Finite Group Databasebeta

← #13166039

Subgroups — #13166039

Order 1728.

681 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 60 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
1C2×C3⋊(C3⋊GL(2,𝔽3))C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
4C4⋊(C3⋊GL(2,𝔽3))—31nomaximal
5#10593215—33nomaximal
6#10591617—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6292—41nomaximal
9C3⋊(C3⋊GL(2,𝔽3))—21no—
10C3×C3⋊GL(2,𝔽3)—21no—
11C2×C3⋊GL(2,𝔽3)—31no—
12C2×C3⋊GL(2,𝔽3)—33no—
13C2×C3⋊GL(2,𝔽3)—61no—
14C2×C3⋊GL(2,𝔽3)—63no—
15C2×C3⋊GL(2,𝔽3)D311yescharacteristic, complemented
16C2×C3×GL(2,𝔽3)—33no—
17C2×#3755—31no—
18C3×Q8×Dic3D311yescharacteristic, complemented
19C2×C3×#215—31no—
20C3×C4×SL(2,𝔽3)—31no—
21#63911—31no—
22Dic3×SL(2,𝔽3)D313yescomplemented
23Dic3⋊(C3⋊C8)—31no—
24#63564—31no—
25#63562—31no—
26C22×C3⋊(C3⋊D3)—41no—
27C22×C3×C3⋊D3—41no—
28C2×C32×Dic3—41no—
29C32×SL(2,𝔽3)D411yescharacteristic, complemented
30#6292—44no—
31C4⋊GL(2,𝔽3)—93no—
32#5163—91no—
33#5027—91no—
34C2×C32×Q8C2×D311yescharacteristic
35C2×C4⋊(C3⋊D3)—31no—
36C2×C4⋊(C3⋊D3)—121no—
37C2×C3×D12—31no—
38C2×C3×SL(2,𝔽3)—21no—
39C2×C3×SL(2,𝔽3)—23no—
40C2×C3×SL(2,𝔽3)C2×D313yes—
41C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
42C2×#626—61no—
43C2×#626—61no—
44C2×#626—123no—
45C3⋊GL(2,𝔽3)—21no—
46C3⋊GL(2,𝔽3)—62no—
47C3⋊GL(2,𝔽3)—66no—
48C3⋊GL(2,𝔽3)—63no—
49C3⋊GL(2,𝔽3)—61no—
50C3×GL(2,𝔽3)—63no—
51#3755—61no—
52#3755—61no—
53C2×C3⋊(C3⋊C8)—31no—
54C3×#215—61no—
55C3×#215—61no—
56C3×C4⋊Dic3—31no—
57C3×C4×Dic3—31no—
58C2×C3×C3⋊C8—31no—
59C22×C33—41no—
60C2×C3⋊(C3⋊D3)—42no—
61C2×C3⋊(C3⋊D3)—81no—
62C2×C3⋊(C3⋊D3)—81no—
63C2×C3×C3⋊D3—42no—
64C2×C3×C3⋊D3—81no—
65C2×C3×C3⋊D3—81no—
66C32×Dic3—42no—
67C2×GL(2,𝔽3)—33no—
68C2×GL(2,𝔽3)—93no—
69C2×GL(2,𝔽3)—183no—
70C2×C3×SD16—91no—
71C3×C4×Q8—31no—
72Q8×Dic3C3⋊D311yescharacteristic, complemented
73C2×#215—31no—
74C2×#215—91no—
75C2×#215—91no—
76C2×#215—181no—
77#940—91no—
78#874—91no—
79C4×SL(2,𝔽3)—33no—
80#832—91no—
81#821—91no—
82Dic3⋊C8—91no—
83#809—91no—
84#809—91no—
85C4⋊(C3⋊C8)—91no—
86C22×C3⋊D3—41no—
87C22×C3⋊D3—61no—
88C22×C3⋊D3—121no—
89C22×C3⋊D3—123no—
90C22×C3⋊D3—241no—
91C22×C3⋊D3—243no—
92C22×C3×D3—61no—
93C22×C3×D3—123no—
94C32×Q8—31no—
95C32×Q8D1211yescharacteristic, complemented
96C2×C32×C4—31no—
97C2×C32×C4—121no—
98C4⋊(C3⋊D3)—61no—
99C4⋊(C3⋊D3)—61no—
100C4⋊(C3⋊D3)—124no—
101C2×C3×Dic3—31no—
102C2×C3×Dic3—43no—
103C2×C3×Dic3S411yescharacteristic
104C3×D12—61no—
105C3×D12—61no—
106C3×SL(2,𝔽3)—21no—
107C3×SL(2,𝔽3)—23no—
108C3×SL(2,𝔽3)D1213yescomplemented
109C3×SL(2,𝔽3)#6711yescharacteristic, complemented
110#626—1212no—
111#626—122no—
112#626—122no—
113C3⋊(C3⋊C8)—61no—
114C3×C3⋊C8—61no—
115#460—271noSylow
116C2×C33—41no—
117C2×C33—41no—
118C2×C33—41no—
119C3⋊(C3⋊D3)—82no—
120C3×C3⋊D3—82no—
121C2×C3×Q8—21no—
122C2×C3×Q8D3211yescharacteristic
123C2×C3×Q8C2×C3⋊D311yescharacteristic
124C2×C3×D4—91no—
125C2×#67—181no—
126C2×#67—181no—
