Finite Group Databasebeta

← #86776⋊C3 #11923274

Subgroups — #86776⋊C3 #11923274

Order 1152.

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

normal subgroups only

This group has 579 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×C4⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
2#5831⋊C3C212yesmaximal, complemented
3#5827⋊C3C212yesmaximal, complemented
4C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
6#86776C311yesmaximal, characteristic, complemented
7#3522⋊C3—31nomaximal
8C2×C3×#41⋊C3C2212yes—
9C2×D3×SL(2,𝔽3)C2211yescharacteristic
10C3×#897—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×#550C2×C311yescharacteristic, complemented
15#5831C2×C312yes—
16#5827C2×C312yescomplemented
17#5716—31no—
18#5713—31no—
19#5611—32no—
20#5603—32no—
21C4×D3×Q8C2×C311yescharacteristic
22Q8×Dic6C2×C311yescharacteristic, complemented
23#5557—31no—
24#5543—31no—
25Q8×SL(2,𝔽3)—31no—
26C4⋊(#41⋊C3)—21no—
27C4⋊(#41⋊C3)D311yescharacteristic, complemented
28#548⋊C3—32no—
29#520⋊C3—32no—
30C2×C4×SL(2,𝔽3)—31no—
31C2×C3×C4×D3—41no—
32C2×C3×Dic6—41no—
33C3×#41⋊C3—22no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35D3×SL(2,𝔽3)—21no—
36C32×C4⋊C4—41no—
37C3×(C2×D3)⋊C4—42no—
38C3×C4⋊Dic3—42no—
39#3522—31noSylow
40C2×C3×#41C22×C312yes—
41C2×D3×Q8C22×C311yescharacteristic
42C2×#237—32no—
43C2×#41⋊C3—22no—
44C2×#41⋊C3C2×D312yes—
45C22×SL(2,𝔽3)—31no—
46C3×#127—31no—
47C3×C22⋊Q8—121no—
48C3×#121—32no—
49C3×C4×Q8C22×C311yescharacteristic
50C3×C4×D4—31no—
51Q8×Dic3C22×C312yes—
52Q8×Dic3C22×C311yescharacteristic
53#938—32no—
54#938—31no—
55#936—32no—
56#936—31no—
57#936—32no—
58#935—34no—
59D4×Dic3—32no—
60D4×Dic3—32no—
61D4×Dic3—32no—
62C4×#67—32no—
63#926—32no—
64C2×C4⋊Dic3—32no—
65#924—32no—
66#897A411yescharacteristic
67D3×C4⋊C4—31no—
68#890—31no—
69#888—32no—
70#887—31no—
71#883—32no—
72#879—32no—
73#878—34no—
74C42×D3—31no—
75#869—31no—
76C4×Dic6—31no—
77C4×Dic6—31no—
78C4×SL(2,𝔽3)—21no—
79C4×SL(2,𝔽3)—31no—
80C4×SL(2,𝔽3)—32no—
81C4×SL(2,𝔽3)C2×D311yescharacteristic
82C22×C3×D3—41no—
83C2×C32×C4—41no—
84C2×C32×C4—42no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—42no—
87C3×C4×D3—82no—
88C3×Dic6—82no—
89C3×SL(2,𝔽3)#4111yescharacteristic, complemented
90#559—91no—
91Q82—31no—
92#554—91no—
93#550C3×D311yescharacteristic, complemented
94#548—32no—
95#547—92no—
96#542—92no—
97#532—91no—
98#529—91no—
99#520—32no—
100C2×C4×Q8—31no—
101C3×#41—22no—
102C3×#41—62no—
103C2×C3×Q8—121no—
104C2×C3×Q8C23×C311yescharacteristic
105C2×C3×D4—32no—
106C2×C3×D4—32no—
107C2×C3×D4—31no—
108C22×C3×C4—32no—
