Finite Group Databasebeta

← #87043⋊C3 #11923294

Subgroups — #87043⋊C3 #11923294

Order 1152.

583 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 583 conjugacy classes of subgroups — too many to draw. Only its 81 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×Q8×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2#5576⋊C3C212yesmaximal, complemented
3C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
5#5574⋊C3C212yesmaximal, complemented
6#87043C311yesmaximal, characteristic, complemented
7#3522⋊C3—31nomaximal
8C2×D3×SL(2,𝔽3)C2211yescharacteristic
9C2×C3⋊(#41⋊C3)C2212yes—
10C3×#946—41nomaximal
11C3×C4×SL(2,𝔽3)C2212yes—
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×Q82C2×C311yescharacteristic, complemented
15#5733—32no—
16#5730—31no—
17#5726—32no—
18#5725—31no—
19#5585—31no—
20#5580—31no—
21#5577C2×C311yescharacteristic, complemented
22#5576C2×C312yes—
23#5574C2×C312yescomplemented
24C4×D3×Q8C2×C311yescharacteristic
25Q8×SL(2,𝔽3)—21no—
26Q8×SL(2,𝔽3)D311yescharacteristic, complemented
27C4⋊(#41⋊C3)—31no—
28#548⋊C3—32no—
29#520⋊C3—32no—
30C2×C4×SL(2,𝔽3)—31no—
31C2×C32×Q8—41no—
32C2×C3×C4×D3—41no—
33C2×C3×SL(2,𝔽3)C2311yescharacteristic
34D3×SL(2,𝔽3)—21no—
35C3⋊(#41⋊C3)—22no—
36C3×(C2×D3)⋊C4—42no—
37C3×C4⋊Dic3—41no—
38C3×Dic3⋊C4—42no—
39#3522—31noSylow
40C2×#239—32no—
41C2×#239C22×C312yes—
42C2×D3×Q8C22×C311yescharacteristic
43C2×#41⋊C3—32no—
44C22×SL(2,𝔽3)—31no—
45C3×C4⋊Q8—31no—
46C3×C4⋊Q8—32no—
47C3×C22⋊Q8—121no—
48C3×C4×Q8—31no—
49C3×C4×Q8C22×C311yescharacteristic
50C3×C4×Q8C22×C312yes—
51#947—32no—
52#947—31no—
53#946—34no—
54#946A411yescharacteristic
55Q8×Dic3C22×C311yescharacteristic
56#898—32no—
57#897—32no—
58#895—32no—
59#895—31no—
60#895—32no—
61#894—34no—
62#894—32no—
63#893—32no—
64#893—32no—
65D3×C4⋊C4—31no—
66D3×C4⋊C4—32no—
67#890—31no—
68#887—31no—
69#875—32no—
70#874—31no—
71C4×D12—32no—
72C4×D12—32no—
73C4×D12—31no—
74#872—32no—
75C42×D3—31no—
76C4×Dic6—31no—
77C4×SL(2,𝔽3)—21no—
78C4×SL(2,𝔽3)—22no—
79C4×SL(2,𝔽3)—31no—
80C4×SL(2,𝔽3)C2×D311yescharacteristic
81C4×SL(2,𝔽3)C2×D312yes—
82C22×C3×D3—41no—
83C32×Q8—82no—
84C2×C32×C4—41no—
85C2×C32×C4—42no—
86C2×C3×Dic3—41no—
87C2×C3×Dic3—42no—
88C3×C4×D3—82no—
89C3×SL(2,𝔽3)#4111yescharacteristic, complemented
90#559—91no—
91Q82C3×D311yescharacteristic, complemented
92#554—91no—
93#550—31no—
94#548—32no—
95#547—92no—
96#542—92no—
97#532—91no—
98#529—91no—
99#520—32no—
100C2×C4×Q8—31no—
101C2×C3×Q8—32no—
102C2×C3×Q8—41no—
103C2×C3×Q8—81no—
104C2×C3×Q8C2×A411yescharacteristic
105C2×C3×Q8C23×C311yescharacteristic
106C22×C3×C4—121no—
107#239—22no—
108#239—62no—
109#239—124no—
110D3×Q8—21no—
