Finite Group Databasebeta

← #13153241

Subgroups — #13153241

Order 1728.

942 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 202 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
1H(3,𝔽3)×#126C211yesmaximal, characteristic, complemented
2#64104⋊C3C213yesmaximal, complemented
3#64103⋊C3C213yesmaximal, complemented
4#64093⋊C3C213yesmaximal, complemented
5#64091⋊C3C213yesmaximal, complemented
6#11721387C211yesmaximal, characteristic, complemented
7#11721385C211yesmaximal, characteristic, complemented
8#10596182C311yesmaximal, characteristic, complemented
9C3×#5711—31nomaximal
10C23×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×#638⋊C3C2213yes—
12C2×C4⋊((C3⋊D3)⋊C3)C2213yes—
13C2×C4⋊((C3⋊D3)⋊C3)C2213yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2213yescomplemented
15C2×C4×(C3⋊D3)⋊C3C2213yescomplemented
16C4⋊C4×H(3,𝔽3)C2213yescomplemented
17C22⋊C4×H(3,𝔽3)C2213yes—
18C42×H(3,𝔽3)C2211yescharacteristic, complemented
19#3752⋊C3C2213yes—
20#3752⋊C3C2213yes—
21#3752⋊C3C2213yescomplemented
22#3750⋊C3C2213yescomplemented
23C32×#126—21no—
24C32×#126D311yescharacteristic, complemented
25C32×#126C2×C311yescharacteristic, complemented
26#64104C2×C313yescomplemented
27#64103C2×C313yescomplemented
28#64093C2×C313yescomplemented
29#64091C2×C313yescomplemented
30#64085C2×C311yescharacteristic, complemented
31#64083C2×C311yescharacteristic, complemented
32C3×#895—33no—
33C3×#894—33no—
34C3×#884—33no—
35C3×#882—33no—
36C3×#876—31no—
37C3×#874—31no—
38C23×H(3,𝔽3)C2311yescharacteristic
39C22×(C3⋊D3)⋊C3—43no—
40C22×(C3⋊D3)⋊C3C2311yescharacteristic
41C22×(C3⋊D3)⋊C3C2311yescharacteristic
42C22×(C3⋊D3)⋊C3C2313yes—
43C2×C4×H(3,𝔽3)C2313yes—
44C2×C4×H(3,𝔽3)C2313yes—
45#638⋊C3—43no—
46C2×(C3⋊Dic3)⋊C3C2313yes—
47C4⋊((C3⋊D3)⋊C3)—43no—
48C4⋊((C3⋊D3)⋊C3)—43no—
49C4⋊((C3⋊D3)⋊C3)—43no—
50C4×(C3⋊D3)⋊C3—43no—
51C3×#560—91no—
52#5711—31no—
53#5711—91no—
54C23×C3⋊D3C22×C311yescharacteristic
55C23×C3×D3—31no—
56C2×#638C22×C313yes—
57C2×C4⋊(C3⋊D3)C22×C313yes—
58C2×C4⋊(C3⋊D3)C22×C313yes—
59C2×C4⋊(C3⋊D3)C22×C313yescomplemented
60C2×C4×C3⋊D3C22×C313yescomplemented
61C2×C3×#67—33no—
62C2×C3×D12—33no—
63C2×C3×D12—33no—
64C2×C3×D12—33no—
65C2×C3×C4×D3—33no—
66C32×C4⋊C4—23no—
67C32×C4⋊C4C2×D313yescomplemented
68C32×C4⋊C4C22×C313yescomplemented
69C32×C22⋊C4—23no—
70C32×C22⋊C4C2×D313yes—
71C32×C22⋊C4C22×C313yes—
72C32×C42—21no—
73C32×C42C2×D311yescharacteristic, complemented
