Finite Group Databasebeta

← #10592834⋊C3 #13143218

Subgroups — #10592834⋊C3 #13143218

Order 1728.

523 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 523 conjugacy classes of subgroups — too many to draw. Only its 102 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×M16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2#64139⋊C3C212yesmaximal, complemented
3C2×#3745⋊C3C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#63644⋊C3C211yesmaximal, characteristic, complemented
6#10592834C311yesmaximal, characteristic, complemented
7C3×#5135—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C2×#638⋊C3—22no—
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11M16×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—
17C2×C32×M16—21no—
18C2×C32×M16D311yescharacteristic, complemented
19C2×C32×M16C2×C311yescharacteristic, complemented
20#64139C2×C312yescomplemented
21C2×#3745C2×C311yescharacteristic, complemented
22C3×#930—32no—
23C2×C3×D24—31no—
24#63648C2×C312yescomplemented
25#63644C2×C311yescharacteristic, complemented
26C3×#821—32no—
27C3×#817—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×#469—91no—
41#5135—31no—
42#5135—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—
50C32×M16—24no—
51C32×M16—22no—
52C32×M16—22no—
53C2×C32×C8—21no—
54C2×C32×C8—21no—
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)—22no—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C2×(C3⋊D3)⋊C3—42no—
69C2×(C3⋊D3)⋊C3—42no—
70C2×(C3⋊D3)⋊C3—42no—
71C4×H(3,𝔽3)—21no—
72C4×H(3,𝔽3)C2×D411yescharacteristic
73C4×H(3,𝔽3)#4111yescharacteristic
74(C3⋊Dic3)⋊C3—42no—
75C2×C3×D8—91no—
76C2×C3×M16—31no—
77C2×C3×M16—31no—
78C2×C3×M16—61no—
79C2×C3×M16C3×D311yescharacteristic
80C3×#121—92no—
81#930—32no—
82#930—92no—
83C2×D24—31no—
84C2×D24—91no—
85C3×#106—91no—
86C3×#102—92no—
87#821—32no—
88#821—92no—
89#817—31no—
90#817—91no—
91C23×C32—21no—
92C23×C32D1211yescharacteristic
93C23×C32C3×D411yescharacteristic
94C22×C3⋊D3—22no—
95C22×C3×D3—62no—
96C2×C32×C4—22no—
97C2×C32×C4—22no—
98C2×C32×C4—21no—
99C2×C32×C4—21no—
100C2×C32×C4—21no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4D1211yescharacteristic
105C2×C32×C4C3×D411yescharacteristic
106C2×C32×C4C22×D311yescharacteristic
107C2×C32×C4C23×C311yescharacteristic
108#638—44no—
109C2×C3⋊Dic3—22no—
110C4⋊(C3⋊D3)—22no—
111C4⋊(C3⋊D3)—22no—
112C3×#67—124no—
113C2×C3×Dic3—62no—
114C3×D12—62no—
115C3×D12—62no—
116C32×C8—22no—
117C32×C8—21no—
118C32×C8—21no—
119C32×C8—22no—
120C32×C8#6712yescomplemented
121C32×C8C3×D412yescomplemented
122#469—91noSylow
123C2×H(3,𝔽3)—41no—
124C2×H(3,𝔽3)#12111yescharacteristic
125C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
126(C3⋊D3)⋊C3—82no—
127C2×C3×D4—92no—
128C2×C3×D4—182no—
129C22×C3×C4—31no—
130C22×C3×C4—31no—
131C22×C3×C4—61no—
132C22×C3×C4C2×C3×D311yescharacteristic
133C2×#67—62no—
134C2×#67—182no—
135C2×D12—32no—
136C2×D12—92no—
137C3×D8—182no—
138C3×M16—22no—
139C3×M16—62no—
140C3×M16—62no—
141C3×M16—64no—
142C2×C3×C8—31no—
143C2×C3×C8—31no—
144C2×C3×C8—31no—
145C2×C3×C8—31no—
146C2×C3×C8—61no—
147C2×C3×C8—61no—
148C2×C3×C8C2×C3×D311yescharacteristic
149C2×C3×C8C2×C3×D311yescharacteristic
150C3×C4⋊C4—92no—
151C3×C22⋊C4—182no—
152(C2×D3)⋊C4—62no—
153(C2×D3)⋊C4—182no—
154C4⋊Dic3—32no—
155C4⋊Dic3—92no—
156D24—62no—
157D24—182no—
158C22×C32—22no—
159C22×C32—22no—
160C22×C32—21no—
161C22×C32—21no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—42no—
165C22×C32C2×D1211yescharacteristic
166C22×C32C2×C3×D411yescharacteristic
167C2×C3⋊D3—42no—
168C2×C3⋊D3—42no—
169C2×C3⋊D3—42no—
170C2×C3×D3—122no—
171C2×C3×D3—122no—
172C2×C3×D3—122no—
173C32×C4—22no—
174C32×C4—21no—
175C32×C4—21no—
176C32×C4—21no—
177C32×C4—21no—
178C32×C4#23511yescharacteristic
179C32×C4C2×#6711yescharacteristic
180C32×C4C2×C3×D411yescharacteristic
181C32×C4C3×#4111yescharacteristic
182C3⋊Dic3—42no—
183C3×Dic3—122no—
184C2×D8—91no—
185C2×M16(C3⋊D3)⋊C311yescharacteristic, complemented
