Finite Group Databasebeta

← #13166041

Subgroups — #13166041

Order 1728.

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

normal subgroups only

This group has 614 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×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3×C3⋊GL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3⋊(C3⋊#859)C211yesmaximal, characteristic, complemented
4#10593655—31nomaximal
5#10593217—33nomaximal
6#10591583—31nomaximal
7C2×C32×SL(2,𝔽3)C2211yescharacteristic
8C2×#6290—41nomaximal
9C3×D3×SL(2,𝔽3)—21no—
10C3×C3⋊GL(2,𝔽3)—21no—
11C2×C3×D3×Q8D311yescharacteristic, complemented
12C22×C3×SL(2,𝔽3)—31no—
13C2×D3×SL(2,𝔽3)D313yescomplemented
14C2×C3⋊GL(2,𝔽3)D311yescharacteristic, complemented
15C2×C3×GL(2,𝔽3)—33no—
16C2×C3×#215—31no—
17#63908—31no—
18C2×#3715—31no—
19C3⋊#859—31no—
20C3⋊#859—33no—
21C3⋊#859—61no—
22C3⋊#859—63no—
23#63662—31no—
24#63566—31no—
25(C2×D3)⋊(C3⋊C8)—31no—
26C22×C3×C3⋊D3—41no—
27C22×C32×D3—41no—
28C2×C3⋊(C3⋊Dic3)—41no—
29C32×SL(2,𝔽3)D411yescharacteristic, complemented
30#6290—44no—
31#5421—93no—
32#5227—91no—
33#5170—91no—
34C2×C32×Q8C2×D311yescharacteristic
35C2×#638—121no—
36C3×D3×Q8—21no—
37C3×D3×Q8—61no—
38C2×C3×D12—31no—
39C2×C3×C4×D3—31no—
40C2×C3×Dic6—31no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)—23no—
43C2×C3×SL(2,𝔽3)—61no—
44C2×C3×SL(2,𝔽3)C2×D313yes—
45C2×C3×SL(2,𝔽3)C2×D311yescharacteristic
46C2×#625—61no—
47C2×#625—123no—
48D3×SL(2,𝔽3)—23no—
49C3⋊GL(2,𝔽3)—21no—
50C3×GL(2,𝔽3)—63no—
51C4⋊(C3⋊Dic3)—31no—
52C2×C3⋊(C3⋊C8)—31no—
53C3×#215—61no—
54C3×#215—61no—
55C2×C3×C3⋊C8—31no—
56(C2×D3)⋊Dic3—61no—
57#3715—62no—
58C22×C33—41no—
59C2×C3×C3⋊D3—42no—
60C2×C3×C3⋊D3—81no—
61C2×C3×C3⋊D3—81no—
62C2×C32×D3—42no—
63C2×C32×D3—81no—
64C2×C32×D3—81no—
65C3⋊(C3⋊Dic3)—42no—
66C22×C3×Q8—31no—
67C2×D3×Q8C3⋊D311yescharacteristic, complemented
68C22×SL(2,𝔽3)—33no—
69C2×GL(2,𝔽3)—33no—
70C2×C3×SD16—91no—
71C2×#215—31no—
72C2×#215—91no—
73#938—91no—
74C2×#214—91no—
75#930—91no—
76#859—93no—
77#859—183no—
78#835—91no—
79#835—91no—
80#835—91no—
81#835—181no—
82#830—91no—
83(C2×D3)⋊C8—91no—
84#816—91no—
85C23×C32—121no—
86C22×C3⋊D3—41no—
87C22×C3×D3—43no—
88C22×C3×D3—61no—
89C22×C3×D3—123no—
90C22×C3×D3S411yescharacteristic
91C32×Q8—31no—
92C32×Q8#6711yescharacteristic, complemented
93C2×C32×C4—31no—
94#638—124no—
95C2×C3⋊Dic3—61no—
96C2×C3⋊Dic3—121no—
97C2×C3⋊Dic3—123no—
98C2×C3⋊Dic3—241no—
99C2×C3⋊Dic3—243no—
100C2×C3×Dic3—31no—
101C3×D12—61no—
102C3×D12—61no—
103C3×C4×D3—61no—
104C3×C4×D3—61no—
105C3×Dic6—32no—
106C3×Dic6—61no—
107C3×SL(2,𝔽3)—21no—
108C3×SL(2,𝔽3)—23no—
109C3×SL(2,𝔽3)#6713yescomplemented
110C3×SL(2,𝔽3)#6711yescharacteristic, complemented
111#625—1212no—
112#625—122no—
113C3⋊(C3⋊C8)—61no—
114C3×C3⋊C8—61no—
115#448—271noSylow