111D3×Q8—61no—
112C2×D12—32no—
113C2×D12—32no—
114C2×D12—31no—
115C2×D12—32no—
116C2×C4×D3—31no—
117C2×C4×D3—31no—
118C2×C4×D3—32no—
119C2×C4×D3—34no—
120C2×C4×D3C2×A411yescharacteristic
121C2×Dic6—31no—
122#41⋊C3—62no—
123C2×SL(2,𝔽3)—21no—
124C2×SL(2,𝔽3)—61no—
125C2×SL(2,𝔽3)C22×D311yescharacteristic
126C3×C4⋊C4—31no—
127C3×C4⋊C4—32no—
128C3×C4⋊C4—31no—
129C3×C4⋊C4—32no—
130C3×C4⋊C4—122no—
131C3×C4⋊C4—121no—
132C3×C22⋊C4—122no—
133C3×C42—31no—
134C3×C42—32no—
135(C2×D3)⋊C4—34no—
136(C2×D3)⋊C4—32no—
137(C2×D3)⋊C4—32no—
138(C2×D3)⋊C4—32no—
139(C2×D3)⋊C4—32no—
140(C2×D3)⋊C4C2×A412yes—
141C4⋊Dic3—32no—
142C4⋊Dic3—31no—
143C4⋊Dic3C2×A411yescharacteristic
144Dic3⋊C4—32no—
145Dic3⋊C4—32no—
146Dic3⋊C4C2×A412yes—
147C4×Dic3—31no—
148C4×Dic3—31no—
149C4×Dic3—31no—
150C22×C32—41no—
151C2×C3×D3—42no—
152C2×C3×D3—81no—
153C2×C3×D3—81no—
154C32×C4—42no—
155C32×C4—82no—
156C3×Dic3—81no—
157C3×Dic3—82no—
158C2×#41—32no—
159C2×#41—92no—
160C22×Q8—31no—
161C4⋊Q8—31no—
162C4⋊Q8—32no—
163#127—91no—
164#126—92no—
165#125—91no—
166#124—92no—
167#124—92no—
168#124—91no—
169#123—92no—
170#123—94no—
171C22⋊Q8—31no—
172C22⋊Q8—92no—
173C22⋊Q8—94no—
174#121—91no—
175#121—92no—
176#121—92no—
177C4×Q8—31no—
178C4×Q8—31no—
179C4×Q8—91no—
180C4×Q8—91no—
181C4×Q8C2×C3×D311yescharacteristic
182C4×Q8C2×C3×D312yes—
183C4×D4—92no—
184C4×D4—91no—
185C4×D4—92no—
186C4×D4—92no—
187C4×D4—92no—
188#117—92no—
189C2×C4⋊C4—91no—
190C2×C4⋊C4—92no—
191C2×C42—91no—
192C23×C3—121no—
193C22×D3—32no—
194C22×D3C22×A411yescharacteristic
195C3×Q8—22no—
196C3×Q8—31no—
197C3×Q8—82no—
198C3×Q8—84no—
199C3×Q8—122no—
200C3×Q8C3×#4111yescharacteristic
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—32no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—82no—
207C2×C3×C4—81no—
208C2×C3×C4—121no—
209C2×C3×C4—122no—
210C2×C3×C4—242no—
211C2×C3×C4C22×A412yes—
212C2×C3×C4C22×A411yescharacteristic
213C2×Dic3—31no—
214C2×Dic3—31no—
215C2×Dic3C22×A411yescharacteristic
216C2×Dic3C22×A412yes—
217D12—62no—
218D12—62no—
219D12—64no—
220D12—62no—
221D12—122no—
222C4×D3—22no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—61no—
226C4×D3—61no—
227C4×D3—62no—
228C4×D3—124no—
229Dic6—61no—
230Dic6—61no—
231SL(2,𝔽3)—21no—
232SL(2,𝔽3)#23911yescharacteristic, complemented
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C3×D3—82no—
237#41—62no—
238#41—182no—
239#41—364no—
240C2×Q8—32no—
241C2×Q8—61no—
242C2×Q8—91no—
243C2×Q8—181no—
244C2×Q8D3×A411yescharacteristic
245C2×Q8C22×C3×D311yescharacteristic
246C2×D4—92no—
247C2×D4—92no—
248C2×D4—92no—
249C2×D4—91no—
250C22×C4—31no—
251C22×C4—92no—
252C22×C4—91no—
253C22×C4—91no—
254C22×C4—94no—
