Finite Group Databasebeta

← #87018⋊C3 #11923278

Subgroups — #87018⋊C3 #11923278

Order 1152.

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

normal subgroups only

This group has 549 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×#557⋊C3C211yesmaximal, characteristic, complemented
2#5570⋊C3C211yesmaximal, characteristic, complemented
3#5574⋊C3C212yesmaximal
4C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic
5SL(2,𝔽3)×Dic6C212yesmaximal, complemented
6#87018C311yesmaximal, characteristic, complemented
7#3503⋊C3—31nomaximal
8C2×C3⋊(#41⋊C3)C2211yescharacteristic
9C3×#888—41nomaximal
10C3×C4×SL(2,𝔽3)C2212yes—
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C3×#557C2×C311yescharacteristic
15#5727—31no—
16D3×C4⋊Q8—31no—
17#5690—32no—
18#5689—32no—
19#5578—31no—
20Q8×D12—31no—
21#5574C2×C312yes—
22C4×#239C2×C311yescharacteristic
23#5570C2×C311yescharacteristic
24Q8×Dic6C2×C312yes—
25Q8×SL(2,𝔽3)—32no—
26#557⋊C3—21no—
27#557⋊C3—31no—
28#557⋊C3D311yescharacteristic, complemented
29#520⋊C3—32no—
30C4×#41⋊C3—31no—
31C2×C3×Dic6—42no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊(#41⋊C3)Q812yescomplemented
34C32×C4⋊C4—41no—
35C3×C4⋊Dic3—41no—
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—121no—
44C3×#125—32no—
45C3×C4×Q8—31no—
46C3×C4×Q8C22×C312yes—
47C3×C4×Q8C22×C311yescharacteristic
48#946—31no—
49Q8×Dic3C22×C311yescharacteristic
50Q8×Dic3C22×C312yes—
51#944—31no—
52#897—34no—
53#897—31no—
54#896—34no—
55#896—32no—
56#893—31no—
57#893—32no—
58#892—31no—
59D3×C4⋊C4—32no—
60D3×C4⋊C4—32no—
61#890—32no—
62#889—32no—
63#888—32no—
64#888—32no—
65#888—32no—
66#888A411yescharacteristic
67C4×D12—31no—
68C4×D12—32no—
69#872—32no—
70C42×D3—31no—
71#869—31no—
72#869—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—
81C2×C32×C4—41no—
82C2×C32×C4—42no—
83C2×C3×Dic3—41no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—42no—
86C3×Dic6—84no—
87C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
88Q82—32no—
89#557—31no—
90#557C3×D311yescharacteristic
91#554—92no—
92D4×Q8—91no—
93D4×Q8—91no—
94#544—92no—
95#533—91no—
96C2×C4⋊Q8—91no—
97#520—32no—
98C4×#41—31no—
99C2×C3×Q8—31no—
100C2×C3×Q8—122no—
101C2×C3×Q8C23×C311yescharacteristic
102#239—32no—
103#239C3×Q812yescomplemented
104D3×Q8—124no—
105C2×D12—31no—
106C2×C4×D3—31no—
107C2×C4×D3—31no—
108C2×C4×D3—31no—
109C2×C4×D3—34no—
110C2×Dic6—31no—
111C2×Dic6—34no—
112C2×Dic6C2×A412yes—
113#41⋊C3—32no—
114C2×SL(2,𝔽3)—21no—
115C2×SL(2,𝔽3)C22×D311yescharacteristic
116C3×C4⋊C4—32no—
117C3×C4⋊C4—31no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—32no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—81no—
122C3×C4⋊C4—122no—
123C3×C4⋊C4—121no—
124C3×C4⋊C4C2×A411yescharacteristic
125C3×C42—31no—
126C3×C42—32no—
127C3×C42—121no—
128(C2×D3)⋊C4—32no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4—32no—
131C4⋊Dic3—32no—
132C4⋊Dic3—32no—
133C4⋊Dic3—31no—
134C4⋊Dic3—32no—
135C4⋊Dic3C2×A411yescharacteristic
136Dic3⋊C4—32no—
137Dic3⋊C4—32no—
138Dic3⋊C4—32no—
139Dic3⋊C4—32no—
140Dic3⋊C4C2×A412yes—
141C4×Dic3—31no—
142C4×Dic3—32no—
143C4×Dic3—31no—
144C4×Dic3C2×A411yescharacteristic
145C22×C32—41no—
146C32×C4—42no—
147C32×C4—82no—
148C3×Dic3—42no—
149C3×Dic3—42no—
150C3×Dic3—82no—
151C2×#41—31no—
152C22×Q8—92no—
153C4⋊Q8—31no—
154C4⋊Q8—31no—
155C4⋊Q8—91no—
156C4⋊Q8—91no—
157C4⋊Q8—92no—
158C4⋊Q8—92no—
159C4⋊Q8—92no—
160C4⋊Q8—92no—
161#125—32no—
162#125—92no—
163#125—92no—
164C22⋊Q8—91no—
165C22⋊Q8—92no—
166C22⋊Q8—94no—
167C22⋊Q8—94no—
168C22⋊Q8—91no—
169C4×Q8—31no—
170C4×Q8—31no—
