Finite Group Databasebeta

← #10592814⋊C3 #13143130

Subgroups — #10592814⋊C3 #13143130

Order 1728.

545 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C22×C8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64139⋊C3C212yesmaximal, complemented
3C2×#3745⋊C3C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#63645⋊C3C211yesmaximal, characteristic, complemented
6#10592814C311yesmaximal, characteristic, complemented
7C3×#5115—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—22no—
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C2×C8×H(3,𝔽3)—22no—
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C2×C8×H(3,𝔽3)C2211yescharacteristic
14#3752⋊C3—22no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16#3745⋊C3—22no—
17C22×C32×C8—21no—
18C22×C32×C8D311yescharacteristic, complemented
19C22×C32×C8C2×C311yescharacteristic, complemented
20#64139C2×C312yescomplemented
21C2×#3745C2×C311yescharacteristic, complemented
22C3×#930—32no—
23C2×C3×D24—31no—
24#63648C2×C312yescomplemented
25#63645C2×C311yescharacteristic, complemented
26C3×#821—32no—
27C3×#818—31no—
28C23×H(3,𝔽3)D411yescharacteristic
29C22×(C3⋊D3)⋊C3—22no—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)D411yescharacteristic
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34#638⋊C3—44no—
35C2×(C3⋊Dic3)⋊C3—22no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37C4⋊((C3⋊D3)⋊C3)—22no—
38C8×H(3,𝔽3)—21no—
39C8×H(3,𝔽3)D412yescomplemented
40C3×#466—91no—
41#5115—31no—
42#5115—91no—
43C22×C32×C4—21no—
44C22×C32×C4C2×D311yescharacteristic
45C22×C32×C4C22×C311yescharacteristic
46C2×#638—22no—
47C2×C4⋊(C3⋊D3)C22×C312yes—
48C2×C3×#67—62no—
49C2×C3×D12—32no—
50C2×C32×C8—24no—
51C2×C32×C8—21no—
52C2×C32×C8—21no—
53C2×C32×C8—22no—
54C2×C32×C8—22no—
55C2×C32×C8C2×D311yescharacteristic
56C2×C32×C8C2×D311yescharacteristic
57C2×C32×C8C22×C311yescharacteristic
58C2×C32×C8C22×C311yescharacteristic
59#3752—22no—
60C4⋊(C3⋊Dic3)C22×C312yes—
61#3745—22no—
62C3×(C2×D3)⋊C4—62no—
63C3×C4⋊Dic3—32no—
64C3×D24—62no—
65C22×H(3,𝔽3)—21no—
66C22×H(3,𝔽3)—21no—
67C22×H(3,𝔽3)D812yescomplemented
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C2×(C3⋊D3)⋊C3—42no—
70C2×(C3⋊D3)⋊C3—42no—
71C2×(C3⋊D3)⋊C3—42no—
72C4×H(3,𝔽3)—21no—
73C4×H(3,𝔽3)C2×D411yescharacteristic
74C4×H(3,𝔽3)#4111yescharacteristic
75(C3⋊Dic3)⋊C3—42no—
76C2×C3×D8—91no—
77C22×C3×C8—31no—
78C22×C3×C8—31no—
79C22×C3×C8—61no—
80C22×C3×C8C3×D311yescharacteristic
81C3×#121—92no—
82#930—32no—
83#930—92no—
84C2×D24—31no—
85C2×D24—91no—
86C3×#107—91no—
87C3×#102—92no—
88#821—32no—
89#821—92no—
90#818—31no—
91#818—91no—
92C23×C32—21no—
93C23×C32D1211yescharacteristic
94C23×C32C3×D411yescharacteristic
95C22×C3⋊D3—22no—
96C22×C3×D3—62no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4D1211yescharacteristic
106C2×C32×C4C3×D411yescharacteristic
107C2×C32×C4C22×D311yescharacteristic
108C2×C32×C4C23×C311yescharacteristic
109#638—44no—
110C2×C3⋊Dic3—22no—
111C4⋊(C3⋊D3)—22no—
112C4⋊(C3⋊D3)—22no—
113C3×#67—124no—