74C32×C42C22×C311yescharacteristic, complemented
75#3752C22×C313yes—
76#3752C22×C313yes—
77#3752C22×C313yescomplemented
78#3750C22×C313yescomplemented
79C3×(C2×D3)⋊C4—33no—
80C3×(C2×D3)⋊C4—33no—
81C3×(C2×D3)⋊C4—33no—
82C3×Dic3⋊C4—33no—
83C22×H(3,𝔽3)—23no—
84C22×H(3,𝔽3)C2411yescharacteristic
85C2×(C3⋊D3)⋊C3—23no—
86C2×(C3⋊D3)⋊C3—23no—
87C2×(C3⋊D3)⋊C3—23no—
88C2×(C3⋊D3)⋊C3—23no—
89C2×(C3⋊D3)⋊C3—23no—
90C2×(C3⋊D3)⋊C3—44no—
91C4×H(3,𝔽3)—23no—
92C4×H(3,𝔽3)—23no—
93(C3⋊Dic3)⋊C3—23no—
94C3×#127—91no—
95C3×#126—31no—
96C3×#126—31no—
97C3×#126—61no—
98C3×#126—91no—
99C3×#126C3×D311yescharacteristic
100C3×#123—93no—
101C3×#121—93no—
102C3×#121—93no—
103C3×#120—93no—
104#895—33no—
105#895—93no—
106#894—33no—
107#894—93no—
108#884—33no—
109#884—93no—
110#882—33no—
111#882—93no—
112#876—31no—
113#876—91no—
114#874—31no—
115#874—91no—
116C23×C32—21no—
117C23×C32C22×D311yescharacteristic
118C23×C32C23×C311yescharacteristic
119C22×C3⋊D3—43no—
120C22×C3⋊D3C23×C311yescharacteristic
121C22×C3⋊D3C23×C311yescharacteristic
122C22×C3⋊D3C23×C313yes—
123C22×C3×D3—31no—
124C22×C3×D3—31no—
125C22×C3×D3—33no—
126C22×C3×D3—123no—
127C2×C32×C4—23no—
128C2×C32×C4—23no—
129C2×C32×C4C22×D313yes—
130C2×C32×C4C22×D313yes—
131C2×C32×C4C23×C313yes—
132C2×C32×C4C23×C313yes—
133#638—43no—
134C2×C3⋊Dic3C23×C313yes—
135C4⋊(C3⋊D3)—43no—
136C4⋊(C3⋊D3)—43no—
137C4⋊(C3⋊D3)—43no—
138C4×C3⋊D3—43no—
139C3×#67—123no—
140C2×C3×Dic3—33no—
141C3×D12—123no—
142C3×D12—123no—
143C3×D12—123no—
144C3×C4×D3—123no—
145#560—91noSylow
146C2×H(3,𝔽3)—41no—
147C2×H(3,𝔽3)H(5,𝔽2)13yes—
148(C3⋊D3)⋊C3—43no—
149(C3⋊D3)⋊C3—41no—
150(C3⋊D3)⋊C3—41no—
151C24×C3—91no—
152C23×D3—31no—
153C23×D3—91no—
154C2×C3×D4—93no—
155C2×C3×D4—93no—
156C2×C3×D4—93no—
157C2×C3×D4—93no—
158C22×C3×C4—93no—
159C2×#67—33no—
160C2×#67—93no—
161C2×D12—33no—
162C2×D12—33no—
163C2×D12—33no—
164C2×D12—93no—
165C2×D12—93no—
166C2×D12—93no—
167C2×C4×D3—33no—
168C2×C4×D3—93no—
169C3×C4⋊C4—33no—
170C3×C4⋊C4—33no—
171C3×C4⋊C4—63no—
172C3×C4⋊C4—93no—
173C3×C4⋊C4C2×C3×D313yes—
174C3×C22⋊C4—33no—
175C3×C22⋊C4—33no—
176C3×C22⋊C4—63no—
177C3×C22⋊C4—93no—
178C3×C22⋊C4—93no—
179C3×C22⋊C4—93no—
180C3×C22⋊C4C2×C3×D313yes—
181C3×C42—31no—
182C3×C42—31no—
183C3×C42—61no—
184C3×C42C2×C3×D311yescharacteristic
185(C2×D3)⋊C4—33no—
186(C2×D3)⋊C4—33no—
187(C2×D3)⋊C4—33no—
188(C2×D3)⋊C4—93no—
189(C2×D3)⋊C4—93no—
190(C2×D3)⋊C4—93no—
