Finite Group Databasebeta

← #87024⋊C3 #11923285

Subgroups — #87024⋊C3 #11923285

Order 1152.

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

normal subgroups only

This group has 573 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×#557⋊C3C211yesmaximal, characteristic, complemented
2#5572⋊C3C212yesmaximal, complemented
3SL(2,𝔽3)×D12C212yesmaximal, complemented
4SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
5C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#87024C311yesmaximal, characteristic, complemented
7#3523⋊C3—31nomaximal
8C2×D3×SL(2,𝔽3)C2211yescharacteristic
9C2×D3×SL(2,𝔽3)C2212yes—
10C3×#895—41nomaximal
11C3×C4×SL(2,𝔽3)C2211yescharacteristic
12C3×C4×SL(2,𝔽3)C2212yes—
13Dic3×SL(2,𝔽3)C2211yescharacteristic
14C3×#557C2×C311yescharacteristic, complemented
15#5732—31no—
16#5728—31no—
17#5698—32no—
18#5694—32no—
19#5580—31no—
20Q8×D12C2×C312yescomplemented
21Q8×D12C2×C311yescharacteristic, complemented
22#5572C2×C312yescomplemented
23C4×D3×Q8C2×C311yescharacteristic, complemented
24#5570—31no—
25#557⋊C3—21no—
26#557⋊C3D311yescharacteristic, complemented
27D4×SL(2,𝔽3)—31no—
28D4×SL(2,𝔽3)—32no—
29#519⋊C3—32no—
30C2×C4×SL(2,𝔽3)—31no—
31C2×C3×D12—41no—
32C2×C3×D12—42no—
33C2×C3×C4×D3—41no—
34C2×C3×SL(2,𝔽3)C2311yescharacteristic
35D3×SL(2,𝔽3)—21no—
36D3×SL(2,𝔽3)—22no—
37C32×C4⋊C4—41no—
38C3×(C2×D3)⋊C4—42no—
39#3523—31noSylow
40C2×D3×Q8C22×C311yescharacteristic
41C2×D3×Q8C22×C312yes—
42C22×SL(2,𝔽3)—31no—
43C22×SL(2,𝔽3)—32no—
44C3×C4⋊Q8—31no—
45C3×#125—32no—
46C3×#121—121no—
47C3×C4×Q8—31no—
48C3×C4×Q8C22×C311yescharacteristic
49C3×C4×Q8C22×C312yes—
50#947—31no—
51#946—31no—
52Q8×Dic3—32no—
53Q8×Dic3C22×C311yescharacteristic
54#898—34no—
55#898—32no—
56#897—32no—
57#897—32no—
58#897—32no—
59#895A411yescharacteristic
60#894—32no—
61#892—32no—
62#892—32no—
63D3×C4⋊C4—31no—
64#890—34no—
65#890—32no—
66#890—31no—
67#887—32no—
68#887—31no—
69C4×D12—32no—
70C4×D12—31no—
71#872—32no—
72C42×D3—31no—
73#869—32no—
74#869—31no—
75C4×Dic6—32no—
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—
83C22×C3×D3—42no—
84C2×C32×C4—41no—
85C2×C32×C4—42no—
86C2×C3×Dic3—41no—
87C3×D12—82no—
88C3×D12—84no—
89C3×C4×D3—82no—
90C3×SL(2,𝔽3)#4111yescharacteristic
91#557—91no—
92#557C3×D311yescharacteristic, complemented
93#556—91no—
94#552—92no—
95D4×Q8—32no—
96D4×Q8—31no—
97#541—92no—
98#533—91no—
99#529—91no—
100#519—32no—
101C2×C4×Q8—31no—
102C2×C3×Q8—31no—
103C2×C3×Q8C23×C311yescharacteristic
104C2×C3×D4—122no—
105C2×C3×D4—121no—
106C22×C3×C4—121no—
107D3×Q8—21no—
108D3×Q8—22no—
109D3×Q8—61no—