114C2×C3×Dic3—62no—
115C3×D12—62no—
116C3×D12—62no—
117C32×C8—22no—
118C32×C8—21no—
119C32×C8—21no—
120C32×C8—22no—
121C32×C8#6712yescomplemented
122C32×C8C3×D412yescomplemented
123#466—91noSylow
124C2×H(3,𝔽3)—22no—
125C2×H(3,𝔽3)#12111yescharacteristic
126C2×H(3,𝔽3)C2×D811yescharacteristic
127C2×H(3,𝔽3)#13511yescharacteristic
128(C3⋊D3)⋊C3—82no—
129C2×C3×D4—92no—
130C2×C3×D4—182no—
131C22×C3×C4—31no—
132C22×C3×C4—31no—
133C22×C3×C4—61no—
134C22×C3×C4C2×C3×D311yescharacteristic
135C2×#67—62no—
136C2×#67—182no—
137C2×D12—32no—
138C2×D12—92no—
139C3×D8—182no—
140C2×C3×C8—22no—
141C2×C3×C8—31no—
142C2×C3×C8—31no—
143C2×C3×C8—31no—
144C2×C3×C8—31no—
145C2×C3×C8—62no—
146C2×C3×C8—62no—
147C2×C3×C8—64no—
148C2×C3×C8—61no—
149C2×C3×C8—61no—
150C2×C3×C8C2×C3×D311yescharacteristic
151C2×C3×C8C2×C3×D311yescharacteristic
152C3×C4⋊C4—92no—
153C3×C22⋊C4—182no—
154(C2×D3)⋊C4—62no—
155(C2×D3)⋊C4—182no—
156C4⋊Dic3—32no—
157C4⋊Dic3—92no—
158D24—62no—
159D24—182no—
160C22×C32—22no—
161C22×C32—22no—
162C22×C32—21no—
163C22×C32—21no—
164C22×C32—21no—
165C22×C32—21no—
166C22×C32—22no—
167C22×C32—21no—
168C22×C32D2412yescomplemented
169C22×C32C3×D812yescomplemented
170C22×C32C2×D1211yescharacteristic
171C22×C32C2×C3×D411yescharacteristic
172C2×C3⋊D3—42no—
173C2×C3⋊D3—42no—
174C2×C3⋊D3—42no—
175C2×C3×D3—122no—
176C2×C3×D3—122no—
177C2×C3×D3—122no—
178C32×C4—22no—
179C32×C4—21no—
180C32×C4—21no—
181C32×C4—21no—
182C32×C4—21no—
183C32×C4#23511yescharacteristic
184C32×C4C2×#6711yescharacteristic
185C32×C4C2×C3×D411yescharacteristic
186C32×C4C3×#4111yescharacteristic
187C3⋊Dic3—42no—
188C3×Dic3—122no—
189C2×D8—91no—
190C22×C8(C3⋊D3)⋊C311yescharacteristic, complemented
191#121—92no—
192#107—91no—
193#102—92no—
194H(3,𝔽3)#46611yescharacteristic, complemented, Sylow
195C23×C3—31no—
196C23×C3—31no—
197C23×C3—61no—
198C23×C3—182no—
199C23×C3C3×D1211yescharacteristic
200C22×D3—62no—
201C22×D3—182no—
202C3×D4—182no—
203C3×D4—182no—
204C3×D4—364no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—61no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—61no—
215C2×C3×C4—62no—
216C2×C3×C4—62no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—182no—
220C2×C3×C4C3×D1211yescharacteristic
221C2×C3×C4C22×C3×D311yescharacteristic
222#67—124no—
223#67—364no—
224C2×Dic3—62no—
225C2×Dic3—182no—
226D12—62no—
227D12—62no—
228D12—182no—
229D12—182no—
230C3×C8—21no—
231C3×C8—32no—
232C3×C8—32no—
233C3×C8—61no—
234C3×C8—61no—
235C3×C8—62no—
236C3×C8—62no—
237C3×C8C3×#6712yes—
238C2×C32—24no—
239C2×C32—22no—
240C2×C32—22no—
241C2×C32—21no—
242C2×C32—21no—
243C2×C32—21no—
244C2×C32C2×D2411yescharacteristic
245C2×C32#90411yescharacteristic
246C2×C32#93011yescharacteristic
247C2×C32C3×#12111yescharacteristic
248C2×C32C2×C3×D811yescharacteristic
249C2×C32C3×#13511yescharacteristic
250C3⋊D3—82no—
251C3×D3—242no—
252C2×D4—92no—
253C2×D4—182no—
254C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
255D8—182no—
256C2×C8—22no—
257C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