191Dic3⋊C4—33no—
192Dic3⋊C4—93no—
193C22×C32—23no—
194C22×C32—23no—
195C22×C32—21no—
196C22×C32—43no—
197C22×C32C23×D311yescharacteristic
198C22×C32C24×C311yescharacteristic
199C2×C3⋊D3—23no—
200C2×C3⋊D3—23no—
201C2×C3⋊D3—23no—
202C2×C3⋊D3—23no—
203C2×C3⋊D3—23no—
204C2×C3⋊D3—44no—
205C2×C3×D3—63no—
206C2×C3×D3—63no—
207C2×C3×D3—63no—
208C2×C3×D3—63no—
209C2×C3×D3—63no—
210C2×C3×D3—124no—
211C32×C4—23no—
212C32×C4—23no—
213C32×C4—23no—
214C32×C4—23no—
215C32×C4—43no—
216C32×C4—43no—
217C3⋊Dic3—23no—
218C3×Dic3—63no—
219#127—91no—
220#126—91no—
221#126(C3⋊D3)⋊C311yescharacteristic, complemented
222#123—93no—
223#121—93no—
224#121—93no—
225#120—93no—
226H(3,𝔽3)#56011yescharacteristic, complemented, Sylow
227C23×C3—31no—
228C23×C3—31no—
229C23×C3—61no—
230C23×C3—91no—
231C23×C3—91no—
232C23×C3—93no—
233C23×C3—363no—
234C23×C3C22×C3×D311yescharacteristic
235C22×D3—31no—
236C22×D3—31no—
237C22×D3—33no—
238C22×D3—91no—
239C22×D3—91no—
240C22×D3—93no—
241C22×D3—123no—
242C22×D3—363no—
243C3×D4—363no—
244C3×D4—363no—
245C3×D4—363no—
246C3×D4—363no—
247C2×C3×C4—33no—
248C2×C3×C4—33no—
249C2×C3×C4—33no—
250C2×C3×C4—33no—
251C2×C3×C4—63no—
252C2×C3×C4—63no—
253C2×C3×C4—93no—
254C2×C3×C4—363no—
255C2×C3×C4C22×C3×D313yes—
256C2×C3×C4C22×C3×D313yes—
257#67—123no—
258#67—363no—
259C2×Dic3—33no—
260C2×Dic3—93no—
261D12—123no—
262D12—123no—
263D12—123no—
264D12—363no—
265D12—363no—
266D12—363no—
267C4×D3—123no—
268C4×D3—363no—
269C2×C32—23no—
270C2×C32—41no—
271C2×C32—41no—
272C2×C32—81no—
273C2×C32#100913yes—
274C2×C32C3×H(5,𝔽2)13yes—
275C3⋊D3—43no—
276C3⋊D3—41no—
277C3⋊D3—41no—
278C3×D3—123no—
279C3×D3—121no—
280C3×D3—121no—
281C24—91no—
282C2×D4—93no—
283C2×D4—93no—
284C2×D4—93no—
285C2×D4—93no—
286C22×C4—93no—
287C4⋊C4—93no—
288C4⋊C4C2×(C3⋊D3)⋊C313yescomplemented
289C22⋊C4—93no—
290C22⋊C4—93no—
291C22⋊C4—93no—
292C22⋊C4C2×(C3⋊D3)⋊C313yes—
293C42C2×(C3⋊D3)⋊C311yescharacteristic, complemented
294C22×C3—23no—
295C22×C3—31no—
296C22×C3—31no—
297C22×C3—63no—
298C22×C3—63no—
299C22×C3—61no—
300C22×C3—123no—
301C22×C3—183no—
302C22×C3—183no—
303C22×C3—183no—
304C22×C3—183no—
305C22×C3—183no—
306C22×C3—364no—
307C22×C3C23×C3×D311yescharacteristic
308C2×D3—63no—
309C2×D3—63no—
310C2×D3—63no—
311C2×D3—63no—
312C2×D3—63no—
313C2×D3—124no—
314C2×D3—183no—
315C2×D3—183no—
316C2×D3—183no—
317C2×D3—183no—
318C2×D3—183no—
319C2×D3—364no—
320C3×C4—23no—
