Finite Group Databasebeta

← #87260⋊C3 #11923380

Subgroups — #87260⋊C3 #11923380

Order 1152.

773 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 773 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic
2#5827⋊C3C212yesmaximal, complemented
3Dic3×#41⋊C3C212yesmaximal, complemented
4#5812⋊C3C212yesmaximal, complemented
5#87260C311yesmaximal, characteristic, complemented
6#3428⋊C3—31nomaximal
7C2×C3×#41⋊C3C414yescomplemented
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×C3×#41⋊C3C2211yescharacteristic
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C3×#926—41nomaximal
12Dic3×SL(2,𝔽3)C2214yescomplemented
13C22×C3×#41C2×C311yescharacteristic
14C22×#41⋊C3—21no—
15C22×#41⋊C3D311yescharacteristic
16#5827—32no—
17#5827C2×C312yescomplemented
18Dic3×#41—32no—
19Dic3×#41C2×C312yescomplemented
20#5812C2×C312yes—
21#5798—32no—
22C2×#926—31no—
23C2×#926—31no—
24#520⋊C3—32no—
25#519⋊C3—32no—
26C4×#41⋊C3—32no—
27C22×C32×C4—41no—
28C3×#41⋊C3C2×C412yescomplemented
29C3×#41⋊C3C2×C412yescomplemented
30C2×C3×SL(2,𝔽3)C2×C412yescomplemented
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3×#217—42no—
33C3×C4⋊Dic3—42no—
34C3×C4×Dic3—42no—
35#3428—31noSylow
36C2×C3×#41—34no—
37C2×C3×#41—31no—
38C2×C3×#41—31no—
39C2×C3×#41C3×C414yescomplemented
40C2×C3×#41C22×C311yescharacteristic
41C2×C3×#41C22×C311yescharacteristic
42C22×C3×Q8C22×C311yescharacteristic
43C22×C3×D4—31no—
44C23×C3×C4—31no—
45C2×#41⋊C3—24no—
46C2×#41⋊C3—21no—
47C2×#41⋊C3—21no—
48C2×#41⋊C3Dic314yescomplemented
49C2×#41⋊C3C2×D311yescharacteristic
50C2×#41⋊C3C2×D311yescharacteristic
51C22×SL(2,𝔽3)—21no—
52C22×SL(2,𝔽3)C2×D311yescharacteristic
53C3×#117—121no—
54C2×#217—32no—
55C2×#217—32no—
56Q8×Dic3—34no—
57Q8×Dic3C22×C314yes—
58D4×Dic3—38no—
59D4×Dic3—34no—
60D4×Dic3—34no—
61#926—32no—
62#926—31no—
63#926—32no—
64#926—32no—
65#926—32no—
66#926—32no—
67#926—32no—
68#926A411yescharacteristic
69C2×C4⋊Dic3—32no—
70C2×C4⋊Dic3—32no—
71C2×C4×Dic3—32no—
72C2×C4×Dic3—32no—
73C4×SL(2,𝔽3)—34no—
74C23×C32—41no—
75C2×C32×C4—44no—
76C2×C32×C4—41no—
77C2×C32×C4—41no—
78C2×C3×Dic3—44no—
79C3×SL(2,𝔽3)C22×C411yescharacteristic
80C22×#41C3×D311yescharacteristic
81#520—32no—
82#520—92no—
83#519—32no—
84#518—92no—
85C4×#41—32no—
86C4×#41—92no—
87C2×#117—91no—
88C2×#117—91no—
89C24×C3—31no—
90C3×#41—34no—
91C3×#41—34no—
92C3×#41—34no—
93C3×#41—34no—
94C3×#41—32no—
95C3×#41—32no—
96C3×#41C2×C3×C412yes—
97C3×#41C2×C3×C412yes—
98C2×C3×Q8—31no—
99C2×C3×Q8—32no—
100C2×C3×Q8C2×C3×C412yes—
101C2×C3×Q8C23×C311yescharacteristic
102C2×C3×D4—31no—
103C2×C3×D4—31no—
104C2×C3×D4—34no—
105C2×C3×D4—32no—
106C2×C3×D4—34no—
107C22×C3×C4—31no—
108C22×C3×C4—34no—
