Finite Group Databasebeta

← #10592721⋊C3 #13142792

Subgroups — #10592721⋊C3 #13142792

Order 1728.

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

normal subgroups only

This group has 385 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
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic
3#63662⋊C3C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic
5#63635⋊C3C211yesmaximal, characteristic
6#63634⋊C3C211yesmaximal, characteristic
7C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic
8#10592721C311yesmaximal, characteristic, complemented
9C3×#5022—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C4⋊C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C42×H(3,𝔽3)—21no—
14C42×H(3,𝔽3)C2211yescharacteristic
15C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
16C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
17#3750⋊C3—21no—
18#3750⋊C3—21no—
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C32×C4×Q8—21no—
22C32×C4×Q8D311yescharacteristic
23C32×C4×Q8C2×C311yescharacteristic
24#64079C2×C311yescharacteristic
25C3×#870—31no—
26#63662C2×C311yescharacteristic
27#63662C2×C311yescharacteristic
28#63635C2×C311yescharacteristic
29#63634C2×C311yescharacteristic
30C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic
31C3×#835—31no—
32C3×#835—31no—
33C3×#808—31no—
34C3×#807—31no—
35C3×C4⋊(C3⋊C8)—31no—
36Q8×H(3,𝔽3)—21no—
37Q8×H(3,𝔽3)Q812yes—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)D411yescharacteristic
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3—21no—
44C2×(C3⋊Dic3)⋊C3—21no—
45(C3⋊(C3⋊C8))⋊C3—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47C3×#479—91no—
48#5022—31no—
49#5022—91no—
50C2×C32×Q8—21no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C32×Q8C22×C311yescharacteristic
53C32×C4⋊C4—22no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4C2×D311yescharacteristic
58C32×C4⋊C4C22×C311yescharacteristic
59C32×C42—22no—
60C32×C42—21no—
61C32×C42—21no—
62C32×C42—21no—
63C32×C42C2×D311yescharacteristic
64C32×C42C22×C311yescharacteristic
65C4⋊(C3⋊Dic3)C22×C311yescharacteristic
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67#3750—21no—
68#3750—21no—
69C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71C3×C4⋊Dic3—31no—
72C3×C4⋊Dic3—31no—
73C3×Dic3⋊C4—61no—
74C3×Dic3⋊C4—61no—
75C2×C3×C3⋊C8—31no—
76C2×C3×C3⋊C8—31no—
77C22×H(3,𝔽3)C2×D411yescharacteristic
78C4×H(3,𝔽3)—22no—
79C4×H(3,𝔽3)—21no—
80C4×H(3,𝔽3)—21no—
81C4×H(3,𝔽3)—41no—
82C4×H(3,𝔽3)C2×Q811yescharacteristic
83C4×H(3,𝔽3)#4111yescharacteristic
84(C3⋊Dic3)⋊C3—41no—
85(C3⋊Dic3)⋊C3—41no—
86C3×#125—91no—
87C3×C4×Q8—31no—
88C3×C4×Q8—31no—
89C3×C4×Q8—61no—
90C3×C4×Q8C3×D311yescharacteristic
91#870—31no—
92#870—91no—
93C3×#107—91no—
94C3×#106—91no—
95C3×C4⋊C8—91no—
96C3×#103—91no—
97C3×#103—91no—
98#835—31no—
99#835—31no—
100#835—91no—
101#835—91no—
102#808—31no—
103#808—91no—
104#807—31no—
105#807—91no—
106C4⋊(C3⋊C8)—31no—
107C4⋊(C3⋊C8)—91no—
108C32×Q8—22no—
109C32×Q8—21no—
110C32×Q8—21no—
111C32×Q8—22no—
112C32×Q8Dic612yes—
113C32×Q8C3×Q812yes—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—22no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4#6711yescharacteristic
124C2×C32×C4#6711yescharacteristic
125C2×C32×C4C3×D411yescharacteristic
126C2×C32×C4C3×D411yescharacteristic
127C2×C32×C4C22×D311yescharacteristic
128C2×C32×C4C23×C311yescharacteristic
129C2×C3⋊Dic3—21no—
130C2×C3⋊Dic3—21no—
131C2×C3×Dic3—61no—
132C2×C3×Dic3—61no—
133C3⋊(C3⋊C8)—21no—
134C3⋊(C3⋊C8)—21no—
135C3×C3⋊C8—61no—
136C3×C3⋊C8—61no—
137#479—91noSylow
138C2×H(3,𝔽3)C22⋊Q811yescharacteristic
139C2×H(3,𝔽3)#13511yescharacteristic
140C2×H(3,𝔽3)#13711yescharacteristic
141C2×C3×Q8—31no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×Q8C2×C3×D311yescharacteristic
145C2×C3×C8—91no—
146C2×C3×C8—91no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—31no—
149C3×C4⋊C4—31no—
150C3×C4⋊C4—61no—
151C3×C4⋊C4—61no—
152C3×C4⋊C4—62no—
153C3×C4⋊C4—61no—
154C3×C4⋊C4—91no—
155C3×C4⋊C4—91no—
156C3×C4⋊C4—181no—
157C3×C4⋊C4—181no—
158C3×C4⋊C4C2×C3×D311yescharacteristic
159C3×C42—21no—
160C3×C42—31no—
161C3×C42—31no—
162C3×C42—61no—
163C3×C42—61no—
164C3×C42—62no—