127C2×D12—31no—
128C2×D12—91no—
129C2×D12—91no—
130C2×D12—181no—
131C2×D12—181no—
132C2×D12—181no—
133C2×D12—363no—
134C2×SL(2,𝔽3)—23no—
135C2×SL(2,𝔽3)D3213yes—
136GL(2,𝔽3)—63no—
137GL(2,𝔽3)—186no—
138GL(2,𝔽3)—183no—
139C3×SD16—181no—
140C3×SD16—181no—
141C2×C3×C8—91no—
142C3×C4⋊C4—91no—
143C3×C42—91no—
144#215—61no—
145#215—61no—
146#215—182no—
147#215—181no—
148#215—181no—
149#215—182no—
150#215—181no—
151#215—181no—
152C4⋊Dic3—31no—
153C4×Dic3—31no—
154C2×C3⋊C8—31no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—91no—
157C2×C3⋊C8—181no—
158C22×C32—41no—
159C22×C32—43no—
160C22×C32—83no—
161C22×C32—81no—
162C22×C32C2×S411yescharacteristic
163C2×C3⋊D3—42no—
164C2×C3⋊D3—81no—
165C2×C3⋊D3—81no—
166C2×C3⋊D3—122no—
167C2×C3⋊D3—121no—
168C2×C3⋊D3—126no—
169C2×C3⋊D3—121no—
170C2×C3⋊D3—121no—
171C2×C3⋊D3—243no—
172C2×C3⋊D3—241no—
173C2×C3⋊D3—243no—
174C2×C3⋊D3—241no—
175C2×C3⋊D3—246no—
176C2×C3⋊D3—242no—
177C2×C3⋊D3—242no—
178C2×C3⋊D3—246no—
179C2×C3⋊D3—242no—
180C2×C3⋊D3—246no—
181C2×C3×D3—126no—
182C2×C3×D3—121no—
183C2×C3×D3—121no—
184C2×C3×D3—121no—
185C2×C3×D3—243no—
186C2×C3×D3—243no—
187C32×C4—31no—
188C32×C4—31no—
189C32×C4—122no—
190C3×Dic3—46no—
191C3×Dic3—61no—
192C3×Dic3GL(2,𝔽3)12yescomplemented
193C2×SD16—91no—
194C2×SD16—271no—
195#127—271no—
196C4×Q8—31no—
197C4⋊C8—271no—
198#102—271no—
199#102—271no—
200C33—41noSylow
201C23×C3—181no—
202C22×D3—61no—
203C22×D3—123no—
204C22×D3—181no—
205C22×D3—181no—
206C22×D3—361no—
207C22×D3—363no—
208C22×D3—723no—
209C3×Q8—21no—
210C3×Q8—31no—
211C3×Q8—31no—
212C3×Q8—61no—
213C3×Q8#62611yescharacteristic, complemented
214C3×Q8C4⋊(C3⋊D3)11yescharacteristic, complemented
215C3×D4—181no—
216C3×D4—181no—
217C2×C3×C4—31no—
218C2×C3×C4—31no—
219C2×C3×C4—31no—
220C2×C3×C4—61no—
221C2×C3×C4—91no—
222C2×C3×C4—123no—
223#67—362no—
224#67—362no—
225C2×Dic3—31no—
226C2×Dic3C3⋊S411yescharacteristic
227D12—61no—
228D12—61no—
229D12—181no—
230D12—181no—
231D12—181no—
232D12—181no—
233D12—182no—
234D12—361no—
235D12—362no—
236D12—3612no—
237D12—362no—
238SL(2,𝔽3)—23no—
239SL(2,𝔽3)#62613yescomplemented
240C3×C8—181no—
241C3⋊C8—61no—
242C3⋊C8—181no—
243C3⋊C8—181no—
244C3⋊C8—182no—
245C2×C32—41no—
246C2×C32—41no—
247C2×C32—41no—
248C2×C32—43no—
249C2×C32—43no—
250C2×C32—43no—
251C2×C32—83no—
252C2×C32—83no—
253C2×C32—83no—
254C2×C32—81no—
255C2×C32—81no—
256C2×C32—81no—
257C2×C32C4⋊S411yescharacteristic
258C2×C32C2×GL(2,𝔽3)11yescharacteristic
259C2×C32#98611yescharacteristic
260C3⋊D3—82no—
261C3⋊D3—242no—
262C3⋊D3—246no—
263C3⋊D3—2412no—
264C3⋊D3—244no—
265C3⋊D3—241no—
266C3×D3—246no—
267C3×D3—241no—
268C2×Q8D3×C3⋊D311yescharacteristic
269C2×D4—91no—
270C2×D4—271no—
271C2×D4—541no—
272C2×D4—541no—
273SD16—181no—
274SD16—181no—
275SD16—541no—
276SD16—541no—
277C2×C8—91no—
278C2×C8—271no—
279C4⋊C4—91no—
280C42—91no—
281C22×C3—21no—
282C22×C3—43no—
283C22×C3—83no—
284C22×C3—361no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3D3×S411yescharacteristic
