Finite Group Databasebeta

← #85802⋊C3 #11923180

Subgroups — #85802⋊C3 #11923180

Order 1152.

599 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C2×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5817⋊C3C212yesmaximal, complemented
3#5570⋊C3C211yesmaximal, characteristic, complemented
4C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
5C4×C3⋊(#41⋊C3)C212yesmaximal, complemented
6#85802C311yesmaximal, characteristic, complemented
7#3480⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊(#41⋊C3)C2212yes—
10C3×#926—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C2×C3×C4×Q8C2×C311yescharacteristic
15#5817C2×C312yes—
16#5585—31no—
17#5584—31no—
18#5580—32no—
19#5577C2×C311yescharacteristic, complemented
20C4×#239C2×C312yescomplemented
21#5570C2×C311yescharacteristic, complemented
22#5514—32no—
23#5477—31no—
24C4×#235—31no—
25#557⋊C3—31no—
26C4⋊(#41⋊C3)—31no—
27#548⋊C3—32no—
28C4×#41⋊C3—32no—
29C2×C4×SL(2,𝔽3)—21no—
30C2×C4×SL(2,𝔽3)D311yescharacteristic, complemented
31C22×C32×C4—41no—
32C2×C3×SL(2,𝔽3)—21no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34C3⋊(#41⋊C3)—22no—
35C3×#217—42no—
36C3×C4⋊Dic3—41no—
37C3×C4⋊Dic3—41no—
38C3×C4×Dic3—42no—
39#3480—31noSylow
40C22×C3×Q8C22×C311yescharacteristic
41C2×#239C22×C312yes—
42C2×#235—31no—
43C2×#41⋊C3—32no—
44C22×SL(2,𝔽3)—21no—
45C22×SL(2,𝔽3)C2×D311yescharacteristic
46C3×C4×Q8—31no—
47C3×C4×Q8—31no—
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×Q8C22×C311yescharacteristic
50C3×#117—121no—
51C2×C3×C4⋊C4—31no—
52C2×C3×C42—31no—
53#947—32no—
54#946—32no—
55Q8×Dic3C22×C312yes—
56#930—32no—
57#930—32no—
58C4×#67—32no—
59C4×#67—32no—
60#926A411yescharacteristic
61#898—34no—
62#895—32no—
63#894—34no—
64#893—32no—
65#892—32no—
66#890—32no—
67#875—32no—
68#874—31no—
69C4×D12—32no—
70C4×D12—32no—
71C4×D12—31no—
72C4×D12—31no—
73#872—32no—
74C42×D3—32no—
75#869—31no—
76C4×Dic6—31no—
77C4×Dic6—31no—
78C4×SL(2,𝔽3)—21no—
79C4×SL(2,𝔽3)—21no—
80C4×SL(2,𝔽3)—32no—
81C4×SL(2,𝔽3)C2×D311yescharacteristic
82C4×SL(2,𝔽3)C2×D311yescharacteristic
83C23×C32—41no—
84C2×C32×C4—44no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C3×Dic3—42no—
88C2×C3×Dic3—42no—
89C3×SL(2,𝔽3)#4111yescharacteristic, complemented
90#559—91no—
91#557—31no—
92#556—91no—
93#553—92no—
94#550—31no—
95#548—32no—
96#532—91no—
97#529—92no—
98C4×#41—32no—
99C4×#41—91no—
100C2×C4×Q8C3×D311yescharacteristic
101C2×C3×Q8—21no—
102C2×C3×Q8—31no—
103C2×C3×Q8—61no—
104C2×C3×Q8C23×C311yescharacteristic
105C22×C3×C4—31no—
106C22×C3×C4—31no—
107C22×C3×C4—41no—
108C22×C3×C4—81no—
109C22×C3×C4C2×A411yescharacteristic
110C2×#67—32no—
111#239—22no—
112#239—62no—
113#235—64no—
114C2×D12—32no—
115C2×D12—31no—
116C2×D12—32no—
