Finite Group Databasebeta

← #87015⋊C3 #11923276

Subgroups — #87015⋊C3 #11923276

Order 1152.

555 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 93 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#5569⋊C3C212yesmaximal, complemented
3C4×C3⋊(#41⋊C3)C212yesmaximal
4C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#87015C311yesmaximal, characteristic, complemented
7Q8×#41⋊C3—31nomaximal
8C2×C3⋊(#41⋊C3)C2211yescharacteristic
9C3×#887—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
15#5728—31no—
16#5727—31no—
17#5690—32no—
18#5688—32no—
19#5579—31no—
20C4×#239C2×C312yes—
21C4×#239C2×C311yescharacteristic
22C4×D3×Q8—31no—
23#5569C2×C312yes—
24Q8×Dic6C2×C311yescharacteristic
25Q8×SL(2,𝔽3)—31no—
26#557⋊C3—21no—
27#557⋊C3—32no—
28#557⋊C3D311yescharacteristic, complemented
29C4×#41⋊C3—31no—
30C4×#41⋊C3—32no—
31C2×C3×Dic6—41no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊(#41⋊C3)Q812yescomplemented
34C32×C4⋊C4—41no—
35C3×Dic3⋊C4—42no—
36C3×C4×Dic3—42no—
37C3×C4×Dic3—41no—
38Q8×#41—31noSylow
39C2×#239C22×C311yescharacteristic
40C2×D3×Q8—31no—
41C2×#41⋊C3—31no—
42C3×C4⋊Q8—31no—
43C3×#125—32no—
44C3×C4×Q8—31no—
45C3×C4×Q8—121no—
46C3×C4×Q8C22×C311yescharacteristic
47C3×C4×Q8C22×C312yes—
48#946—32no—
49Q8×Dic3—31no—
50Q8×Dic3C22×C312yes—
51Q8×Dic3C22×C311yescharacteristic
52#897—32no—
53#896—34no—
54#896—34no—
55#893—31no—
56#893—32no—
57#892—31no—
58#892—32no—
59D3×C4⋊C4—31no—
60D3×C4⋊C4—32no—
61#890—32no—
62#888—32no—
63#888—32no—
64#887—32no—
65#887—32no—
66#887A411yescharacteristic
67C4×D12—32no—
68C4×D12—31no—
69C42×D3—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×D312yes—
80C4×SL(2,𝔽3)C2×D311yescharacteristic
81C2×C32×C4—41no—
82C2×C32×C4—42no—
83C2×C3×Dic3—42no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—41no—
86C3×Dic6—44no—
87C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
88Q82—31no—
89#557—32no—
90#557C3×D311yescharacteristic
91#554—91no—
92#554—92no—
93D4×Q8—91no—
94#533—92no—
95#533—91no—
96C4×#41—31no—
97C4×#41—32no—
98C2×C4×Q8—91no—
99C2×C3×Q8—31no—
100C2×C3×Q8—121no—
101C2×C3×Q8C23×C311yescharacteristic
102#239—32no—
103#239C3×Q812yescomplemented
104D3×Q8—64no—
105C2×D12—31no—
106C2×C4×D3—31no—
107C2×C4×D3—31no—
108C2×C4×D3—32no—
109C2×C4×D3—32no—
110C2×C4×D3—31no—
111C2×Dic6—32no—
112C2×Dic6—32no—
113C2×Dic6C2×A411yescharacteristic
114#41⋊C3—32no—
115C2×SL(2,𝔽3)—21no—
116C2×SL(2,𝔽3)C22×D311yescharacteristic
117C3×C4⋊C4—31no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—32no—
120C3×C4⋊C4—32no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—81no—
123C3×C4⋊C4—122no—
124C3×C4⋊C4C2×A411yescharacteristic
125C3×C42—31no—
126C3×C42—32no—
127C3×C42—121no—
128C3×C42—122no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4—34no—
131C4⋊Dic3—31no—
132C4⋊Dic3—32no—
133C4⋊Dic3—32no—
134C4⋊Dic3—31no—
135Dic3⋊C4—34no—
136Dic3⋊C4—32no—
137Dic3⋊C4—32no—
138Dic3⋊C4C2×A412yes—
139C4×Dic3—32no—
140C4×Dic3—31no—
141C4×Dic3—31no—
142C4×Dic3—32no—
143C4×Dic3C2×A412yes—
144C4×Dic3C2×A411yescharacteristic
145C22×C32—41no—
146C32×C4—42no—
147C32×C4—82no—
148C3×Dic3—42no—
149C3×Dic3—44no—
150C3×Dic3—81no—
151C2×#41—31no—
152C22×Q8—91no—
153C4⋊Q8—31no—
154C4⋊Q8—91no—
155C4⋊Q8—92no—
156C4⋊Q8—92no—
157#125—32no—
158#125—92no—
159#125—92no—
160C22⋊Q8—94no—
161C22⋊Q8—94no—
162C22⋊Q8—92no—
163C22⋊Q8—92no—
164C4×Q8—31no—
165C4×Q8—31no—
166C4×Q8—31no—
167C4×Q8—32no—
168C4×Q8—92no—
169C4×Q8—92no—
170C4×Q8—91no—
171C4×Q8—91no—
172C4×Q8—92no—