109C22×C3×C4—121no—
110C2×#67—32no—
111C22×Dic3—34no—
112D3×Q8—21no—
113D3×Q8—61no—
114#237—124no—
115C2×C4×D3—31no—
116C2×C4×D3—31no—
117C2×C4×D3C2×A411yescharacteristic
118C2×Dic6—31no—
119C2×Dic6—32no—
120C2×Dic6C2×A411yescharacteristic
121#41⋊C3—22no—
122#41⋊C3—42no—
123C2×SL(2,𝔽3)—21no—
124C2×SL(2,𝔽3)—61no—
125C2×SL(2,𝔽3)C22×D311yescharacteristic
126C3×C4⋊C4—31no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—81no—
129C3×C4⋊C4—122no—
130C3×C4⋊C4C2×A411yescharacteristic
131C3×C22⋊C4—32no—
132C3×C22⋊C4—122no—
133C3×C42—31no—
134#217—34no—
135#217—32no—
136#217—32no—
137#217—32no—
138(C2×D3)⋊C4C2×A412yes—
139C4⋊Dic3—31no—
140C4⋊Dic3—32no—
141C4⋊Dic3—31no—
142C4⋊Dic3—32no—
143C4⋊Dic3C2×A412yes—
144Dic3⋊C4—32no—
145Dic3⋊C4—32no—
146Dic3⋊C4—32no—
147C4×Dic3—31no—
148C4×Dic3—32no—
149C4×Dic3—31no—
150C4×Dic3—31no—
151C22×C32—41no—
152C2×C3×D3—42no—
153C2×C3×D3—81no—
154C2×C3×D3—81no—
155C32×C4—42no—
156C32×C4—82no—
157C3×Dic3—82no—
158C3×Dic3—81no—
159C2×#41—92no—
160C2×#41C2×C3×D312yes—
161C22×Q8—31no—
162C4⋊Q8—91no—
163C4⋊Q8—92no—
164#127—31no—
165#126—92no—
166#125—91no—
167#124—91no—
168#124—92no—
169#124—92no—
170#123—92no—
171#123—94no—
172C22⋊Q8—31no—
173C22⋊Q8—94no—
174C22⋊Q8—92no—
175#121—32no—
176#121—91no—
177#121—92no—
178C4×Q8—31no—
179C4×Q8—32no—
180C4×Q8—91no—
181C4×Q8—91no—
182C4×Q8—91no—
183C4×Q8C2×C3×D311yescharacteristic
184C4×D4—31no—
185C4×D4—92no—
186C4×D4—92no—
187C4×D4—92no—
188C4×D4—92no—
189#117—92no—
190C2×C4⋊C4—91no—
191C2×C4⋊C4—92no—
192C2×C42—91no—
193C23×C3—32no—
194C23×C3—121no—
195C22×D3C22×A411yescharacteristic
196C3×Q8—31no—
197C3×Q8—242no—
198C3×Q8C3×#4111yescharacteristic
199C3×D4—62no—
200C3×D4—62no—
201C3×D4—62no—
202C3×D4—122no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—62no—
208C2×C3×C4—62no—
209C2×C3×C4—81no—
210C2×C3×C4—82no—
211C2×C3×C4—122no—
212C2×C3×C4—121no—
213C2×C3×C4—242no—
214C2×C3×C4C22×A412yes—
215C2×C3×C4C22×A411yescharacteristic
216#67—64no—
217C2×Dic3—31no—
218C2×Dic3—31no—
219C2×Dic3—32no—
220C2×Dic3—124no—
221C2×Dic3C22×A411yescharacteristic
222C2×Dic3C22×A412yes—
223C4×D3—22no—
224C4×D3—61no—
225C4×D3—61no—
226C4×D3—62no—
227Dic6—22no—
228Dic6—61no—
229Dic6—61no—
230Dic6—122no—
231SL(2,𝔽3)—21no—
232SL(2,𝔽3)#23711yescharacteristic, complemented
233C2×C32—41no—
234C2×C32—41no—
235C2×C32—41no—
236C3×D3—82no—
237#41—22no—
238#41—62no—
239#41—364no—
240C2×Q8—31no—
241C2×Q8—61no—
242C2×Q8—92no—
243C2×Q8—91no—
244C2×Q8—181no—
245C2×Q8C22×C3×D311yescharacteristic
246C2×D4—32no—
247C2×D4—32no—
248C2×D4—31no—
249C2×D4—92no—
