2×A412yes—
129C4⋊Dic3—34no—
130C4⋊Dic3—31no—
131C4⋊Dic3—32no—
132C4⋊Dic3—31no—
133C4⋊Dic3C2×A412yes—
134Dic3⋊C4—32no—
135Dic3⋊C4—32no—
136Dic3⋊C4—32no—
137Dic3⋊C4—32no—
138Dic3⋊C4—32no—
139Dic3⋊C4—32no—
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—82no—
152C3×Dic3—81no—
153C22×Q8—31no—
154C4⋊Q8—31no—
155C4⋊Q8—92no—
156C4⋊Q8—91no—
157C4⋊Q8—94no—
158C4⋊Q8—92no—
159C4⋊Q8—91no—
160C4⋊Q8—91no—
161#125—32no—
162#125—92no—
163#125—92no—
164#125—92no—
165#125—94no—
166C22⋊Q8—31no—
167C22⋊Q8—92no—
168C22⋊Q8—92no—
169C22⋊Q8—91no—
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—31no—
190C3×Q8—62no—
191C3×Q8—242no—
192C3×Q8C2×C3×Q811yescharacteristic
193C2×C3×C4—31no—
194C2×C3×C4—32no—
195C2×C3×C4—31no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—81no—
199C2×C3×C4—82no—
200C2×C3×C4—122no—
201C2×C3×C4—121no—
202C2×C3×C4—242no—
203C2×C3×C4C22×A412yes—
204C2×C3×C4C22×A411yescharacteristic
205C2×Dic3—31no—
206C2×Dic3—32no—
207C2×Dic3—31no—
208C2×Dic3C22×A411yescharacteristic
209C2×Dic3C22×A412yes—
210C4×D3—22no—
211C4×D3—32no—
212C4×D3—32no—
213C4×D3—121no—
214Dic6—22no—
215Dic6—32no—
216Dic6—32no—
217Dic6—122no—
218Dic6—122no—
219SL(2,𝔽3)—21no—
220SL(2,𝔽3)D3×Q811yescharacteristic
221C2×C32—41no—
222C2×C32—41no—
223C2×C32—41no—
224C3×D3—82no—
225C2×Q8—31no—
226C2×Q8—31no—
227C2×Q8—32no—
228C2×Q8—92no—
229C2×Q8—91no—
230C2×Q8—92no—
231C2×Q8—92no—
232C2×Q8C22×C3×D311yescharacteristic
233C22×C4—31no—
234C22×C4—91no—
235C22×C4—91no—
236C4⋊C4—31no—
237C4⋊C4—32no—
238C4⋊C4—32no—
239C4⋊C4—32no—
240C4⋊C4—32no—
241C4⋊C4—92no—
242C4⋊C4—92no—
243C4⋊C4—92no—
244C4⋊C4—92no—
245C4⋊C4—91no—
246C4⋊C4—92no—
247C4⋊C4—92no—
248C4⋊C4—94no—
249C4⋊C4—92no—
250C4⋊C4—91no—
251C4⋊C4D3×A411yescharacteristic
252C22⋊C4—32no—
253C22⋊C4—92no—
254C42—31no—
255C42—32no—
256C42—91no—
257C42—91no—
258C42—92no—
259C42—94no—
260C42—91no—
261C22×C3—41no—
262C22×C3—81no—
263C22×C3—122no—
264C22×C3—241no—
265C22×C3—241no—
266C22×C3C23×A411yescharacteristic
267C2×D3—21no—
268C2×D3—21no—
269C2×D3Q8×A412yes—
270C3×C4—22no—
271C3×C4—31no—
272C3×C4—31no—
273C3×C4—42no—
274C3×C4—62no—
275C3×C4—61no—
276C3×C4—82no—
277C3×C4—82no—
278C3×C4—162no—
279C3×C4—241no—
280C3×C4—242no—
281C3×C42−1+4⋊C312yes—
282Dic3—22no—
283Dic3—21no—
284Dic3—32no—
285Dic3—62no—
286Dic3—61no—
287C32—41noSylow
288C23—31no—
289Q8—31no—
290Q8—62no—
291Q8—62no—
292Q8—92no—
293Q8—92no—
294Q8—362no—
295Q8—362no—
296Q8C3×D3×Q811yescharacteristic
297C2×C4—31no—
298C2×C4—32no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—32no—
302C2×C4—62no—
303C2×C4—92no—
304C2×C4—92no—
305C2×C4—91no—
306C2×C4—92no—
307C2×C4—91no—
308C2×C4—361no—
309C2×C4C2×D3×A412yes—
310C2×C4C2×D3×A411yescharacteristic
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—81no—
315C2×C3—81no—
316C2×C3—81no—
317C2×C3—242no—
318C2×C3C2×Q8×A411yescharacteristic
319C2×C3C2×2−1+4⋊C311yescharacteristic
320C2×C3#594511yescharacteristic
321D3—22no—
322C22—32no—
323C22—61no—
324C22—61no—
325C22C22×D3×A411yescharacteristic
326C4—22no—
327C4—31no—
328C4—31no—
329C4—61no—
330C4—62no—
331C4—62no—
332C4—61no—
333C4—92no—
334C4—181no—
335C4—182no—
336C4#1010⋊C312yes—
337C3—41no—
338C3—81no—
339C3#3502⋊C311yescharacteristic, complemented
340C2—62no—
341C2D3×Q8×A411yescharacteristic
342C2C2×#1010⋊C311yescharacteristic
343C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#557C2×C3
socleC22×C3C23×A4
prime radicalC2×D3×SL(2,𝔽3)C22
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
4. #5570⋊C3C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #87031C3semidirect
14. C3×#557C2×C3semidirect
22. #5570C2×C3semidirect
24. Q8×Dic6C2×C3semidirect
28. #557⋊C3D3semidirect
34. D3×SL(2,𝔽3)Q8semidirect
94. #557C3×D3semidirect
105. D3×Q8C3×Q8semidirect
339. C3#3502⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#5572⋊C32
C4×D3×SL(2,𝔽3)2
#5570⋊C32
SL(2,𝔽3)×Dic62
#870313
#3502⋊C33
C3×#8974