173C4×Q8C2×C3×D312yes—
174C4×Q8C2×C3×D311yescharacteristic
175C4×D4—91no—
176C4×D4—92no—
177C4×D4—92no—
178C4×D4—91no—
179#117—91no—
180#117—92no—
181C2×C4⋊C4—92no—
182C2×C4⋊C4—91no—
183C2×C42—92no—
184C2×C42—91no—
185C22×D3—31no—
186C3×Q8—31no—
187C3×Q8—62no—
188C3×Q8—124no—
189C3×Q8C2×C3×Q811yescharacteristic
190C2×C3×C4—31no—
191C2×C3×C4—32no—
192C2×C3×C4—31no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
195C2×C3×C4—81no—
196C2×C3×C4—82no—
197C2×C3×C4—122no—
198C2×C3×C4—121no—
199C2×C3×C4—121no—
200C2×C3×C4C22×A412yes—
201C2×C3×C4C22×A411yescharacteristic
202C2×Dic3—31no—
203C2×Dic3—32no—
204C2×Dic3C22×A411yescharacteristic
205C2×Dic3C22×A411yescharacteristic
206C2×Dic3C22×A412yes—
207D12—32no—
208D12—32no—
209C4×D3—32no—
210C4×D3—32no—
211C4×D3—62no—
212C4×D3—64no—
213C4×D3—121no—
214C4×D3—122no—
215Dic6—64no—
216Dic6—122no—
217Dic6#41⋊C314yescomplemented
218SL(2,𝔽3)—21no—
219SL(2,𝔽3)D3×Q811yescharacteristic, complemented
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223#41—32no—
224#41—92no—
225C2×Q8—31no—
226C2×Q8—31no—
227C2×Q8—92no—
228C2×Q8—92no—
229C2×Q8—184no—
230C2×Q8C22×C3×D311yescharacteristic
231C2×D4—91no—
232C22×C4—91no—
233C22×C4—92no—
234C22×C4—91no—
235C22×C4—91no—
236C22×C4—92no—
237C4⋊C4—31no—
238C4⋊C4—32no—
239C4⋊C4—32no—
240C4⋊C4—32no—
241C4⋊C4—32no—
242C4⋊C4—92no—
243C4⋊C4—92no—
244C4⋊C4—91no—
245C4⋊C4—92no—
246C4⋊C4—94no—
247C4⋊C4—92no—
248C4⋊C4—91no—
249C4⋊C4D3×A411yescharacteristic
250C22⋊C4—92no—
251C22⋊C4—94no—
252C42—32no—
253C42—31no—
254C42—32no—
255C42—31no—
256C42—91no—
257C42—92no—
258C42—91no—
259C42—92no—
260C22×C3—41no—
261C22×C3—81no—
262C22×C3C23×A411yescharacteristic
263C2×D3—32no—
264C2×D3—61no—
265C2×D3—61no—
266C3×C4—22no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—42no—
270C3×C4—62no—
271C3×C4—61no—
272C3×C4—82no—
273C3×C4—82no—
274C3×C4—124no—
275C3×C4—122no—
276C3×C4—162no—
277C3×C4—241no—
278C3×C4C2×#41⋊C312yes—
279Dic3—21no—
280Dic3—61no—
281Dic3—62no—
282Dic3Q8×A412yes—
283Dic3C2×#41⋊C314yes—
284C32—41noSylow
285C23—91no—
286Q8—31no—
287Q8—34no—
288Q8—62no—
289Q8—184no—
290Q8—362no—
291Q8C3×D3×Q811yescharacteristic
292D4—92no—
293D4—92no—
294C2×C4—31no—
295C2×C4—31no—
296C2×C4—32no—
297C2×C4—31no—
298C2×C4—32no—
299C2×C4—31no—
300C2×C4—92no—
301C2×C4—92no—
302C2×C4—91no—
303C2×C4—92no—
304C2×C4—184no—
305C2×C4—182no—
306C2×C4—362no—
307C2×C4—361no—
308C2×C4C2×D3×A412yes—
309C2×C4C2×D3×A411yescharacteristic
310C2×C3—41no—
311C2×C3—41no—
312C2×C3—41no—
313C2×C3—81no—
314C2×C3—81no—
315C2×C3—81no—
316C2×C3C2×Q8×A411yescharacteristic
317C2×C3C22×#41⋊C311yescharacteristic
318C2×C3#594511yescharacteristic
319D3—62no—
320C22—92no—
321C22—181no—
322C22—181no—
323C22C22×D3×A411yescharacteristic
324C4—22no—
325C4—31no—
326C4—31no—
327C4—32no—
328C4—34no—
329C4—62no—
330C4—61no—
331C4—61no—
332C4—181no—
333C4—182no—
334C4D3×#41⋊C312yes—
335C3—41no—
336C3—81no—
337C3Q8×#41⋊C311yescharacteristic, complemented
338C2—182no—
339C2D3×Q8×A411yescharacteristic
340C2C2×D3×#41⋊C311yescharacteristic
341C2#5968⋊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. #5569⋊C3C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #87015C3semidirect
28. #557⋊C3D3semidirect
33. C3⋊(#41⋊C3)Q8semidirect
87. C3×SL(2,𝔽3)C2×Q8semidirect
103. #239C3×Q8semidirect
217. Dic6#41⋊C3semidirect
219. SL(2,𝔽3)D3×Q8semidirect
337. C3Q8×#41⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#5569⋊C32
C4×C3⋊(#41⋊C3)2
SL(2,𝔽3)×Dic62
#870153
Q8×#41⋊C33
C3×#8874