109C22×C3×C4—34no—
110C22×C3×C4—31no—
111C22×C3×C4—31no—
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—41no—
115C22×C3×C4—81no—
116C22×C3×C4C2×A411yescharacteristic
117C22×Dic3—34no—
118C22×Dic3—34no—
119#41⋊C3—22no—
120#41⋊C3—22no—
121#41⋊C3C2×Dic312yescomplemented
122#41⋊C3C2×Dic312yescomplemented
123C2×SL(2,𝔽3)—22no—
124C2×SL(2,𝔽3)—21no—
125C2×SL(2,𝔽3)C2×Dic312yescomplemented
126C2×SL(2,𝔽3)C22×D311yescharacteristic
127C3×C4⋊C4—122no—
128C3×C22⋊C4—122no—
129C3×C42—122no—
130#217—34no—
131#217—32no—
132#217—34no—
133#217C2×A412yes—
134C4⋊Dic3—32no—
135C4⋊Dic3—34no—
136C4⋊Dic3—34no—
137C4⋊Dic3C2×A412yes—
138C4×Dic3—34no—
139C4×Dic3—32no—
140C4×Dic3—34no—
141C4×Dic3C2×A412yes—
142C22×C32—42no—
143C22×C32—41no—
144C22×C32—82no—
145C32×C4—42no—
146C32×C4—42no—
147C3×Dic3—84no—
148C2×#41—31no—
149C2×#41—31no—
150C2×#41—34no—
151C2×#41C3×Dic314yescomplemented
152C2×#41C2×C3×D311yescharacteristic
153C2×#41C2×C3×D311yescharacteristic
154C22×Q8C2×C3×D311yescharacteristic
155C22×D4—31no—
156C23×C4—31no—
157C4×Q8—34no—
158C4×Q8—94no—
159C4×D4—98no—
160C4×D4—94no—
161C4×D4—94no—
162#117—31no—
163#117—92no—
164#117—92no—
165#117—92no—
166#117—92no—
167#117—91no—
168#117—92no—
169#117—92no—
170C2×C4⋊C4—92no—
171C2×C4⋊C4—92no—
172C2×C22⋊C4—92no—
173C2×C22⋊C4—92no—
174C2×C42—92no—
175C2×C42—92no—
176C23×C3—31no—
177C23×C3—31no—
178C23×C3—34no—
179C23×C3—41no—
180C23×C3—81no—
181C23×C3—122no—
182C23×C3C22×A411yescharacteristic
183C3×Q8—32no—
184C3×Q8—31no—
185C3×Q8—32no—
186C3×Q8C22×C3×C411yescharacteristic
187C3×D4—32no—
188C3×D4—32no—
189C3×D4—32no—
190C3×D4—32no—
191C3×D4—38no—
192C2×C3×C4—32no—
193C2×C3×C4—32no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—34no—
199C2×C3×C4—34no—
200C2×C3×C4—32no—
201C2×C3×C4—32no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—44no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4—84no—
208C2×C3×C4—124no—
209C2×C3×C4C4×A414yes—
210C2×C3×C4C22×A411yescharacteristic
211C2×C3×C4C22×A411yescharacteristic
212C2×Dic3—34no—
213C2×Dic3—124no—
214C2×Dic3—124no—
215C2×Dic3C22×A414yes—
216SL(2,𝔽3)—21no—
217SL(2,𝔽3)C22×Dic311yescharacteristic
218C2×C32—41no—
219C2×C32—42no—
220C2×C32—82no—
221C24—31no—
222#41—34no—
223#41—34no—
224#41—34no—
225#41—34no—
226#41—32no—
227#41—32no—
228#41C2×C3×Dic312yes—
229#41C2×C3×Dic312yes—
230C2×Q8—31no—
231C2×Q8—32no—
232C2×Q8C2×C3×Dic312yes—
233C2×Q8C22×C3×D311yescharacteristic
234C2×D4—31no—
235C2×D4—31no—
236C2×D4—34no—
237C2×D4—32no—
238C2×D4—34no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—34no—
245C22×C4—34no—
246C22×C4—94no—
247C22×C4—94no—
248C22×C4D3×A411yescharacteristic
249C4⋊C4—32no—
