Finite Group Databasebeta

← #87023⋊C3 #11923280

Subgroups — #87023⋊C3 #11923280

Order 1152.

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

normal subgroups only

This group has 553 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×#557⋊C3C211yesmaximal, characteristic, complemented
2C4⋊(C3⋊(#41⋊C3))C212yesmaximal, complemented
3C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
4#5574⋊C3C212yesmaximal, complemented
5C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
6#87023C311yesmaximal, characteristic, complemented
7#3517⋊C3—31nomaximal
8C2×C3⋊(#41⋊C3)C2211yescharacteristic
9C2×C3⋊(#41⋊C3)C2212yes—
10C3×#890—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×#557C2×C311yescharacteristic
15#5726—31no—
16#5725—31no—
17#5697—32no—
18#5695—32no—
19#5580—31no—
20#5577—31no—
21#5577C2×C312yes—
22#5577C2×C311yescharacteristic
23#5574C2×C312yes—
24C4×#239C2×C311yescharacteristic
25#557⋊C3—21no—
26#557⋊C3D311yescharacteristic, complemented
27C4⋊(#41⋊C3)—31no—
28C4⋊(#41⋊C3)—32no—
29#520⋊C3—32no—
30C4×#41⋊C3—31no—
31C2×C3×SL(2,𝔽3)C2311yescharacteristic
32C3⋊(#41⋊C3)—21no—
33C3⋊(#41⋊C3)—22no—
34C32×C4⋊C4—41no—
35C3×C4⋊Dic3—42no—
36C3×C4⋊Dic3—41no—
37C3×Dic3⋊C4—42no—
38C3×C4×Dic3—41no—
39#3517—31noSylow
40C2×#239—32no—
41C2×#239C22×C311yescharacteristic
42C2×#239C22×C312yes—
43C2×#41⋊C3—31no—
44C2×#41⋊C3—32no—
45C3×C4⋊Q8—31no—
46C3×#125—32no—
47C3×#125—121no—
48C3×C4×Q8—31no—
49C3×C4×Q8C22×C311yescharacteristic
50C3×C4×Q8C22×C312yes—
51#947—31no—
52#946—31no—
53Q8×Dic3C22×C311yescharacteristic
54#898—32no—
55#895—34no—
56#895—32no—
57#895—31no—
58#895—32no—
59#895—32no—
60#895—32no—
61#894—34no—
62#894—32no—
63#893—32no—
64#893—31no—
65#893—32no—
66#892—31no—
67D3×C4⋊C4—32no—
68#890A411yescharacteristic
69#874—32no—
70#874—31no—
71C4×D12—31no—
72C4×D12—32no—
73C4×D12—32no—
74C4×D12—31no—
75#872—32no—
76C42×D3—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—
82C2×C32×C4—41no—
83C2×C32×C4—42no—
84C2×C3×Dic3—41no—
85C2×C3×Dic3—41no—
86C2×C3×Dic3—42no—
87C3×SL(2,𝔽3)#4111yescharacteristic, complemented
88#557C3×D311yescharacteristic
89#553—92no—
90#550—31no—
91#550—32no—
92#550—91no—
93#547—91no—
94#540—92no—
95#532—91no—
96#529—91no—
97#520—32no—
98C4×#41—31no—
99C2×C3×Q8—31no—
100C2×C3×Q8C23×C311yescharacteristic
101#239—21no—
102#239—22no—
103#239—61no—
104#239—62no—
105#239—124no—
106C2×D12—31no—
107C2×D12—32no—
108C2×D12—31no—
109C2×D12—32no—
110C2×D12—34no—
111C2×D12—32no—
112C2×C4×D3—31no—
113C2×C4×D3—31no—
114C2×C4×D3—32no—
115C2×C4×D3—34no—
116C2×C4×D3—31no—
117#41⋊C3—61no—
118#41⋊C3—62no—
119C2×SL(2,𝔽3)—21no—
120C2×SL(2,𝔽3)C22×D311yescharacteristic
121C3×C4⋊C4—32no—
122C3×C4⋊C4—31no—
123C3×C4⋊C4—32no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—41no—
126C3×C4⋊C4—81no—
127C3×C4⋊C4—122no—
128C3×C4⋊C4—121no—
129C3×C4⋊C4—122no—
130C3×C4⋊C4C2×A411yescharacteristic
131C3×C42—31no—
132C3×C42—32no—
133C3×C42—121no—
134(C2×D3)⋊C4—32no—
135(C2×D3)⋊C4—32no—
136(C2×D3)⋊C4—32no—
137(C2×D3)⋊C4—32no—
138(C2×D3)⋊C4—32no—
139(C2×D3)⋊C4—32no—
140C4⋊Dic3—31no—
141C4⋊Dic3C2×A412yes—
142C4⋊Dic3C2×A411yescharacteristic
143Dic3⋊C4—32no—
144Dic3⋊C4C2×A412yes—
145C4×Dic3—31no—
146C4×Dic3—31no—
147C4×Dic3C2×A411yescharacteristic
148C22×C32—41no—
149C32×C4—42no—
150C32×C4—82no—
151C3×Dic3—81no—
152C3×Dic3—81no—
153C3×Dic3—82no—
154C2×#41—31no—
155C2×#41—32no—
156C2×#41—92no—
157C4⋊Q8—31no—
158#127—91no—
159#127—92no—
160#126—92no—
161#125—32no—
162#125—31no—
163#124—91no—
164#123—92no—
165#123—94no—
166C22⋊Q8—91no—
167#121—92no—
168#121—94no—
169#121—91no—
170#121—92no—
171#121—92no—
172#121—92no—
173C4×Q8—31no—
174C4×Q8—31no—