321C3×C4—23no—
322C3×C4—63no—
323C3×C4—63no—
324C3×C4—63no—
325C3×C4—63no—
326C3×C4—123no—
327C3×C4—123no—
328C3×C4—183no—
329Dic3—63no—
330Dic3—183no—
331C32—21no—
332C32#571111yescharacteristic, complemented
333C32C3×#56011yescharacteristic, complemented
334C23—91no—
335C23—91no—
336C23—93no—
337C23—363no—
338C23C22×(C3⋊D3)⋊C311yescharacteristic
339D4—363no—
340D4—363no—
341D4—363no—
342D4—363no—
343C2×C4—93no—
344C2×C4—363no—
345C2×C4C22×(C3⋊D3)⋊C313yes—
346C2×C4C22×(C3⋊D3)⋊C313yes—
347C2×C3—33no—
348C2×C3—33no—
349C2×C3—41no—
350C2×C3—63no—
351C2×C3—121no—
352C2×C3—121no—
353C2×C3—241no—
354C2×C3—361no—
355C2×C3—361no—
356C2×C3—363no—
357C2×C3C3×#100913yes—
358D3—121no—
359D3—121no—
360D3—123no—
361D3—361no—
362D3—361no—
363D3—363no—
364C22—23no—
365C22—183no—
366C22—183no—
367C22—183no—
368C22—183no—
369C22—183no—
370C22—364no—
371C22C23×(C3⋊D3)⋊C311yescharacteristic
372C4—23no—
373C4—23no—
374C4—183no—
375C3—31no—
376C3—31no—
377C3—61no—
378C3C3×#571111yescharacteristic
379C2—41no—
380C2—361no—
381C2—361no—
382C2—363no—
383C2#1172263813yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#126C2
hypercenter#126(C3⋊D3)⋊C3
hypocenterC32C3×#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#126C2semidirect
2. #64104⋊C3C2semidirect
3. #64103⋊C3C2semidirect
4. #64093⋊C3C2semidirect
5. #64091⋊C3C2semidirect
6. #11721387C2semidirect
7. #11721385C2semidirect
8. #10596182C3semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
15. C2×C4×(C3⋊D3)⋊C3C22semidirect
16. C4⋊C4×H(3,𝔽3)C22semidirect
18. C42×H(3,𝔽3)C22semidirect
21. #3752⋊C3C22semidirect
22. #3750⋊C3C22semidirect
24. C32×#126D3semidirect
25. C32×#126C2×C3semidirect
26. #64104C2×C3semidirect
27. #64103C2×C3semidirect
28. #64093C2×C3semidirect
29. #64091C2×C3semidirect
30. #64085C2×C3semidirect
31. #64083C2×C3semidirect
59. C2×C4⋊(C3⋊D3)C22×C3semidirect
60. C2×C4×C3⋊D3C22×C3semidirect
67. C32×C4⋊C4C2×D3semidirect
68. C32×C4⋊C4C22×C3semidirect
73. C32×C42C2×D3semidirect
74. C32×C42C22×C3semidirect
77. #3752C22×C3semidirect
78. #3750C22×C3semidirect
221. #126(C3⋊D3)⋊C3semidirect
226. H(3,𝔽3)#560semidirect
288. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
293. C42C2×(C3⋊D3)⋊C3semidirect
332. C32#5711semidirect
333. C32C3×#560semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1262
#64104⋊C32
#64103⋊C32
#64093⋊C32
#64091⋊C32
#117213872
#117213852
#105961823
C3×#57113