258C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
259C4⋊C4—92no—
260C22⋊C4—182no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—31no—
264C22×C3—31no—
265C22×C3—32no—
266C22×C3—32no—
267C22×C3—61no—
268C22×C3—61no—
269C22×C3—61no—
270C22×C3—61no—
271C22×C3—62no—
272C22×C3—62no—
273C22×C3—61no—
274C22×C3—62no—
275C22×C3—362no—
276C22×C3—362no—
277C22×C3—362no—
278C22×C3C3×D2412yes—
279C22×C3C2×C3×D1211yescharacteristic
280C2×D3—122no—
281C2×D3—122no—
282C2×D3—122no—
283C2×D3—362no—
284C2×D3—362no—
285C2×D3—362no—
286C3×C4—21no—
287C3×C4—31no—
288C3×C4—31no—
289C3×C4—31no—
290C3×C4—31no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—62no—
294C3×C4—61no—
295C3×C4—61no—
296C3×C4—362no—
297C3×C4C3×#23511yescharacteristic
298C3×C4C2×C3×#6711yescharacteristic
299Dic3—122no—
300Dic3—362no—
301C32—21no—
302C32#511511yescharacteristic, complemented
303C32C3×#46611yescharacteristic, complemented
304C23—182no—
305C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
306D4—182no—
307D4—182no—
308D4—364no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—182no—
312C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
313C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
314C8—21no—
315C8#638⋊C312yescomplemented
316C2×C3—22no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—31no—
323C2×C3—62no—
324C2×C3—62no—
325C2×C3—64no—
326C2×C3—61no—
327C2×C3—61no—
328C2×C3—61no—
329C2×C3—722no—
330C2×C3C2×C3×D2411yescharacteristic
331C2×C3C3×#90411yescharacteristic
332C2×C3C3×#93011yescharacteristic
333D3—242no—
334D3—722no—
335C22—21no—
336C22—21no—
337C22—362no—
338C22—362no—
339C22—362no—
340C22#3745⋊C312yescomplemented
341C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
342C4—21no—
343C4—362no—
344C4#3828⋊C311yescharacteristic
345C4C2×#638⋊C311yescharacteristic
346C3—31no—
347C3—31no—
348C3—61no—
349C3C3×#511511yescharacteristic
350C2—22no—
351C2—722no—
352C2C2×#3745⋊C311yescharacteristic
353C2#64113⋊C311yescharacteristic
354C2#64139⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC22×C8×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
hypercenterC22×C8(C3⋊D3)⋊C3
hypocenterC32C3×#466
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C22×C8×H(3,𝔽3)C2semidirect
2. #64139⋊C3C2semidirect
3. C2×#3745⋊C3C2semidirect
4. #63648⋊C3C2semidirect
5. #63645⋊C3C2semidirect
6. #10592814C3semidirect
18. C22×C32×C8D3semidirect
19. C22×C32×C8C2×C3semidirect
20. #64139C2×C3semidirect
21. C2×#3745C2×C3semidirect
24. #63648C2×C3semidirect
25. #63645C2×C3semidirect
39. C8×H(3,𝔽3)D4semidirect
67. C22×H(3,𝔽3)D8semidirect
121. C32×C8#67semidirect
122. C32×C8C3×D4semidirect
168. C22×C32D24semidirect
169. C22×C32C3×D8semidirect
190. C22×C8(C3⋊D3)⋊C3semidirect
194. H(3,𝔽3)#466semidirect
302. C32#5115semidirect
303. C32C3×#466semidirect
315. C8#638⋊C3semidirect
340. C22#3745⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C8×H(3,𝔽3)2
#64139⋊C32
C2×#3745⋊C32
#63648⋊C32
#63645⋊C32
#105928143
C3×#51153