186#121—92no—
187#106—91no—
188#102—92no—
189H(3,𝔽3)#46911yescharacteristic, complemented, Sylow
190C23×C3—31no—
191C23×C3—31no—
192C23×C3—61no—
193C23×C3—182no—
194C23×C3C3×D1211yescharacteristic
195C22×D3—62no—
196C22×D3—182no—
197C3×D4—182no—
198C3×D4—182no—
199C3×D4—364no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—31no—
203C2×C3×C4—31no—
204C2×C3×C4—31no—
205C2×C3×C4—31no—
206C2×C3×C4—61no—
207C2×C3×C4—61no—
208C2×C3×C4—61no—
209C2×C3×C4—61no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4—61no—
213C2×C3×C4—61no—
214C2×C3×C4—182no—
215C2×C3×C4C3×D1211yescharacteristic
216C2×C3×C4C22×C3×D311yescharacteristic
217#67—124no—
218#67—364no—
219C2×Dic3—62no—
220C2×Dic3—182no—
221D12—62no—
222D12—62no—
223D12—182no—
224D12—182no—
225C3×C8—21no—
226C3×C8—32no—
227C3×C8—32no—
228C3×C8—61no—
229C3×C8—61no—
230C3×C8—62no—
231C3×C8—62no—
232C3×C8C3×#6712yes—
233C2×C32—22no—
234C2×C32—21no—
235C2×C32—42no—
236C2×C32—41no—
237C2×C32—41no—
238C2×C32#90812yes—
239C2×C32#93011yescharacteristic
240C2×C32C3×#12111yescharacteristic
241C2×C32C3×AGL(1,ℤ/8)12yes—
242C3⋊D3—82no—
243C3×D3—242no—
244C2×D4—92no—
245C2×D4—182no—
246C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
247D8—182no—
248M16—22no—
249C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
250C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
251C4⋊C4—92no—
252C22⋊C4—182no—
253C22×C3—22no—
254C22×C3—21no—
255C22×C3—31no—
256C22×C3—31no—
257C22×C3—62no—
258C22×C3—61no—
259C22×C3—61no—
260C22×C3—62no—
261C22×C3—62no—
262C22×C3—61no—
263C22×C3—122no—
264C22×C3—362no—
265C22×C3—362no—
266C22×C3—362no—
267C22×C3C2×C3×D1211yescharacteristic
268C2×D3—122no—
269C2×D3—122no—
270C2×D3—122no—
271C2×D3—362no—
272C2×D3—362no—
273C2×D3—362no—
274C3×C4—21no—
275C3×C4—31no—
276C3×C4—31no—
277C3×C4—31no—
278C3×C4—31no—
279C3×C4—61no—
280C3×C4—61no—
281C3×C4—62no—
282C3×C4—61no—
283C3×C4—61no—
284C3×C4—362no—
285C3×C4C3×#23511yescharacteristic
286C3×C4C2×C3×#6711yescharacteristic
287Dic3—122no—
288Dic3—362no—
289C32—21no—
290C32#513511yescharacteristic, complemented
291C32C3×#46911yescharacteristic, complemented
292C23—182no—
293C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
294D4—182no—
295D4—182no—
296D4—364no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—182no—
300C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
301C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
302C8—21no—
303C8#638⋊C312yescomplemented
304C2×C3—32no—
305C2×C3—32no—
306C2×C3—31no—
307C2×C3—31no—
308C2×C3—41no—
309C2×C3—62no—
310C2×C3—61no—
311C2×C3—121no—
312C2×C3—121no—
313C2×C3—122no—
314C2×C3—722no—
315C2×C3C3×#90812yes—
316C2×C3C3×#93011yescharacteristic
317D3—242no—
318D3—722no—
319C22—22no—
320C22—21no—
321C22—362no—
322C22—362no—
323C22—362no—
324C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
325C4—21no—
326C4—362no—
327C4#3828⋊C311yescharacteristic
328C4C2×#638⋊C311yescharacteristic
329C3—31no—
330C3—31no—
331C3—61no—
332C3C3×#513511yescharacteristic
333C2—41no—
334C2—722no—
335C2#64117⋊C312yes—
336C2#64139⋊C311yescharacteristic
subgroupG/H
centerC22C2×C4⋊((C3⋊D3)⋊C3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×H(3,𝔽3)C2
socleC22×C3C2×C3×D12
hypercenterC2×M16(C3⋊D3)⋊C3
hypocenterC32C3×#469
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×M16×H(3,𝔽3)C2semidirect
2. #64139⋊C3C2semidirect
3. C2×#3745⋊C3C2semidirect
4. #63648⋊C3C2semidirect
5. #63644⋊C3C2semidirect
6. #10592834C3semidirect
18. C2×C32×M16D3semidirect
19. C2×C32×M16C2×C3semidirect
20. #64139C2×C3semidirect
21. C2×#3745C2×C3semidirect
24. #63648C2×C3semidirect
25. #63644C2×C3semidirect
39. C8×H(3,𝔽3)D4semidirect
120. C32×C8#67semidirect
121. C32×C8C3×D4semidirect
185. C2×M16(C3⋊D3)⋊C3semidirect
189. H(3,𝔽3)#469semidirect
290. C32#5135semidirect
291. C32C3×#469semidirect
303. C8#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×M16×H(3,𝔽3)2
#64139⋊C32
C2×#3745⋊C32
#63648⋊C32
#63644⋊C32
#105928343
C3×#51353