255C4⋊C4—32no—
256C4⋊C4—31no—
257C4⋊C4—32no—
258C4⋊C4—31no—
259C4⋊C4—31no—
260C4⋊C4—32no—
261C4⋊C4—92no—
262C4⋊C4—92no—
263C4⋊C4—91no—
264C4⋊C4—92no—
265C22⋊C4—32no—
266C22⋊C4—92no—
267C22⋊C4—92no—
268C22⋊C4—94no—
269C22⋊C4—92no—
270C22⋊C4—92no—
271C42—31no—
272C42—32no—
273C42—91no—
274C42—91no—
275C42—91no—
276C22×C3—41no—
277C22×C3—81no—
278C22×C3—122no—
279C22×C3—241no—
280C22×C3—241no—
281C22×C3C23×A411yescharacteristic
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—62no—
285C2×D3—62no—
286C2×D3—62no—
287C2×D3#99412yes—
288C3×C4—22no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—42no—
292C3×C4—61no—
293C3×C4—62no—
294C3×C4—82no—
295C3×C4—84no—
296C3×C4—82no—
297C3×C4—241no—
298C3×C4—242no—
299C3×C42−1+4⋊C312yes—
300Dic3—22no—
301Dic3—21no—
302Dic3—61no—
303Dic3—61no—
304C32—41noSylow
305C23—31no—
306C23—92no—
307Q8—22no—
308Q8—31no—
309Q8—122no—
310Q8—181no—
311Q8—181no—
312Q8C3×#23911yescharacteristic
313D4—182no—
314D4—182no—
315D4—184no—
316D4—182no—
317D4—362no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—32no—
321C2×C4—31no—
322C2×C4—32no—
323C2×C4—62no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—182no—
327C2×C4—182no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—182no—
331C2×C4—364no—
332C2×C4C2×D3×A412yes—
333C2×C4C2×D3×A411yescharacteristic
334C2×C3—41no—
335C2×C3—41no—
336C2×C3—41no—
337C2×C3—81no—
338C2×C3—81no—
339C2×C3—81no—
340C2×C3—242no—
341C2×C3A4×#4111yescharacteristic
342C2×C3C2×#99411yescharacteristic
343C2×C3C2×2−1+4⋊C311yescharacteristic
344D3—22no—
345D3—122no—
346C22—32no—
347C22—61no—
348C22—61no—
349C22—182no—
350C22—182no—
351C22—182no—
352C22C22×D3×A411yescharacteristic
353C4—22no—
354C4—31no—
355C4—31no—
356C4—61no—
357C4—62no—
358C4—61no—
359C4—62no—
360C4—181no—
361C4—181no—
362C4#1007⋊C312yes—
363C3—41no—
364C3—81no—
365C3#3522⋊C311yescharacteristic, complemented
366C2—62no—
367C2—362no—
368C2C2×#1007⋊C311yescharacteristic
369C2A4×#23911yescharacteristic
370C2D3×#99411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×Q82C2×C3
socleC22×C3C23×A4
hypercenterC2×Q8D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×SL(2,𝔽3)C2semidirect
2. #5576⋊C3C2semidirect
3. C4×D3×SL(2,𝔽3)C2semidirect
4. C4⋊(C3⋊(#41⋊C3))C2semidirect
5. #5574⋊C3C2semidirect
6. #87043C3semidirect
14. C3×Q82C2×C3semidirect
21. #5577C2×C3semidirect
23. #5574C2×C3semidirect
26. Q8×SL(2,𝔽3)D3semidirect
89. C3×SL(2,𝔽3)#41semidirect
91. Q82C3×D3semidirect
232. SL(2,𝔽3)#239semidirect
365. C3#3522⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×SL(2,𝔽3)2
#5576⋊C32
C4×D3×SL(2,𝔽3)2
C4⋊(C3⋊(#41⋊C3))2
#5574⋊C32
#870433
#3522⋊C33
C3×#9464