Finite Group Databasebeta

← #87041⋊C3 #11923288

Subgroups — #87041⋊C3 #11923288

Order 1152.

555 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 87 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#5569⋊C3C212yesmaximal, complemented
3#5574⋊C3C212yesmaximal
4C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#87041C311yesmaximal, characteristic, complemented
7#3503⋊C3—31nomaximal
8C2×C3⋊(#41⋊C3)C2211yescharacteristic
9C3×#944—41nomaximal
10C3×C4×SL(2,𝔽3)C2211yescharacteristic
11C3×C4×SL(2,𝔽3)C2212yes—
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×Q82C2×C311yescharacteristic
15#5730—31no—
16#5729—32no—
17#5727—32no—
18D3×C4⋊Q8—31no—
19#5579—31no—
20#5578—31no—
21#5574C2×C312yes—
22C4×#239C2×C311yescharacteristic
23#5569C2×C312yes—
24Q8×Dic6C2×C311yescharacteristic
25Q8×SL(2,𝔽3)—21no—
26Q8×SL(2,𝔽3)—31no—
27Q8×SL(2,𝔽3)D311yescharacteristic, complemented
28#557⋊C3—32no—
29#520⋊C3—32no—
30C4×#41⋊C3—31no—
31C2×C32×Q8—41no—
32C2×C3×Dic6—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊(#41⋊C3)Q812yescomplemented
35C3×Dic3⋊C4—42no—
36C3×Dic3⋊C4—42no—
37C3×C4×Dic3—41no—
38#3503—31noSylow
39C2×#239C22×C311yescharacteristic
40C2×D3×Q8—32no—
41C2×#41⋊C3—31no—
42C3×C4⋊Q8—31no—
43C3×C4⋊Q8—32no—
44C3×C4⋊Q8—121no—
45C3×C4×Q8—31no—
46C3×C4×Q8C22×C311yescharacteristic
47C3×C4×Q8C22×C312yes—
48#946—34no—
49#946—31no—
50Q8×Dic3C22×C312yes—
51Q8×Dic3C22×C311yescharacteristic
52#944—32no—
53#944A411yescharacteristic
54#897—31no—
55#896—34no—
56#896—32no—
57#893—31no—
58#893—32no—
59#892—31no—
60D3×C4⋊C4—32no—
61D3×C4⋊C4—32no—
62#890—32no—
63#889—32no—
64#888—32no—
65#888—32no—
66#888—31no—
67C4×D12—31no—
68C4×D12—32no—
69#872—32no—
70C42×D3—31no—
71#870—32no—
72#869—31no—
73C4×Dic6—31no—
74C4×Dic6—32no—
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—
81C32×Q8—82no—
82C2×C32×C4—41no—
83C2×C32×C4—42no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—42no—
87C3×Dic6—82no—
88C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
89Q82—31no—
90Q82C3×D311yescharacteristic
91#557—32no—
92#554—91no—
93#554—91no—
94D4×Q8—92no—
95#544—92no—
96#533—91no—
97C2×C4⋊Q8—91no—
98#520—32no—
99C4×#41—31no—
100C2×C3×Q8—32no—
101C2×C3×Q8—41no—
102C2×C3×Q8—81no—
103C2×C3×Q8—121no—
104C2×C3×Q8C2×A411yescharacteristic
105C2×C3×Q8C23×C311yescharacteristic
106#239—32no—
107#239C3×Q812yescomplemented
108D3×Q8—124no—
109C2×D12—31no—
110C2×C4×D3—31no—
111C2×C4×D3—31no—
112C2×C4×D3—31no—
113C2×C4×D3—34no—
114C2×Dic6—32no—
115C2×Dic6—32no—
116C2×Dic6C2×A411yescharacteristic
117#41⋊C3—32no—
118C2×SL(2,𝔽3)—21no—
119C2×SL(2,𝔽3)C22×D311yescharacteristic
120C3×C4⋊C4—31no—
121C3×C4⋊C4—32no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—122no—
125C3×C4⋊C4—122no—
126C3×C42—31no—
127C3×C42—32no—
128C3×C42—121no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4—32no—
131(C2×D3)⋊C4—32no—
132C4⋊Dic3—31no—
133C4⋊Dic3—32no—
134C4⋊Dic3—32no—
135C4⋊Dic3—31no—
136Dic3⋊C4—34no—
137Dic3⋊C4—32no—
138Dic3⋊C4—32no—
139Dic3⋊C4—32no—
140Dic3⋊C4C2×A412yes—
141Dic3⋊C4C2×A412yes—
142C4×Dic3—31no—
143C4×Dic3—32no—
144C4×Dic3—31no—
145C4×Dic3C2×A411yescharacteristic
146C22×C32—41no—
147C32×C4—42no—
148C32×C4—82no—
149C3×Dic3—42no—
150C3×Dic3—42no—
151C3×Dic3—82no—
152C2×#41—31no—
153C22×Q8—92no—
154C4⋊Q8—31no—
155C4⋊Q8—31no—
156C4⋊Q8—32no—
157C4⋊Q8—91no—
158C4⋊Q8—91no—
159C4⋊Q8—92no—
160C4⋊Q8—92no—
161C4⋊Q8—92no—
162#125—92no—
163#125—92no—
164#125—92no—
165C22⋊Q8—92no—
166C22⋊Q8—91no—
167C22⋊Q8—94no—
168C22⋊Q8—94no—
169C22⋊Q8—91no—
170C4×Q8—31no—
171C4×Q8—31no—
172C4×Q8—32no—
