Finite Group Databasebeta

← #85844⋊C3 #11923203

Subgroups — #85844⋊C3 #11923203

Order 1152.

611 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 611 conjugacy classes of subgroups — too many to draw. Only its 91 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×C4×#41⋊C3C211yesmaximal, characteristic, complemented
2#5831⋊C3C212yesmaximal, complemented
3SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
4C4×D3×SL(2,𝔽3)C212yesmaximal, complemented
5#5570⋊C3C211yesmaximal, characteristic, complemented
6#85844C311yesmaximal, characteristic, complemented
7#3482⋊C3—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9C2×D3×SL(2,𝔽3)C2212yes—
10C3×C4×D12—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C3×C4×#41C2×C311yescharacteristic, complemented
15#5831C2×C312yescomplemented
16Q8×D12C2×C311yescharacteristic
17C4×D3×Q8C2×C312yescomplemented
18#5570C2×C311yescharacteristic, complemented
19#5557—32no—
20#5551—31no—
21#5542—31no—
22#5529—32no—
23C4×#235—31no—
24#5470—31no—
25#557⋊C3—31no—
26D4×SL(2,𝔽3)—31no—
27#548⋊C3—32no—
28C4×#41⋊C3—21no—
29C4×#41⋊C3D311yescharacteristic, complemented
30C2×C4×SL(2,𝔽3)—32no—
31C2×C3×D12—41no—
32C2×C3×C4×D3—42no—
33C3×#41⋊C3—21no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35D3×SL(2,𝔽3)—22no—
36C32×C42—41no—
37C3×(C2×D3)⋊C4—42no—
38C3×C4⋊Dic3—41no—
39#3482—31noSylow
40C2×C3×#41C22×C311yescharacteristic
41C2×D3×Q8C22×C312yes—
42C2×#235—31no—
43C2×#41⋊C3—21no—
44C2×#41⋊C3C2×D311yescharacteristic
45C22×SL(2,𝔽3)—32no—
46C3×C4×Q8C22×C311yescharacteristic
47C3×C4×Q8C22×C311yescharacteristic
48C3×C4×D4—31no—
49C3×C4×D4—31no—
50C3×C4×D4—121no—
51C3×#117—31no—
52C2×C3×C42—31no—
53Q8×Dic3C22×C312yes—
54#938—32no—
55#936—32no—
56C4×#67—32no—
57C4×#67—32no—
58#926—32no—
59#926—31no—
60#924—32no—
61#924—32no—
62#897—32no—
63D3×C4⋊C4—32no—
64#890—32no—
65#887—32no—
66#883—34no—
67#879—34no—
68C4×D12—31no—
69C4×D12A411yescharacteristic
70#872—32no—
71C42×D3—32no—
72#870—32no—
73#869—31no—
74#869—31no—
75C4×Dic6—31no—
76C4×Dic6—31no—
77C4×Dic6—32no—
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
83C22×C3×D3—42no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4—41no—
87C2×C3×Dic3—42no—
88C3×D12—44no—
89C3×C4×D3—84no—
90C3×SL(2,𝔽3)#4111yescharacteristic, complemented
91#557—31no—
92#556—91no—
93#552—92no—
94D4×Q8—31no—
95#548—32no—
96#547—91no—
97#533—91no—
98#529—92no—
99C4×#41—91no—
100C4×#41C3×D311yescharacteristic, complemented
101C2×C4×Q8—32no—
102C3×#41—21no—
103C3×#41—61no—
104C2×C3×Q8C23×C311yescharacteristic
105C2×C3×D4—31no—
106C2×C3×D4—121no—
107C22×C3×C4—31no—
108C22×C3×C4—31no—
109C22×C3×C4—31no—