288C22×C3C2×C3⋊S411yescharacteristic
289C2×D3—126no—
290C2×D3—121no—
291C2×D3—121no—
292C2×D3—121no—
293C2×D3—243no—
294C2×D3—243no—
295C2×D3—362no—
296C2×D3—366no—
297C2×D3—361no—
298C2×D3—361no—
299C2×D3—361no—
300C2×D3—361no—
301C2×D3—361no—
302C2×D3—361no—
303C2×D3—721no—
304C2×D3—721no—
305C2×D3—723no—
306C2×D3—723no—
307C2×D3—726no—
308C2×D3—726no—
309C2×D3—726no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—32no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—61no—
316C3×C4—61no—
317C3×C4—126no—
318C3×C4—181no—
319Dic3—61no—
320Dic3C3⋊GL(2,𝔽3)12yescomplemented
321C32—41no—
322C32—43no—
323C32—83no—
324C32—81no—
325C32C4⋊GL(2,𝔽3)11yescharacteristic, complemented
326C23—181no—
327C23—541no—
328Q8—31no—
329Q8#629211yescharacteristic, complemented
330D4—181no—
331D4—181no—
332D4—541no—
333D4—541no—
334D4—1082no—
335D4—1082no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—91no—
339C8—181no—
340C8—541no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3—21no—
344C2×C3—43no—
345C2×C3—43no—
346C2×C3—43no—
347C2×C3—83no—
348C2×C3—83no—
349C2×C3—83no—
350C2×C3—721no—
351C2×C3#6419911yescharacteristic
352C2×C3D3×GL(2,𝔽3)11yescharacteristic
353C2×C3#6420711yescharacteristic
354C2×C3C4⋊(C3⋊S4)11yescharacteristic
355C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
356C2×C3#6426711yescharacteristic
357D3—246no—
358D3—241no—
359D3—722no—
360D3—726no—
361D3—7212no—
362D3—721no—
363D3—721no—
364C22—361no—
365C22—361no—
366C22—361no—
367C22—1081no—
368C22—1081no—
369C22—1081no—
370C22D3×C3⋊S411yescharacteristic
371C4—32no—
372C4—31no—
373C4—31no—
374C4—181no—
375C3—21no—
376C3—43no—
377C3—83no—
378C3#1059321511yescharacteristic, complemented
379C3C4⋊(C3⋊GL(2,𝔽3))11yescharacteristic, complemented
380C2—721no—
381C2—2161no—
382C2#1172419411yescharacteristic
383C2D3×C3⋊GL(2,𝔽3)11yescharacteristic
384C2#1172420211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D3×C3⋊S4
commutator, cosocleC2×C32×SL(2,𝔽3)C22
FittingC2×C32×Q8C2×D3
socleC22×C32C2×S4
hypocenterC32×SL(2,𝔽3)D4
LCSG ▷ C2×C32×SL(2,𝔽3) ▷ C32×SL(2,𝔽3)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊(C3⋊GL(2,𝔽3))C2semidirect
2. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
3. C3×Dic3×SL(2,𝔽3)C2semidirect
15. C2×C3⋊GL(2,𝔽3)D3semidirect
18. C3×Q8×Dic3D3semidirect
22. Dic3×SL(2,𝔽3)D3semidirect
29. C32×SL(2,𝔽3)D4semidirect
72. Q8×Dic3C3⋊D3semidirect
95. C32×Q8D12semidirect
108. C3×SL(2,𝔽3)D12semidirect
109. C3×SL(2,𝔽3)#67semidirect
192. C3×Dic3GL(2,𝔽3)semidirect
213. C3×Q8#626semidirect
214. C3×Q8C4⋊(C3⋊D3)semidirect
239. SL(2,𝔽3)#626semidirect
320. Dic3C3⋊GL(2,𝔽3)semidirect
325. C32C4⋊GL(2,𝔽3)semidirect
329. Q8#6292semidirect
378. C3#10593215semidirect
379. C3C4⋊(C3⋊GL(2,𝔽3))semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊(C3⋊GL(2,𝔽3))2
C2×C3×C3⋊GL(2,𝔽3)2
C3×Dic3×SL(2,𝔽3)2
C4⋊(C3⋊GL(2,𝔽3))3
#105932153
#105916173
C2×#62924