116C2×C33—41no—
117C2×C33—41no—
118C2×C33—41no—
119C3×C3⋊D3—82no—
120C32×D3—82no—
121C2×C3×Q8—21no—
122C2×C3×Q8—61no—
123C2×C3×Q8—91no—
124C2×C3×Q8—181no—
125C2×C3×Q8D3211yescharacteristic
126C2×C3×Q8C2×C3⋊D311yescharacteristic
127C2×C3×D4—91no—
128C22×C3×C4—91no—
129C2×#67—181no—
130C2×#67—181no—
131C2×#67—363no—
132D3×Q8—21no—
133D3×Q8—61no—
134C2×D12—31no—
135C2×C4×D3—31no—
136C2×Dic6—31no—
137C2×SL(2,𝔽3)—23no—
138C2×SL(2,𝔽3)—63no—
139C2×SL(2,𝔽3)D3213yes—
140GL(2,𝔽3)—63no—
141C3×SD16—181no—
142C3×SD16—181no—
143C2×C3×C8—91no—
144#217—181no—
145#215—61no—
146#215—61no—
147#215—182no—
148#214—182no—
149(C2×D3)⋊C4—181no—
150C4⋊Dic3—91no—
151C4⋊Dic3—91no—
152C4⋊Dic3—181no—
153C2×C3⋊C8—31no—
154C2×C3⋊C8—91no—
155C2×C3⋊C8—91no—
156C2×C3⋊C8—181no—
157C22×C32—41no—
158C22×C32—43no—
159C22×C32—83no—
160C22×C32—81no—
161C22×C32—122no—
162C22×C32—241no—
163C22×C32—241no—
164C22×C32C2×S411yescharacteristic
165C2×C3⋊D3—42no—
166C2×C3⋊D3—81no—
167C2×C3⋊D3—81no—
168C2×C3×D3—21no—
169C2×C3×D3—21no—
170C2×C3×D3—46no—
171C2×C3×D3—83no—
172C2×C3×D3—83no—
173C2×C3×D3—126no—
174C2×C3×D3—121no—
175C2×C3×D3—121no—
176C2×C3×D3—121no—
177C2×C3×D3—243no—
178C2×C3×D3—243no—
179C2×C3×D3GL(2,𝔽3)12yescomplemented
180C32×C4—31no—
181C32×C4—31no—
182C3⋊Dic3—122no—
183C3⋊Dic3—126no—
184C3⋊Dic3—121no—
185C3⋊Dic3—246no—
186C3⋊Dic3—242no—
187C3×Dic3—61no—
188C22×Q8—31no—
189C2×SD16—91no—
190C2×SD16—271no—
191#121—271no—
192#103—271no—
193#103—271no—
194C22⋊C8—271no—
195C33—41noSylow
196C23×C3—31no—
197C23×C3—123no—
198C23×C3—181no—
199C22×D3—61no—
200C22×D3—123no—
201C22×D3C3⋊S411yescharacteristic
202C3×Q8—21no—
203C3×Q8—31no—
204C3×Q8—31no—
205C3×Q8—61no—
206C3×Q8—92no—
207C3×Q8—181no—
208C3×Q8#62511yescharacteristic, complemented
209C3×Q8#63811yescharacteristic, complemented
210C3×D4—181no—
211C3×D4—181no—
212C2×C3×C4—31no—
213C2×C3×C4—31no—
214C2×C3×C4—61no—
215C2×C3×C4—91no—
216C2×C3×C4—181no—
217C2×C3×C4—181no—
218#67—362no—
219#67—3612no—
220#67—362no—
221C2×Dic3—31no—
222C2×Dic3—181no—
223C2×Dic3—181no—
224C2×Dic3—363no—
225C2×Dic3—361no—
226C2×Dic3—723no—
227D12—61no—
228D12—61no—
229C4×D3—61no—
230C4×D3—61no—
231Dic6—32no—
232Dic6—61no—
233SL(2,𝔽3)—23no—
234SL(2,𝔽3)#62513yescomplemented
235C3×C8—181no—
236C3⋊C8—61no—
237C3⋊C8—181no—
238C3⋊C8—181no—
239C3⋊C8—361no—
240C2×C32—41no—
241C2×C32—41no—
242C2×C32—41no—
243C2×C32—43no—
244C2×C32—43no—
245C2×C32—43no—
246C2×C32—83no—
247C2×C32—83no—
248C2×C32—83no—
249C2×C32—81no—
250C2×C32—81no—
251C2×C32—81no—
252C2×C32—242no—
253C2×C32C2×GL(2,𝔽3)11yescharacteristic
254C2×C32#98311yescharacteristic
255C2×C32GL(2,ℤ/4)11yescharacteristic
256C3⋊D3—82no—
257C3×D3—22no—
258C3×D3—86no—
259C3×D3—246no—
260C3×D3—241no—
261C2×Q8—61no—
262C2×Q8—91no—
263C2×Q8—181no—
264C2×Q8D3×C3⋊D311yescharacteristic