110C22×C3×C4—122no—
111C2×#67—32no—
112D3×Q8—22no—
113D3×Q8—62no—
114#235—64no—
115C2×D12C2×A411yescharacteristic
116C2×C4×D3—32no—
117C2×C4×D3—32no—
118C2×C4×D3C2×A412yes—
119C2×Dic6—31no—
120C2×Dic6—32no—
121#41⋊C3—22no—
122#41⋊C3—21no—
123C2×SL(2,𝔽3)—21no—
124C2×SL(2,𝔽3)—62no—
125C2×SL(2,𝔽3)C22×D311yescharacteristic
126C3×C4⋊C4—31no—
127C3×C4⋊C4—31no—
128C3×C4⋊C4—121no—
129C3×C22⋊C4—32no—
130C3×C22⋊C4—122no—
131C3×C42—31no—
132C3×C42—31no—
133C3×C42—31no—
134C3×C42—41no—
135C3×C42—81no—
136C3×C42C2×A411yescharacteristic
137#217—34no—
138#217—32no—
139(C2×D3)⋊C4—32no—
140(C2×D3)⋊C4C2×A412yes—
141C4⋊Dic3—32no—
142C4⋊Dic3—32no—
143C4⋊Dic3—31no—
144C4⋊Dic3—32no—
145C4⋊Dic3C2×A411yescharacteristic
146Dic3⋊C4—34no—
147Dic3⋊C4—32no—
148Dic3⋊C4—32no—
149C4×Dic3—32no—
150C4×Dic3—32no—
151C4×Dic3—32no—
152C22×C32—41no—
153C2×C3×D3—44no—
154C2×C3×D3—82no—
155C2×C3×D3—82no—
156C32×C4—42no—
157C32×C4—42no—
158C32×C4—81no—
159C3×Dic3—82no—
160C2×#41—91no—
161C2×#41C2×C3×D311yescharacteristic
162C22×Q8—32no—
163C4⋊Q8—91no—
164C4⋊Q8—91no—
165#126—94no—
166#125—92no—
167#125—92no—
168#124—92no—
169#123—94no—
170C22⋊Q8—92no—
171C22⋊Q8—92no—
172C22⋊Q8—92no—
173#121—92no—
174C4×Q8—32no—
175C4×Q8—91no—
176C4×Q8—92no—
177C4×Q8—91no—
178C4×Q8—92no—
179C4×Q8C2×C3×D311yescharacteristic
180C4×Q8C2×C3×D311yescharacteristic
181C4×D4—31no—
182C4×D4—31no—
183C4×D4—31no—
184C4×D4—91no—
185C4×D4—92no—
186C4×D4—92no—
187#117—31no—
188#117—92no—
189#117—91no—
190#117—92no—
191C2×C4⋊C4—92no—
192C2×C42—31no—
193C2×C42—92no—
194C23×C3—31no—
195C23×C3—122no—
196C22×D3C22×A412yes—
197C3×Q8—31no—
198C3×Q8C3×#4111yescharacteristic
199C3×D4—61no—
200C3×D4—61no—
201C3×D4—124no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4—81no—
215C2×C3×C4—122no—
216C2×C3×C4—244no—
217C2×C3×C4C22×A411yescharacteristic
218C2×C3×C4C22×A411yescharacteristic
219C2×C3×C4C22×A411yescharacteristic
220#67—64no—
221C2×Dic3—32no—
222C2×Dic3—32no—
223C2×Dic3C22×A412yes—
224D12#41⋊C314yescomplemented
225C4×D3—24no—
226C4×D3—62no—
227C4×D3—62no—
228C4×D3—64no—
229Dic6—62no—
230Dic6—62no—
231Dic6—62no—
232SL(2,𝔽3)—21no—
233SL(2,𝔽3)#23511yescharacteristic, complemented
234C2×C32—41no—
235C2×C32—41no—
236C2×C32—41no—
237C3×D3—84no—
238#41—21no—
239#41—61no—
240#41—184no—
241C2×Q8—62no—
242C2×Q8—92no—
243C2×Q8—91no—
244C2×Q8—182no—
245C2×Q8C22×C3×D311yescharacteristic
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—92no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—32no—
253C22×C4—92no—
254C22×C4—92no—
255C4⋊C4—31no—
256C4⋊C4—31no—
257C4⋊C4—31no—
258C4⋊C4—92no—
259C4⋊C4—92no—
260C4⋊C4—94no—
261C4⋊C4—92no—