250C22×C4—31no—
251C22×C4—32no—
252C22×C4—91no—
253C22×C4—91no—
254C22×C4—94no—
255C4⋊C4—31no—
256C4⋊C4—32no—
257C4⋊C4—92no—
258C4⋊C4—91no—
259C4⋊C4—92no—
260C4⋊C4—92no—
261C4⋊C4—92no—
262C4⋊C4—91no—
263C4⋊C4—92no—
264C4⋊C4D3×A411yescharacteristic
265C22⋊C4—32no—
266C22⋊C4—32no—
267C22⋊C4—92no—
268C22⋊C4—92no—
269C22⋊C4—92no—
270C22⋊C4—94no—
271C42—31no—
272C42—91no—
273C42—91no—
274C42—92no—
275C42—91no—
276C22×C3—41no—
277C22×C3—62no—
278C22×C3—62no—
279C22×C3—62no—
280C22×C3—81no—
281C22×C3—122no—
282C22×C3—241no—
283C22×C3—241no—
284C22×C3C23×A411yescharacteristic
285C2×D3—21no—
286C2×D3—21no—
287C2×D3#99412yes—
288C3×C4—22no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—42no—
292C3×C4—61no—
293C3×C4—82no—
294C3×C4—82no—
295C3×C4—162no—
296C3×C4—241no—
297C3×C4—242no—
298C3×C42−1+4⋊C312yes—
299Dic3—22no—
300Dic3—21no—
301Dic3—61no—
302Dic3—62no—
303Dic3—61no—
304C32—41noSylow
305C23—31no—
306C23—32no—
307Q8—31no—
308Q8—62no—
309Q8—181no—
310Q8—181no—
311Q8—362no—
312Q8C3×#23711yescharacteristic
313D4—62no—
314D4—62no—
315D4—62no—
316D4—122no—
317D4—184no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—32no—
322C2×C4—62no—
323C2×C4—62no—
324C2×C4—62no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—92no—
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—122no—
341C2×C3—242no—
342C2×C3A4×#4111yescharacteristic
343C2×C3C2×#99411yescharacteristic
344C2×C3C2×2−1+4⋊C311yescharacteristic
345D3—22no—
346C22—32no—
347C22—61no—
348C22—61no—
349C22—62no—
350C22—62no—
351C22—62no—
352C22C22×D3×A411yescharacteristic
353C4—22no—
354C4—31no—
355C4—31no—
356C4—61no—
357C4—62no—
358C4—61no—
359C4—181no—
360C4—181no—
361C4—182no—
362C4#1010⋊C312yes—
363C3—41no—
364C3—81no—
365C3#3522⋊C311yescharacteristic, complemented
366C2—62no—
367C2—122no—
368C2A4×#23711yescharacteristic
369C2C2×#1010⋊C311yescharacteristic
370C2D3×#99411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#550C2×C3
socleC22×C3C23×A4
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C3×#41
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×C4⋊(#41⋊C3)C2semidirect
2. #5831⋊C3C2semidirect
3. #5827⋊C3C2semidirect
4. C4×D3×SL(2,𝔽3)C2semidirect
5. SL(2,𝔽3)×Dic6C2semidirect
6. #86776C3semidirect
14. C3×#550C2×C3semidirect
16. #5827C2×C3semidirect
22. Q8×Dic6C2×C3semidirect
27. C4⋊(#41⋊C3)D3semidirect
89. C3×SL(2,𝔽3)#41semidirect
93. #550C3×D3semidirect
232. SL(2,𝔽3)#237semidirect
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×C4⋊(#41⋊C3)2
#5831⋊C32
#5827⋊C32
C4×D3×SL(2,𝔽3)2
SL(2,𝔽3)×Dic62
#867763
#3522⋊C33
C3×#8974