250C4⋊C4—94no—
251C4⋊C4—94no—
252C4⋊C4—92no—
253C22⋊C4—32no—
254C22⋊C4—94no—
255C22⋊C4—94no—
256C22⋊C4—92no—
257C42—32no—
258C42—94no—
259C42—94no—
260C42—92no—
261C22×C3—22no—
262C22×C3—32no—
263C22×C3—32no—
264C22×C3—42no—
265C22×C3—41no—
266C22×C3—62no—
267C22×C3—62no—
268C22×C3—82no—
269C22×C3—84no—
270C22×C3—82no—
271C22×C3—81no—
272C22×C3—124no—
273C22×C3C2×C4×A412yes—
274C22×C3C23×A411yescharacteristic
275C3×C4—31no—
276C3×C4—32no—
277C3×C4—31no—
278C3×C4—42no—
279C3×C4—42no—
280C3×C4—82no—
281C3×C4—82no—
282C3×C4—244no—
283C3×C4C2×C4×A412yes—
284C3×C4C2×C4×A412yes—
285Dic3—24no—
286Dic3—64no—
287C32—41noSylow
288C23—31no—
289C23—31no—
290C23—34no—
291C23—122no—
292C23C2×D3×A411yescharacteristic
293Q8—32no—
294Q8—31no—
295Q8—32no—
296Q8C22×C3×Dic311yescharacteristic
297D4—32no—
298D4—32no—
299D4—32no—
300D4—32no—
301D4—38no—
302C2×C4—32no—
303C2×C4—32no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—32no—
307C2×C4—32no—
308C2×C4—32no—
309C2×C4—34no—
310C2×C4—32no—
311C2×C4—34no—
312C2×C4—34no—
313C2×C4—94no—
314C2×C4—364no—
315C2×C4—364no—
316C2×C4Dic3×A414yes—
317C2×C4C2×D3×A411yescharacteristic
318C2×C4C2×D3×A411yescharacteristic
319C2×C3—22no—
320C2×C3—42no—
321C2×C3—41no—
322C2×C3—62no—
323C2×C3—62no—
324C2×C3—82no—
325C2×C3—84no—
326C2×C3—82no—
327C2×C3—81no—
328C2×C3C22×C4×A411yescharacteristic
329C2×C3#594512yes—
330C22—22no—
331C22—32no—
332C22—32no—
333C22—62no—
334C22—62no—
335C22—124no—
336C22C2×Dic3×A412yes—
337C22C22×D3×A411yescharacteristic
338C4—31no—
339C4—32no—
340C4—31no—
341C4—64no—
342C4—184no—
343C4C2×Dic3×A412yes—
344C4C2×Dic3×A412yes—
345C3—41no—
346C3—81no—
347C3#3428⋊C311yescharacteristic, complemented
348C2—22no—
349C2—62no—
350C2—62no—
351C2#5964⋊C312yes—
352C2C22×Dic3×A411yescharacteristic
subgroupG/H
centerC2×C4C2×D3×A4
commutator, hypocenterC3×Q8C22×C3×C4
FrattiniC22C22×D3×A4
FittingC22×C3×#41C2×C3
socleC22×C3C23×A4
prime radicalC22×C3×#41⋊C3C2
cosocleC2×C3×Q8C23×C3
hypercenterC22×C4D3×A4
LCSG ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. #5827⋊C3C2semidirect
3. Dic3×#41⋊C3C2semidirect
4. #5812⋊C3C2semidirect
5. #87260C3semidirect
7. C2×C3×#41⋊C3C4semidirect
12. Dic3×SL(2,𝔽3)C22semidirect
17. #5827C2×C3semidirect
19. Dic3×#41C2×C3semidirect
28. C3×#41⋊C3C2×C4semidirect
29. C3×#41⋊C3C2×C4semidirect
30. C2×C3×SL(2,𝔽3)C2×C4semidirect
39. C2×C3×#41C3×C4semidirect
48. C2×#41⋊C3Dic3semidirect
121. #41⋊C3C2×Dic3semidirect
122. #41⋊C3C2×Dic3semidirect
125. C2×SL(2,𝔽3)C2×Dic3semidirect
151. C2×#41C3×Dic3semidirect
347. C3#3428⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
#5827⋊C32
Dic3×#41⋊C32
#5812⋊C32
#872603
#3428⋊C33
C3×#9264