262C4⋊C4—92no—
263C4⋊C4—91no—
264C4⋊C4—92no—
265C22⋊C4—32no—
266C22⋊C4—32no—
267C22⋊C4—92no—
268C22⋊C4—92no—
269C22⋊C4—94no—
270C42—31no—
271C42—31no—
272C42—31no—
273C42—92no—
274C42—92no—
275C42—92no—
276C42D3×A411yescharacteristic
277C22×C3—41no—
278C22×C3—61no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3—81no—
282C22×C3—124no—
283C22×C3—242no—
284C22×C3—242no—
285C22×C3C23×A411yescharacteristic
286C2×D3—22no—
287C2×D3—22no—
288C2×D3C2×#41⋊C314yes—
289C3×C4—21no—
290C3×C4—31no—
291C3×C4—31no—
292C3×C4—42no—
293C3×C4—42no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—81no—
297C3×C4—82no—
298C3×C4—82no—
299C3×C4—82no—
300C3×C4—242no—
301C3×C4C2×#41⋊C312yes—
302C3×C42−1+4⋊C312yes—
303Dic3—22no—
304Dic3—62no—
305Dic3—62no—
306C32—41noSylow
307C23—32no—
308C23—31no—
309Q8—31no—
310Q8—182no—
311Q8—182no—
312Q8—182no—
313Q8C3×#23511yescharacteristic
314D4—34no—
315D4—61no—
316D4—61no—
317D4—184no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—32no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—62no—
325C2×C4—64no—
326C2×C4—62no—
327C2×C4—92no—
328C2×C4—92no—
329C2×C4—182no—
330C2×C4—182no—
331C2×C4—184no—
332C2×C4C2×D3×A411yescharacteristic
333C2×C4C2×D3×A411yescharacteristic
334C2×C4C2×D3×A411yescharacteristic
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3—81no—
341C2×C3—121no—
342C2×C3—244no—
343C2×C3C22×#41⋊C311yescharacteristic
344C2×C3A4×#4111yescharacteristic
345C2×C3C2×2−1+4⋊C311yescharacteristic
346D3—24no—
347C22—34no—
348C22—62no—
349C22—62no—
350C22—61no—
351C22—61no—
352C22—61no—
353C22C22×D3×A411yescharacteristic
354C4—21no—
355C4—31no—
356C4—31no—
357C4—61no—
358C4—62no—
359C4—61no—
360C4—182no—
361C4—182no—
362C4D3×#41⋊C312yes—
363C4#1010⋊C312yes—
364C3—41no—
365C3—81no—
366C3#3482⋊C311yescharacteristic, complemented
367C2—64no—
368C2—121no—
369C2A4×#23511yescharacteristic
370C2C2×D3×#41⋊C311yescharacteristic
371C2C2×#1010⋊C311yescharacteristic
subgroupG/H
centerC2×C4C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
hypercenterC42D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×#41⋊C3C2semidirect
2. #5831⋊C3C2semidirect
3. SL(2,𝔽3)×D12C2semidirect
4. C4×D3×SL(2,𝔽3)C2semidirect
5. #5570⋊C3C2semidirect
6. #85844C3semidirect
14. C3×C4×#41C2×C3semidirect
15. #5831C2×C3semidirect
17. C4×D3×Q8C2×C3semidirect
18. #5570C2×C3semidirect
29. C4×#41⋊C3D3semidirect
90. C3×SL(2,𝔽3)#41semidirect
100. C4×#41C3×D3semidirect
224. D12#41⋊C3semidirect
233. SL(2,𝔽3)#235semidirect
366. C3#3482⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
#5831⋊C32
SL(2,𝔽3)×D122
C4×D3×SL(2,𝔽3)2
#5570⋊C32
#858443
#3482⋊C33
C3×C4×D124