110D3×Q8—62no—
111C2×D12C2×A412yes—
112C2×D12C2×A411yescharacteristic
113C2×C4×D3—31no—
114C2×C4×D3—31no—
115C2×C4×D3—32no—
116C2×C4×D3C2×A411yescharacteristic
117C2×Dic6—32no—
118C2×Dic6—31no—
119C2×SL(2,𝔽3)—21no—
120C2×SL(2,𝔽3)—61no—
121C2×SL(2,𝔽3)—62no—
122C2×SL(2,𝔽3)C22×D311yescharacteristic
123C3×C4⋊C4—31no—
124C3×C4⋊C4—32no—
125C3×C4⋊C4—32no—
126C3×C4⋊C4—32no—
127C3×C4⋊C4—41no—
128C3×C4⋊C4—81no—
129C3×C4⋊C4C2×A411yescharacteristic
130C3×C22⋊C4—122no—
131C3×C42—31no—
132C3×C42—32no—
133(C2×D3)⋊C4—32no—
134(C2×D3)⋊C4—32no—
135(C2×D3)⋊C4—32no—
136(C2×D3)⋊C4—32no—
137(C2×D3)⋊C4C2×A412yes—
138C4⋊Dic3—32no—
139C4⋊Dic3—34no—
140C4⋊Dic3—32no—
141C4⋊Dic3—31no—
142C4⋊Dic3—31no—
143Dic3⋊C4—32no—
144Dic3⋊C4—32no—
145Dic3⋊C4—32no—
146C4×Dic3—34no—
147C4×Dic3—31no—
148C4×Dic3—31no—
149C4×Dic3—31no—
150C22×C32—41no—
151C2×C3×D3—42no—
152C2×C3×D3—81no—
153C2×C3×D3—81no—
154C2×C3×D3—82no—
155C2×C3×D3—82no—
156C2×C3×D3—82no—
157C32×C4—42no—
158C32×C4—82no—
159C3×Dic3—81no—
160C22×Q8—31no—
161C22×Q8—32no—
162C4⋊Q8—31no—
163C4⋊Q8—91no—
164C4⋊Q8—92no—
165#126—92no—
166#126—94no—
167#125—32no—
168#125—92no—
169#125—94no—
170#125—91no—
171#124—91no—
172#123—92no—
173C22⋊Q8—91no—
174C22⋊Q8—92no—
175C22⋊Q8—92no—
176C22⋊Q8—92no—
177#121—31no—
178C4×Q8—31no—
179C4×Q8—31no—
180C4×Q8—92no—
181C4×Q8—91no—
182C4×Q8—92no—
183C4×Q8—92no—
184C4×Q8—91no—
185C4×Q8C2×C3×D311yescharacteristic
186C4×Q8C2×C3×D312yes—
187C4×D4—92no—
188C4×D4—91no—
189#117—92no—
190#117—92no—
191#117—92no—
192C2×C4⋊C4—91no—
193C2×C42—91no—
194C23×C3—121no—
195C23×C3—122no—
196C22×D3C22×A412yes—
197C22×D3C22×A411yescharacteristic
198C3×Q8—31no—
199C3×Q8—62no—
200C3×Q8C3×#4111yescharacteristic
201C3×D4—242no—
202C3×D4—244no—
203C2×C3×C4—31no—
204C2×C3×C4—32no—
205C2×C3×C4—31no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—81no—
209C2×C3×C4—82no—
210C2×C3×C4—121no—
211C2×C3×C4—242no—
212C2×C3×C4C22×A412yes—
213C2×C3×C4C22×A411yescharacteristic
214C2×Dic3—31no—
215C2×Dic3—32no—
216C2×Dic3—31no—
217C2×Dic3C22×A411yescharacteristic
218D12—24no—
219D12—22no—
220C4×D3—22no—
221C4×D3—61no—
222C4×D3—62no—
223C4×D3—61no—
224C4×D3—62no—
225C4×D3—62no—
226Dic6—62no—
227Dic6—61no—
228Dic6—62no—
229Dic6—61no—
230SL(2,𝔽3)—21no—
231SL(2,𝔽3)#23911yescharacteristic
232C2×C32—41no—
233C2×C32—41no—
234C2×C32—41no—
235C3×D3—82no—
236C3×D3—162no—
237C2×Q8—31no—
238C2×Q8—61no—
239C2×Q8—62no—
240C2×Q8—92no—
241C2×Q8—91no—
242C2×Q8—181no—
243C2×Q8—182no—
244C2×Q8C22×C3×D311yescharacteristic
245C2×D4—32no—
246C2×D4—31no—
247C22×C4—31no—
248C22×C4—91no—