171C4×Q8—32no—
172C4×Q8—91no—
173C4×Q8—92no—
174C4×Q8C2×C3×D311yescharacteristic
175C4×Q8C2×C3×D312yes—
176C4×D4—91no—
177C4×D4—92no—
178C4×D4—91no—
179C4×D4—92no—
180#117—92no—
181#117—91no—
182C2×C4⋊C4—92no—
183C2×C4⋊C4—92no—
184C2×C42—91no—
185C22×D3—31no—
186C3×Q8—31no—
187C3×Q8—62no—
188C3×Q8—244no—
189C3×Q8C2×C3×Q811yescharacteristic
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—32no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
195C2×C3×C4—81no—
196C2×C3×C4—82no—
197C2×C3×C4—121no—
198C2×C3×C4—121no—
199C2×C3×C4—122no—
200C2×C3×C4C22×A412yes—
201C2×C3×C4C22×A411yescharacteristic
202C2×Dic3—31no—
203C2×Dic3—32no—
204C2×Dic3C22×A412yes—
205C2×Dic3C22×A411yescharacteristic
206C2×Dic3C22×A411yescharacteristic
207D12—32no—
208D12—32no—
209C4×D3—32no—
210C4×D3—32no—
211C4×D3—62no—
212C4×D3—62no—
213C4×D3—124no—
214Dic6—24no—
215Dic6—62no—
216Dic6—124no—
217SL(2,𝔽3)—21no—
218SL(2,𝔽3)D3×Q811yescharacteristic, complemented
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—41no—
222#41—32no—
223#41—92no—
224C2×Q8—31no—
225C2×Q8—32no—
226C2×Q8—94no—
227C2×Q8—91no—
228C2×Q8—364no—
229C2×Q8C22×C3×D311yescharacteristic
230C2×D4—91no—
231C22×C4—91no—
232C22×C4—91no—
233C22×C4—94no—
234C22×C4—91no—
235C4⋊C4—32no—
236C4⋊C4—31no—
237C4⋊C4—32no—
238C4⋊C4—32no—
239C4⋊C4—32no—
240C4⋊C4—31no—
241C4⋊C4—92no—
242C4⋊C4—92no—
243C4⋊C4—92no—
244C4⋊C4—92no—
245C4⋊C4—92no—
246C4⋊C4—91no—
247C4⋊C4—92no—
248C4⋊C4—92no—
249C4⋊C4D3×A411yescharacteristic
250C22⋊C4—92no—
251C22⋊C4—92no—
252C22⋊C4—92no—
253C42—31no—
254C42—31no—
255C42—32no—
256C42—91no—
257C42—91no—
258C42—92no—
259C22×C3—41no—
260C22×C3—81no—
261C22×C3C23×A411yescharacteristic
262C2×D3—32no—
263C2×D3—61no—
264C2×D3—61no—
265C3×C4—22no—
266C3×C4—31no—
267C3×C4—31no—
268C3×C4—42no—
269C3×C4—62no—
270C3×C4—61no—
271C3×C4—82no—
272C3×C4—82no—
273C3×C4—122no—
274C3×C4—122no—
275C3×C4—162no—
276C3×C4—242no—
277C3×C4#99412yes—
278Dic3—22no—
279Dic3—62no—
280Dic3—61no—
281Dic3Q8×A412yes—
282Dic32−1+4⋊C312yes—
283C32—41noSylow
284C23—91no—
285Q8—31no—
286Q8—64no—
287Q8—62no—
288Q8—182no—
289Q8—364no—
290Q8C3×D3×Q811yescharacteristic
291D4—92no—
292D4—92no—
293C2×C4—31no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—32no—
298C2×C4—32no—
299C2×C4—92no—
300C2×C4—92no—
301C2×C4—91no—
302C2×C4—92no—
303C2×C4—182no—
304C2×C4—182no—
305C2×C4—364no—
306C2×C4C2×D3×A412yes—
307C2×C4C2×D3×A411yescharacteristic
308C2×C3—41no—
309C2×C3—41no—
310C2×C3—41no—
311C2×C3—81no—
312C2×C3—81no—
313C2×C3—81no—
314C2×C3C2×Q8×A411yescharacteristic
315C2×C3C2×#99411yescharacteristic
316C2×C3C2×2−1+4⋊C311yescharacteristic
317D3—62no—
318C22—92no—
319C22—181no—
320C22—181no—
321C22C22×D3×A411yescharacteristic
322C4—22no—
323C4—31no—
324C4—31no—
325C4—32no—
326C4—32no—
327C4—62no—
328C4—61no—
329C4—62no—
330C4—181no—
331C4—182no—
332C4C3⋊#99412yes—
333C3—41no—
334C3—81no—
335C3#3503⋊C311yescharacteristic, complemented
336C2—182no—
337C2D3×Q8×A411yescharacteristic
338C2C2×C3⋊#99411yescharacteristic
339C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#557C2×C3
socleC22×C3C23×A4
prime radicalC2×C3⋊(#41⋊C3)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
2. #5570⋊C3C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #87018C3semidirect
28. #557⋊C3D3semidirect
33. C3⋊(#41⋊C3)Q8semidirect
87. C3×SL(2,𝔽3)C2×Q8semidirect
103. #239C3×Q8semidirect
218. SL(2,𝔽3)D3×Q8semidirect
335. C3#3503⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#5570⋊C32
#5574⋊C32
C4×C3⋊(#41⋊C3)2
SL(2,𝔽3)×Dic62
#870183
#3503⋊C33
C3×#8884