Finite Group Databasebeta

← #86773⋊C3 #11923268

Subgroups — #86773⋊C3 #11923268

Order 1152.

555 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 79 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#5827⋊C3C212yesmaximal, complemented
3#5834⋊C3C212yesmaximal, complemented
4#5570⋊C3C211yesmaximal, characteristic, complemented
5C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
6#86773C311yesmaximal, characteristic, complemented
7#3517⋊C3—31nomaximal
8C2×C3×#41⋊C3C2212yes—
9C2×C3⋊(#41⋊C3)C2211yescharacteristic
10C3×#890—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12Dic3×SL(2,𝔽3)C2211yescharacteristic
13Dic3×SL(2,𝔽3)C2212yes—
14C3×#550C2×C311yescharacteristic
15#5834C2×C312yes—
16#5827C2×C312yes—
17#5718—31no—
18#5716—31no—
19#5619—32no—
20#5618—32no—
21C4×#239C2×C311yescharacteristic
22#5570C2×C311yescharacteristic
23#5557—31no—
24#5555—31no—
25#557⋊C3—31no—
26C4⋊(#41⋊C3)—21no—
27C4⋊(#41⋊C3)—32no—
28C4⋊(#41⋊C3)D311yescharacteristic, complemented
29#520⋊C3—32no—
30C4×#41⋊C3—31no—
31C3×#41⋊C3—22no—
32C2×C3×SL(2,𝔽3)C2311yescharacteristic
33C3⋊(#41⋊C3)—21no—
34C32×C4⋊C4—41no—
35C3×C4⋊Dic3—41no—
36C3×C4⋊Dic3—42no—
37C3×Dic3⋊C4—42no—
38C3×C4×Dic3—41no—
39#3517—31noSylow
40C2×C3×#41C22×C312yes—
41C2×#239C22×C311yescharacteristic
42C2×#237—32no—
43C2×#41⋊C3—22no—
44C2×#41⋊C3—31no—
45C2×#41⋊C3C2×D312yes—
46C3×#127—31no—
47C3×#125—121no—
48C3×#121—32no—
49C3×C4×Q8C22×C311yescharacteristic
50C3×C4×D4—31no—
51Q8×Dic3C22×C312yes—
52Q8×Dic3C22×C311yescharacteristic
53#940—32no—
54#939—34no—
55#938—32no—
56#938—32no—
57#938—31no—
58#936—31no—
59D4×Dic3—32no—
60D4×Dic3—32no—
61D4×Dic3—32no—
62#930—32no—
63C4×#67—32no—
64#926—32no—
65C2×C4⋊Dic3—32no—
66#897—31no—
67#893—31no—
68#892—31no—
69#890—32no—
70#890A411yescharacteristic
71#886—34no—
72#883—32no—
73#879—32no—
74C4×D12—31no—
75C42×D3—31no—
76#869—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
82C2×C32×C4—41no—
83C2×C32×C4—42no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—42no—
86C2×C3×Dic3—41no—
87C3×SL(2,𝔽3)#4111yescharacteristic, complemented
88#557—31no—
89#553—92no—
90#550—32no—
91#550—91no—
92#550C3×D311yescharacteristic
93#547—91no—
94#540—92no—
95#532—91no—
96#529—91no—
97#520—32no—
98C4×#41—31no—
99C3×#41—22no—
100C3×#41—62no—
101C2×C3×Q8C23×C311yescharacteristic
102C2×C3×D4—32no—
103C2×C3×D4—32no—
104C2×C3×D4—31no—
105C22×C3×C4—32no—
106C2×#67—34no—
107C2×#67—32no—
108C22×Dic3—34no—
109#239—21no—
110#239—61no—
111#237—124no—
112C2×D12—31no—
113C2×C4×D3—31no—
114C2×C4×D3—31no—
115C2×C4×D3—31no—
116C2×Dic6—31no—
117#41⋊C3—22no—
118#41⋊C3—42no—
119#41⋊C3—61no—
120C2×SL(2,𝔽3)—21no—
121C2×SL(2,𝔽3)C22×D311yescharacteristic
122C3×C4⋊C4—31no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4—81no—
125C3×C4⋊C4—122no—
126C3×C4⋊C4—122no—
127C3×C4⋊C4—121no—
128C3×C4⋊C4C2×A411yescharacteristic
129C3×C22⋊C4—32no—
130C3×C42—31no—
131C3×C42—121no—
132#217—32no—
133#217—32no—
134#217—32no—
135(C2×D3)⋊C4—32no—
136(C2×D3)⋊C4—32no—
137C4⋊Dic3—32no—
138C4⋊Dic3—32no—
139C4⋊Dic3—32no—
140C4⋊Dic3—31no—
141C4⋊Dic3C2×A411yescharacteristic
142C4⋊Dic3C2×A412yes—
143Dic3⋊C4—32no—
144Dic3⋊C4C2×A412yes—
145C4×Dic3—32no—
146C4×Dic3—31no—
147C4×Dic3—31no—
148C4×Dic3C2×A411yescharacteristic
149C22×C32—41no—
150C32×C4—42no—
151C32×C4—82no—
152C3×Dic3—81no—
153C3×Dic3—82no—
154C3×Dic3—81no—
155C2×#41—31no—
156C2×#41—92no—
157C2×#41C2×C3×D312yes—
158C4⋊Q8—91no—
159#127—31no—
160#127—92no—
161#126—92no—
162#125—31no—
163#125—92no—
164#124—91no—
165#123—92no—
166#123—94no—
167C22⋊Q8—91no—
168#121—32no—
169#121—91no—
170#121—92no—
171#121—94no—
172#121—92no—
173#121—92no—
174C4×Q8—31no—