173C4×Q8—92no—
174C4×Q8—91no—
175C4×Q8C2×C3×D311yescharacteristic
176C4×Q8C2×C3×D312yes—
177C4×D4—91no—
178C4×D4—92no—
179C4×D4—91no—
180C4×D4—92no—
181#117—92no—
182#117—91no—
183C2×C4⋊C4—92no—
184C2×C4⋊C4—92no—
185C2×C42—91no—
186C22×D3—31no—
187C3×Q8—22no—
188C3×Q8—31no—
189C3×Q8—82no—
190C3×Q8—84no—
191C3×Q8—122no—
192C3×Q8—242no—
193C3×Q8C2×C3×Q811yescharacteristic
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—41no—
198C2×C3×C4—42no—
199C2×C3×C4—82no—
200C2×C3×C4—81no—
201C2×C3×C4—121no—
202C2×C3×C4—121no—
203C2×C3×C4—122no—
204C2×C3×C4C22×A412yes—
205C2×C3×C4C22×A411yescharacteristic
206C2×Dic3—31no—
207C2×Dic3—32no—
208C2×Dic3C22×A412yes—
209C2×Dic3C22×A411yescharacteristic
210C2×Dic3C22×A411yescharacteristic
211D12—32no—
212D12—32no—
213C4×D3—32no—
214C4×D3—32no—
215C4×D3—62no—
216C4×D3—62no—
217C4×D3—124no—
218Dic6—22no—
219Dic6—64no—
220Dic6—122no—
221SL(2,𝔽3)—21no—
222SL(2,𝔽3)D3×Q811yescharacteristic, complemented
223C2×C32—41no—
224C2×C32—41no—
225C2×C32—41no—
226#41—32no—
227#41—92no—
228C2×Q8—32no—
229C2×Q8—31no—
230C2×Q8—92no—
231C2×Q8—92no—
232C2×Q8—364no—
233C2×Q8D3×A411yescharacteristic
234C2×Q8C22×C3×D311yescharacteristic
235C2×D4—91no—
236C22×C4—91no—
237C22×C4—91no—
238C22×C4—94no—
239C22×C4—91no—
240C4⋊C4—32no—
241C4⋊C4—31no—
242C4⋊C4—31no—
243C4⋊C4—32no—
244C4⋊C4—32no—
245C4⋊C4—32no—
246C4⋊C4—92no—
247C4⋊C4—92no—
248C4⋊C4—92no—
249C4⋊C4—92no—
250C4⋊C4—91no—
251C4⋊C4—94no—
252C4⋊C4—91no—
253C4⋊C4—92no—
254C22⋊C4—92no—
255C22⋊C4—92no—
256C22⋊C4—92no—
257C42—31no—
258C42—31no—
259C42—32no—
260C42—91no—
261C42—91no—
262C42—92no—
263C22×C3—41no—
264C22×C3—81no—
265C22×C3C23×A411yescharacteristic
266C2×D3—32no—
267C2×D3—61no—
268C2×D3—61no—
269C3×C4—22no—
270C3×C4—31no—
271C3×C4—31no—
272C3×C4—42no—
273C3×C4—62no—
274C3×C4—61no—
275C3×C4—82no—
276C3×C4—84no—
277C3×C4—82no—
278C3×C4—122no—
279C3×C4—122no—
280C3×C4—242no—
281C3×C42−1+4⋊C312yes—
282Dic3—22no—
283Dic3—62no—
284Dic3—61no—
285Dic3Q8×A412yes—
286Dic3#99412yes—
287C32—41noSylow
288C23—91no—
289Q8—22no—
290Q8—31no—
291Q8—62no—
292Q8—122no—
293Q8—184no—
294Q8—362no—
295Q8C3×D3×Q811yescharacteristic
296D4—92no—
297D4—92no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—32no—
303C2×C4—32no—
304C2×C4—92no—
305C2×C4—92no—
306C2×C4—91no—
307C2×C4—92no—
308C2×C4—182no—
309C2×C4—182no—
310C2×C4—364no—
311C2×C4C2×D3×A412yes—
312C2×C4C2×D3×A411yescharacteristic
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—81no—
317C2×C3—81no—
318C2×C3—81no—
319C2×C3C2×Q8×A411yescharacteristic
320C2×C3C2×#99411yescharacteristic
321C2×C3C2×2−1+4⋊C311yescharacteristic
322D3—62no—
323C22—92no—
324C22—181no—
325C22—181no—
326C22C22×D3×A411yescharacteristic
327C4—22no—
328C4—31no—
329C4—31no—
330C4—32no—
331C4—32no—
332C4—62no—
333C4—61no—
334C4—62no—
335C4—181no—
336C4—182no—
337C4#1007⋊C312yes—
338C3—41no—
339C3—81no—
340C3#3503⋊C311yescharacteristic, complemented
341C2—182no—
342C2C2×#1007⋊C311yescharacteristic
343C2D3×Q8×A411yescharacteristic
344C2D3×#99411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×Q82C2×C3
socleC22×C3C23×A4
prime radicalC2×C3⋊(#41⋊C3)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
2. #5569⋊C3C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #87041C3semidirect
27. Q8×SL(2,𝔽3)D3semidirect
34. C3⋊(#41⋊C3)Q8semidirect
88. C3×SL(2,𝔽3)C2×Q8semidirect
107. #239C3×Q8semidirect
222. SL(2,𝔽3)D3×Q8semidirect
340. C3#3503⋊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
#5569⋊C32
#5574⋊C32
C4×C3⋊(#41⋊C3)2
SL(2,𝔽3)×Dic62
#870413
#3503⋊C33
C3×#9444