249C22×C4—91no—
250C22×C4—92no—
251C4⋊C4—31no—
252C4⋊C4—32no—
253C4⋊C4—32no—
254C4⋊C4—32no—
255C4⋊C4—92no—
256C4⋊C4—91no—
257C4⋊C4—92no—
258C4⋊C4—92no—
259C4⋊C4—94no—
260C4⋊C4—92no—
261C4⋊C4—91no—
262C4⋊C4—92no—
263C4⋊C4D3×A411yescharacteristic
264C22⋊C4—32no—
265C22⋊C4—92no—
266C22⋊C4—92no—
267C22⋊C4—92no—
268C22⋊C4—92no—
269C42—31no—
270C42—32no—
271C42—91no—
272C42—91no—
273C42—94no—
274C42—91no—
275C22×C3—41no—
276C22×C3—81no—
277C22×C3—122no—
278C22×C3—241no—
279C22×C3—241no—
280C22×C3—242no—
281C22×C3—242no—
282C22×C3—242no—
283C22×C3C23×A411yescharacteristic
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—22no—
287C2×D3—22no—
288C2×D3—22no—
289C2×D32−1+4⋊C312yes—
290C3×C4—22no—
291C3×C4—31no—
292C3×C4—31no—
293C3×C4—42no—
294C3×C4—61no—
295C3×C4—62no—
296C3×C4—82no—
297C3×C4—82no—
298C3×C4—162no—
299C3×C4—241no—
300C3×C42−1+4⋊C312yes—
301Dic3—21no—
302Dic3—61no—
303Dic3—62no—
304Dic3—61no—
305C32—41noSylow
306C23—31no—
307C23—32no—
308Q8—31no—
309Q8—62no—
310Q8—181no—
311Q8—182no—
312Q8—182no—
313Q8—181no—
314Q8C3×#23911yescharacteristic
315D4—64no—
316D4—62no—
317C2×C4—31no—
318C2×C4—32no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—62no—
322C2×C4—91no—
323C2×C4—91no—
324C2×C4—92no—
325C2×C4—181no—
326C2×C4—182no—
327C2×C4—181no—
328C2×C4—182no—
329C2×C4—182no—
330C2×C4C2×D3×A412yes—
331C2×C4C2×D3×A411yescharacteristic
332C2×C3—41no—
333C2×C3—41no—
334C2×C3—41no—
335C2×C3—81no—
336C2×C3—81no—
337C2×C3—81no—
338C2×C3—242no—
339C2×C3—482no—
340C2×C3A4×#4111yescharacteristic
341C2×C3C2×2−1+4⋊C311yescharacteristic
342C2×C3C2×2−1+4⋊C311yescharacteristic
343D3—22no—
344D3—42no—
345C22—32no—
346C22—61no—
347C22—61no—
348C22—62no—
349C22—62no—
350C22—62no—
351C22C22×D3×A411yescharacteristic
352C4—22no—
353C4—31no—
354C4—31no—
355C4—61no—
356C4—62no—
357C4—61no—
358C4—181no—
359C4—182no—
360C4—181no—
361C4#1010⋊C312yes—
362C3—41no—
363C3—81no—
364C3#3523⋊C311yescharacteristic, complemented
365C2—62no—
366C2—122no—
367C2A4×#23911yescharacteristic
368C2C2×#1010⋊C311yescharacteristic
369C2D3×2−1+4⋊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. #5572⋊C3C2semidirect
3. SL(2,𝔽3)×D12C2semidirect
4. SL(2,𝔽3)×D12C2semidirect
5. C4×D3×SL(2,𝔽3)C2semidirect
6. #87024C3semidirect
14. C3×#557C2×C3semidirect
20. Q8×D12C2×C3semidirect
21. Q8×D12C2×C3semidirect
22. #5572C2×C3semidirect
23. C4×D3×Q8C2×C3semidirect
26. #557⋊C3D3semidirect
92. #557C3×D3semidirect
364. C3#3523⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#5572⋊C32
SL(2,𝔽3)×D122
C4×D3×SL(2,𝔽3)2
#870243
#3523⋊C33
C3×#8954