175C4×Q8C2×C3×D311yescharacteristic
176C4×Q8C2×C3×D312yes—
177C4×D4—91no—
178C4×D4—92no—
179C4×D4—91no—
180C4×D4—92no—
181C4×D4—92no—
182C4×D4—91no—
183C4×D4—92no—
184#117—92no—
185#117—91no—
186C2×C4⋊C4—92no—
187C2×C42—91no—
188C22×D3—31no—
189C22×D3—32no—
190C3×Q8—31no—
191C3×Q8—62no—
192C3×Q8C3×#4111yescharacteristic
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—32no—
196C2×C3×C4—41no—
197C2×C3×C4—42no—
198C2×C3×C4—81no—
199C2×C3×C4—82no—
200C2×C3×C4—121no—
201C2×C3×C4—121no—
202C2×C3×C4—122no—
203C2×C3×C4C22×A412yes—
204C2×C3×C4C22×A411yescharacteristic
205C2×Dic3—31no—
206C2×Dic3C22×A411yescharacteristic
207C2×Dic3C22×A411yescharacteristic
208C2×Dic3C22×A412yes—
209D12—61no—
210D12—62no—
211D12—61no—
212D12—62no—
213D12—62no—
214D12—122no—
215D12—122no—
216D12—124no—
217C4×D3—61no—
218C4×D3—62no—
219C4×D3—61no—
220C4×D3—62no—
221C4×D3—62no—
222C4×D3—121no—
223C4×D3—124no—
224SL(2,𝔽3)—21no—
225SL(2,𝔽3)#23911yescharacteristic, complemented
226C2×C32—41no—
227C2×C32—41no—
228C2×C32—41no—
229#41—61no—
230#41—62no—
231#41—181no—
232#41—182no—
233#41—364no—
234C2×Q8—31no—
235C2×Q8C22×C3×D311yescharacteristic
236C2×D4—91no—
237C2×D4—92no—
238C2×D4—92no—
239C2×D4—94no—
240C2×D4—91no—
241C2×D4—92no—
242C22×C4—91no—
243C22×C4—92no—
244C22×C4—91no—
245C22×C4—94no—
246C22×C4—91no—
247C4⋊C4—32no—
248C4⋊C4—31no—
249C4⋊C4—32no—
250C4⋊C4—32no—
251C4⋊C4—31no—
252C4⋊C4—32no—
253C4⋊C4—32no—
254C4⋊C4—92no—
255C4⋊C4—91no—
256C4⋊C4D3×A411yescharacteristic
257C22⋊C4—92no—
258C22⋊C4—92no—
259C22⋊C4—92no—
260C22⋊C4—92no—
261C22⋊C4—92no—
262C22⋊C4—92no—
263C42—31no—
264C42—31no—
265C42—32no—
266C42—91no—
267C42—91no—
268C22×C3—41no—
269C22×C3—81no—
270C22×C3C23×A411yescharacteristic
271C2×D3—61no—
272C2×D3—62no—
273C2×D3—61no—
274C2×D3—62no—
275C2×D3—61no—
276C2×D3—62no—
277C3×C4—22no—
278C3×C4—31no—
279C3×C4—31no—
280C3×C4—42no—
281C3×C4—62no—
282C3×C4—61no—
283C3×C4—82no—
284C3×C4—82no—
285C3×C4—162no—
286C3×C4—241no—
287C3×C4—241no—
288C3×C4—242no—
289C3×C4#99412yes—
290Dic3—21no—
291Dic3—22no—
292Dic3—21no—
293Dic3—61no—
294C32—41noSylow
295C23—91no—
296C23—92no—
297Q8—31no—
298Q8—62no—
299Q8C3×#23911yescharacteristic
300D4—181no—
301D4—182no—
302D4—181no—
303D4—182no—
304D4—182no—
305D4—362no—
306D4—364no—
307D4—362no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—32no—
313C2×C4—32no—
314C2×C4—91no—
315C2×C4—181no—
316C2×C4—182no—
317C2×C4—181no—
318C2×C4—182no—
319C2×C4—182no—
320C2×C4—361no—
321C2×C4—364no—
322C2×C4C2×D3×A412yes—
323C2×C4C2×D3×A411yescharacteristic
324C2×C3—41no—
325C2×C3—41no—
326C2×C3—41no—
327C2×C3—81no—
328C2×C3—81no—
329C2×C3—81no—
330C2×C3A4×#4111yescharacteristic
331C2×C3C2×#99411yescharacteristic
332C2×C3#594511yescharacteristic
333D3—121no—
334D3—122no—
335C22—181no—
336C22—182no—
337C22—181no—
338C22—182no—
339C22—181no—
340C22—182no—
341C22C22×D3×A411yescharacteristic
342C4—22no—
343C4—31no—
344C4—31no—
345C4—61no—
346C4—62no—
347C4—62no—
348C4—61no—
349C4—61no—
350C4—181no—
351C4C3⋊#99412yes—
352C3—41no—
353C3—81no—
354C3#3517⋊C311yescharacteristic, complemented
355C2—361no—
356C2—362no—
357C2A4×#23911yescharacteristic
358C2C2×C3⋊#99411yescharacteristic
359C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#557C2×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×#557⋊C3C2semidirect
2. C4⋊(C3⋊(#41⋊C3))C2semidirect
3. C4⋊(C3⋊(#41⋊C3))C2semidirect
4. #5574⋊C3C2semidirect
5. C4×C3⋊(#41⋊C3)C2semidirect
6. #87023C3semidirect
26. #557⋊C3D3semidirect
87. C3×SL(2,𝔽3)#41semidirect
225. SL(2,𝔽3)#239semidirect
354. C3#3517⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
C4⋊(C3⋊(#41⋊C3))2
#5574⋊C32
C4×C3⋊(#41⋊C3)2
#870233
#3517⋊C33
C3×#8904