117C2×C4×D3—32no—
118C2×C4×D3—32no—
119C2×C4×D3—32no—
120C2×Dic6—31no—
121#41⋊C3—62no—
122C2×SL(2,𝔽3)—22no—
123C2×SL(2,𝔽3)—21no—
124C2×SL(2,𝔽3)—21no—
125C2×SL(2,𝔽3)C22×D311yescharacteristic
126C3×C4⋊C4—31no—
127C3×C4⋊C4—31no—
128C3×C4⋊C4—32no—
129C3×C4⋊C4—121no—
130C3×C4⋊C4—121no—
131C3×C22⋊C4—122no—
132C3×C42—31no—
133C3×C42—31no—
134C3×C42—31no—
135C3×C42—31no—
136C3×C42—122no—
137#217C2×A412yes—
138(C2×D3)⋊C4—34no—
139(C2×D3)⋊C4—34no—
140(C2×D3)⋊C4—32no—
141(C2×D3)⋊C4—32no—
142C4⋊Dic3—32no—
143C4⋊Dic3—32no—
144C4⋊Dic3C2×A411yescharacteristic
145C4⋊Dic3C2×A411yescharacteristic
146Dic3⋊C4—32no—
147Dic3⋊C4—32no—
148C4×Dic3—32no—
149C4×Dic3—32no—
150C4×Dic3C2×A412yes—
151C22×C32—42no—
152C22×C32—41no—
153C22×C32—81no—
154C22×C32—81no—
155C32×C4—42no—
156C32×C4—42no—
157C3×Dic3—82no—
158C3×Dic3—82no—
159C2×#41—32no—
160C2×#41—91no—
161C22×Q8C2×C3×D311yescharacteristic
162C4⋊Q8—91no—
163#127—91no—
164#126—94no—
165#125—92no—
166#124—92no—
167#124—92no—
168#123—94no—
169C22⋊Q8—92no—
170#121—92no—
171#121—92no—
172#121—92no—
173C4×Q8—31no—
174C4×Q8—31no—
175C4×Q8—32no—
176C4×Q8—91no—
177C4×Q8—91no—
178C4×Q8C2×C3×D311yescharacteristic
179C4×Q8C2×C3×D311yescharacteristic
180C4×D4—91no—
181C4×D4—91no—
182C4×D4—92no—
183C4×D4—92no—
184C4×D4—92no—
185C4×D4—92no—
186C4×D4—92no—
187#117—31no—
188#117—92no—
189#117—92no—
190C2×C4⋊C4—31no—
191C2×C42—31no—
192C2×C42—92no—
193C23×C3—41no—
194C23×C3—81no—
195C23×C3C22×A411yescharacteristic
196C22×D3—32no—
197C3×Q8—31no—
198C3×Q8—61no—
199C3×Q8—61no—
200C3×Q8C3×#4111yescharacteristic
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—44no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—81no—
211C2×C3×C4—81no—
212C2×C3×C4—84no—
213C2×C3×C4—122no—
214C2×C3×C4—122no—
215C2×C3×C4—121no—
216C2×C3×C4#41⋊C314yescomplemented
217C2×C3×C4C22×A411yescharacteristic
218C2×C3×C4C22×A411yescharacteristic
219#67—64no—
220C2×Dic3—32no—
221C2×Dic3C22×A412yes—
222C2×Dic3C22×A412yes—
223D12—62no—
224D12—62no—
225D12—62no—
226D12—122no—
227C4×D3—62no—
228C4×D3—62no—
229C4×D3—64no—
230C4×D3—122no—
231Dic6—62no—
232SL(2,𝔽3)—21no—
233SL(2,𝔽3)#23511yescharacteristic, complemented
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—41no—
237C2×C32—82no—
238#41—62no—
239#41—184no—
240#41—182no—
241C2×Q8—21no—
242C2×Q8—31no—
243C2×Q8—61no—
244C2×Q8—91no—
245C2×Q8C22×C3×D311yescharacteristic
246C2×D4—92no—
247C2×D4—92no—
248C2×D4—92no—
249C2×D4—91no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—92no—
253C22×C4—92no—
254C22×C4—92no—
255C22×C4D3×A411yescharacteristic
256C4⋊C4—31no—
257C4⋊C4—31no—
258C4⋊C4—32no—
259C4⋊C4—31no—
260C4⋊C4—31no—
261C4⋊C4—92no—
262C4⋊C4—92no—
263C4⋊C4—92no—
264C4⋊C4—92no—