265C2×D4—91no—
266C2×D4—541no—
267C22×C4—91no—
268SD16—181no—
269SD16—181no—
270SD16—542no—
271C2×C8—91no—
272C2×C8—271no—
273C4⋊C4—271no—
274C22⋊C4—541no—
275C22×C3—21no—
276C22×C3—32no—
277C22×C3—43no—
278C22×C3—61no—
279C22×C3—61no—
280C22×C3—83no—
281C22×C3—126no—
282C22×C3—243no—
283C22×C3—243no—
284C22×C3—361no—
285C22×C3—361no—
286C22×C3—361no—
287C22×C3D3×S411yescharacteristic
288C22×C3C2×C3⋊S411yescharacteristic
289C2×D3—21no—
290C2×D3—21no—
291C2×D3—126no—
292C2×D3—121no—
293C2×D3—121no—
294C2×D3—121no—
295C2×D3—243no—
296C2×D3—243no—
297C2×D3C3⋊GL(2,𝔽3)12yescomplemented
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—61no—
303C3×C4—61no—
304C3×C4—181no—
305Dic3—61no—
306Dic3—366no—
307Dic3—361no—
308Dic3—361no—
309Dic3—721no—
310Dic3—726no—
311C32—41no—
312C32—43no—
313C32—83no—
314C32—81no—
315C32#542111yescharacteristic, complemented
316C23—31no—
317C23—181no—
318Q8—31no—
319Q8—92no—
320Q8—181no—
321Q8#629011yescharacteristic, complemented
322D4—181no—
323D4—181no—
324D4—1082no—
325C2×C4—31no—
326C2×C4—91no—
327C2×C4—181no—
328C2×C4—181no—
329C2×C4—541no—
330C8—181no—
331C8—541no—
332C2×C3—21no—
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—43no—
336C2×C3—43no—
337C2×C3—43no—
338C2×C3—62no—
339C2×C3—83no—
340C2×C3—83no—
341C2×C3—83no—
342C2×C3—246no—
343C2×C3—721no—
344C2×C3#6420111yescharacteristic
345C2×C3D3×GL(2,𝔽3)11yescharacteristic
346C2×C3#6420911yescharacteristic
347C2×C3C2×C3⋊GL(2,𝔽3)11yescharacteristic
348C2×C3#6426411yescharacteristic
349C2×C3#6426911yescharacteristic
350D3—22no—
351D3—246no—
352D3—241no—
353C22—32no—
354C22—61no—
355C22—61no—
356C22—361no—
357C22—361no—
358C22—361no—
359C22D3×C3⋊S411yescharacteristic
360C4—31no—
361C4—31no—
362C4—181no—
363C4—1081no—
364C3—21no—
365C3—43no—
366C3—83no—
367C3#1059321711yescharacteristic, complemented
368C3#1059365511yescharacteristic, complemented
369C2—62no—
370C2—721no—
371C2#1172419611yescharacteristic
372C2D3×C3⋊GL(2,𝔽3)11yescharacteristic
373C2#1172420411yescharacteristic
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×D3×SL(2,𝔽3)C2semidirect
2. C2×C3×C3⋊GL(2,𝔽3)C2semidirect
3. C3⋊(C3⋊#859)C2semidirect
11. C2×C3×D3×Q8D3semidirect
13. C2×D3×SL(2,𝔽3)D3semidirect
14. C2×C3⋊GL(2,𝔽3)D3semidirect
29. C32×SL(2,𝔽3)D4semidirect
67. C2×D3×Q8C3⋊D3semidirect
92. C32×Q8#67semidirect
109. C3×SL(2,𝔽3)#67semidirect
110. C3×SL(2,𝔽3)#67semidirect
179. C2×C3×D3GL(2,𝔽3)semidirect
208. C3×Q8#625semidirect
209. C3×Q8#638semidirect
234. SL(2,𝔽3)#625semidirect
297. C2×D3C3⋊GL(2,𝔽3)semidirect
315. C32#5421semidirect
321. Q8#6290semidirect
367. C3#10593217semidirect
368. C3#10593655semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×SL(2,𝔽3)2
C2×C3×C3⋊GL(2,𝔽3)2
C3⋊(C3⋊#859)2
#105936553
#105932173
#105915833
C2×#62904