165C3×C42—61no—
166C3×C42C2×C3×D311yescharacteristic
167C4⋊Dic3—31no—
168C4⋊Dic3—31no—
169C4⋊Dic3—91no—
170C4⋊Dic3—91no—
171Dic3⋊C4—61no—
172Dic3⋊C4—61no—
173Dic3⋊C4—181no—
174Dic3⋊C4—181no—
175C2×C3⋊C8—31no—
176C2×C3⋊C8—31no—
177C2×C3⋊C8—91no—
178C2×C3⋊C8—91no—
179C22×C32—21no—
180C22×C32C2×#6711yescharacteristic
181C22×C32C2×C3×D411yescharacteristic
182C32×C4—22no—
183C32×C4—24no—
184C32×C4—22no—
185C32×C4—21no—
186C32×C4—21no—
187C32×C4—22no—
188C32×C4—21no—
189C32×C4—21no—
190C32×C4—21no—
191C32×C4—21no—
192C32×C4—41no—
193C32×C4—42no—
194C32×C4—41no—
195C32×C4—41no—
196C32×C4C2×Dic611yescharacteristic
197C32×C4#23511yescharacteristic
198C32×C4C2×C3×Q811yescharacteristic
199C32×C4C3×#4111yescharacteristic
200C3⋊Dic3—41no—
201C3⋊Dic3—41no—
202C3×Dic3—121no—
203C3×Dic3—121no—
204#125—91no—
205C4×Q8(C3⋊D3)⋊C311yescharacteristic
206#107—91no—
207#106—91no—
208C4⋊C8—91no—
209#103—91no—
210#103—91no—
211H(3,𝔽3)#47911yescharacteristic, complemented, Sylow
212C3×Q8—21no—
213C3×Q8—32no—
214C3×Q8—32no—
215C3×Q8—61no—
216C3×Q8—61no—
217C3×Q8—62no—
218C3×Q8—62no—
219C3×Q8C3×Dic612yes—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—31no—
226C2×C3×C4—31no—
227C2×C3×C4—31no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—62no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—62no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—181no—
238C2×C3×C4—181no—
239C2×C3×C4C3×#6711yescharacteristic
240C2×C3×C4C3×#6711yescharacteristic
241C2×C3×C4C22×C3×D311yescharacteristic
242C2×Dic3—61no—
243C2×Dic3—61no—
244C2×Dic3—181no—
245C2×Dic3—181no—
246C3×C8—181no—
247C3×C8—181no—
248C3⋊C8—61no—
249C3⋊C8—61no—
250C3⋊C8—181no—
251C3⋊C8—181no—
252C2×C32—21no—
253C2×C32—21no—
254C2×C32—21no—
255C2×C32#92411yescharacteristic
256C2×C32#94211yescharacteristic
257C2×C32#95011yescharacteristic
258C2×C32C3×C22⋊Q811yescharacteristic
259C2×C32C3×#13511yescharacteristic
260C2×C32C3×#13711yescharacteristic
261C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
262C2×C8—91no—
263C2×C8—91no—
264C4⋊C4—21no—
265C4⋊C4—91no—
266C4⋊C4—91no—
267C4⋊C4—181no—
268C4⋊C4—181no—
269C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
270C42—21no—
271C42C2×(C3⋊D3)⋊C311yescharacteristic
272C22×C3—31no—
273C22×C3—31no—
274C22×C3—61no—
275C22×C3C2×C3×#6711yescharacteristic
276C3×C4—22no—
277C3×C4—21no—
278C3×C4—21no—
279C3×C4—31no—
280C3×C4—31no—
281C3×C4—31no—
282C3×C4—31no—
283C3×C4—41no—
284C3×C4—61no—
285C3×C4—61no—
286C3×C4—61no—
287C3×C4—61no—
288C3×C4—62no—
289C3×C4—62no—
290C3×C4—64no—
291C3×C4—62no—
292C3×C4—61no—
293C3×C4—61no—
294C3×C4—121no—
295C3×C4—121no—
296C3×C4—121no—
297C3×C4—122no—
298C3×C4—361no—
299C3×C4—361no—
300C3×C4C2×C3×Dic611yescharacteristic
301C3×C4C3×#23511yescharacteristic
302Dic3—121no—
303Dic3—121no—
304Dic3—361no—
305Dic3—361no—
306C32—21no—
307C32#502211yescharacteristic, complemented
308C32C3×#47911yescharacteristic, complemented
309Q8—21no—
310Q8#634⋊C312yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—181no—
314C2×C4—181no—
315C2×C4#638⋊C311yescharacteristic
316C2×C4#638⋊C311yescharacteristic
317C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
318C8—181no—
319C8—181no—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—61no—
327C2×C3—61no—
328C2×C3—61no—
329C2×C3C3×#92411yescharacteristic
330C2×C3C3×#94211yescharacteristic
331C2×C3C3×#95011yescharacteristic
332C22C2×#638⋊C311yescharacteristic
333C4—22no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—361no—
338C4—361no—
339C4C2×#634⋊C311yescharacteristic
340C4#3828⋊C311yescharacteristic
341C3—31no—
342C3—31no—
343C3—61no—
344C3C3×#502211yescharacteristic
345C2#64133⋊C311yescharacteristic
346C2#64151⋊C311yescharacteristic
347C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC22×H(3,𝔽3)C2×D4
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#479
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
8. #10592721C3semidirect
211. H(3,𝔽3)#479semidirect
307. C32#5022semidirect
308. C32C3×#479semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×H(3,𝔽3)2
#64079⋊C32
#63662⋊C32
#63635⋊C32
#63634⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927213
C3×#50223