175C4×Q8—32no—
176C4×Q8—91no—
177C4×Q8C2×C3×D311yescharacteristic
178C4×D4—31no—
179C4×D4—91no—
180C4×D4—92no—
181C4×D4—91no—
182C4×D4—92no—
183C4×D4—92no—
184C4×D4—92no—
185#117—92no—
186#117—91no—
187C2×C4⋊C4—92no—
188C2×C42—91no—
189C23×C3—32no—
190C22×D3—31no—
191C3×Q8—31no—
192C3×Q8C3×#4111yescharacteristic
193C3×D4—62no—
194C3×D4—62no—
195C3×D4—62no—
196C3×D4—122no—
197C2×C3×C4—31no—
198C2×C3×C4—31no—
199C2×C3×C4—41no—
200C2×C3×C4—42no—
201C2×C3×C4—62no—
202C2×C3×C4—62no—
203C2×C3×C4—81no—
204C2×C3×C4—82no—
205C2×C3×C4—121no—
206C2×C3×C4—122no—
207C2×C3×C4—121no—
208C2×C3×C4C22×A412yes—
209C2×C3×C4C22×A411yescharacteristic
210#67—122no—
211#67—124no—
212C2×Dic3—31no—
213C2×Dic3—32no—
214C2×Dic3—124no—
215C2×Dic3C22×A411yescharacteristic
216C2×Dic3C22×A411yescharacteristic
217C2×Dic3C22×A412yes—
218D12—61no—
219D12—61no—
220C4×D3—61no—
221C4×D3—61no—
222C4×D3—62no—
223C4×D3—121no—
224Dic6—62no—
225SL(2,𝔽3)—21no—
226SL(2,𝔽3)#23711yescharacteristic, complemented
227C2×C32—41no—
228C2×C32—41no—
229C2×C32—41no—
230#41—22no—
231#41—62no—
232#41—61no—
233#41—181no—
234#41—364no—
235C2×Q8—91no—
236C2×Q8C22×C3×D311yescharacteristic
237C2×D4—32no—
238C2×D4—32no—
239C2×D4—31no—
240C2×D4—91no—
241C2×D4—92no—
242C2×D4—94no—
243C22×C4—32no—
244C22×C4—91no—
245C22×C4—91no—
246C22×C4—94no—
247C22×C4—91no—
248C4⋊C4—31no—
249C4⋊C4—32no—
250C4⋊C4—32no—
251C4⋊C4—31no—
252C4⋊C4—92no—
253C4⋊C4—91no—
254C4⋊C4—92no—
255C4⋊C4—92no—
256C4⋊C4—92no—
257C4⋊C4D3×A411yescharacteristic
258C22⋊C4—32no—
259C22⋊C4—92no—
260C22⋊C4—92no—
261C22⋊C4—92no—
262C22⋊C4—92no—
263C22⋊C4—92no—
264C42—31no—
265C42—31no—
266C42—91no—
267C42—91no—
268C42—92no—
269C22×C3—41no—
270C22×C3—62no—
271C22×C3—62no—
272C22×C3—62no—
273C22×C3—81no—
274C22×C3C23×A411yescharacteristic
275C2×D3—61no—
276C2×D3—61no—
277C2×D3—61no—
278C3×C4—22no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—42no—
282C3×C4—61no—
283C3×C4—82no—
284C3×C4—82no—
285C3×C4—162no—
286C3×C4—241no—
287C3×C4—242no—
288C3×C4—241no—
289C3×C4#99412yes—
290Dic3—21no—
291Dic3—21no—
292Dic3—22no—
293Dic3—62no—
294Dic3—61no—
295C32—41noSylow
296C23—32no—
297C23—91no—
298Q8—31no—
299Q8—182no—
300Q8C3×#23711yescharacteristic
301D4—62no—
302D4—62no—
303D4—62no—
304D4—122no—
305D4—181no—
306D4—181no—
307D4—362no—
308D4—364no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—32no—
314C2×C4—62no—
315C2×C4—62no—
316C2×C4—91no—
317C2×C4—92no—
318C2×C4—181no—
319C2×C4—181no—
320C2×C4—182no—
321C2×C4—361no—
322C2×C4—364no—
323C2×C4C2×D3×A412yes—
324C2×C4C2×D3×A411yescharacteristic
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—41no—
328C2×C3—81no—
329C2×C3—81no—
330C2×C3—81no—
331C2×C3—122no—
332C2×C3A4×#4111yescharacteristic
333C2×C3C2×#99411yescharacteristic
334C2×C3#594511yescharacteristic
335D3—121no—
336C22—62no—
337C22—62no—
338C22—62no—
339C22—181no—
340C22—181no—
341C22—181no—
342C22C22×D3×A411yescharacteristic
343C4—22no—
344C4—31no—
345C4—31no—
346C4—61no—
347C4—61no—
348C4—61no—
349C4—62no—
350C4—181no—
351C4—182no—
352C4C3⋊#99412yes—
353C3—41no—
354C3—81no—
355C3#3517⋊C311yescharacteristic, complemented
356C2—122no—
357C2—361no—
358C2A4×#23711yescharacteristic
359C2C2×C3⋊#99411yescharacteristic
360C2#1059662311yescharacteristic
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. #5827⋊C3C2semidirect
3. #5834⋊C3C2semidirect
4. #5570⋊C3C2semidirect
5. C4×C3⋊(#41⋊C3)C2semidirect
6. #86773C3semidirect
28. C4⋊(#41⋊C3)D3semidirect
87. C3×SL(2,𝔽3)#41semidirect
226. SL(2,𝔽3)#237semidirect
355. C3#3517⋊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
#5827⋊C32
#5834⋊C32
#5570⋊C32
C4×C3⋊(#41⋊C3)2
#867733
#3517⋊C33
C3×#8904