265C22⋊C4—32no—
266C22⋊C4—92no—
267C22⋊C4—92no—
268C22⋊C4—94no—
269C22⋊C4—94no—
270C42—31no—
271C42—31no—
272C42—31no—
273C42—31no—
274C42—32no—
275C42—92no—
276C42—92no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—42no—
280C22×C3—41no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—82no—
284C22×C3—82no—
285C22×C3—82no—
286C22×C3—81no—
287C22×C3C2×#41⋊C312yes—
288C22×C3C23×A411yescharacteristic
289C2×D3—62no—
290C2×D3—62no—
291C2×D3—62no—
292C3×C4—31no—
293C3×C4—31no—
294C3×C4—42no—
295C3×C4—42no—
296C3×C4—61no—
297C3×C4—61no—
298C3×C4—61no—
299C3×C4—82no—
300C3×C4—82no—
301C3×C4—242no—
302C3×C4—242no—
303C3×C4C2×#41⋊C312yes—
304C3×C4C2×#41⋊C312yes—
305Dic3—22no—
306Dic3—22no—
307Dic3—62no—
308C32—41noSylow
309C23—92no—
310C23C2×D3×A411yescharacteristic
311Q8—31no—
312Q8—61no—
313Q8—61no—
314Q8—182no—
315Q8C3×#23511yescharacteristic
316D4—182no—
317D4—182no—
318D4—184no—
319D4—182no—
320D4—362no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—31no—
325C2×C4—32no—
326C2×C4—32no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—92no—
330C2×C4—121no—
331C2×C4—182no—
332C2×C4—182no—
333C2×C4—184no—
334C2×C4—362no—
335C2×C4C3⋊(#41⋊C3)14yescomplemented
336C2×C4C2×D3×A411yescharacteristic
337C2×C4C2×D3×A411yescharacteristic
338C2×C3—22no—
339C2×C3—41no—
340C2×C3—41no—
341C2×C3—41no—
342C2×C3—82no—
343C2×C3—84no—
344C2×C3—81no—
345C2×C3—81no—
346C2×C3—81no—
347C2×C3C22×#41⋊C311yescharacteristic
348C2×C3A4×#4111yescharacteristic
349C2×C3#594511yescharacteristic
350D3—122no—
351C22—21no—
352C22—21no—
353C22—182no—
354C22—182no—
355C22—182no—
356C22C2×C3⋊(#41⋊C3)12yes—
357C22C22×D3×A411yescharacteristic
358C4—31no—
359C4—31no—
360C4—61no—
361C4—62no—
362C4—61no—
363C4—61no—
364C4—62no—
365C4—182no—
366C4C2×C3⋊(#41⋊C3)12yes—
367C4C2×C3⋊(#41⋊C3)12yes—
368C3—41no—
369C3—81no—
370C3#3480⋊C311yescharacteristic, complemented
371C2—22no—
372C2—362no—
373C2A4×#23511yescharacteristic
374C2C22×C3⋊(#41⋊C3)11yescharacteristic
375C2#5964⋊C311yescharacteristic
subgroupG/H
centerC2×C4C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC2×C3×C4×Q8C2×C3
socleC22×C3C23×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×SL(2,𝔽3)C2semidirect
2. #5817⋊C3C2semidirect
3. #5570⋊C3C2semidirect
4. C4⋊(C3⋊(#41⋊C3))C2semidirect
5. C4×C3⋊(#41⋊C3)C2semidirect
6. #85802C3semidirect
19. #5577C2×C3semidirect
20. C4×#239C2×C3semidirect
21. #5570C2×C3semidirect
30. C2×C4×SL(2,𝔽3)D3semidirect
89. C3×SL(2,𝔽3)#41semidirect
216. C2×C3×C4#41⋊C3semidirect
233. SL(2,𝔽3)#235semidirect
335. C2×C4C3⋊(#41⋊C3)semidirect
370. C3#3480⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
#5817⋊C32
#5570⋊C32
C4⋊(C3⋊(#41⋊C3))2
C4×C3⋊(#41⋊C3)2
#858023
#3480⋊C33
C3×#9264