Finite Group Databasebeta

← #10592708⋊C3 #13142743

Subgroups — #10592708⋊C3 #13142743

Order 1728.

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

normal subgroups only

This group has 451 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×D4×H(3,𝔽3)C211yesmaximal, characteristic
2#64079⋊C3C211yesmaximal, characteristic, complemented
3#63659⋊C3C211yesmaximal, characteristic
4#63659⋊C3C211yesmaximal, characteristic
5#63635⋊C3C211yesmaximal, characteristic, complemented
6#63634⋊C3C211yesmaximal, characteristic, complemented
7C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
8#10592708C311yesmaximal, characteristic, complemented
9C3×#5009—31nomaximal
10C2×D4×H(3,𝔽3)C2211yescharacteristic
11C22×C4×H(3,𝔽3)—21no—
12C4⋊C4×H(3,𝔽3)C2211yescharacteristic
13C22⋊C4×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×D4—21no—
22C32×C4×D4D311yescharacteristic
23C32×C4×D4C2×C311yescharacteristic
24#64079C2×C311yescharacteristic, complemented
25C3×#870—31no—
26#63659C2×C311yescharacteristic
27#63659C2×C311yescharacteristic
28#63635C2×C311yescharacteristic, complemented
29#63634C2×C311yescharacteristic, complemented
30C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
31C3×#832—31no—
32C3×#832—31no—
33C3×#808—31no—
34C3×#807—31no—
35C3×C4⋊(C3⋊C8)—31no—
36C23×H(3,𝔽3)—21no—
37D4×H(3,𝔽3)—21no—
38D4×H(3,𝔽3)Q812yes—
39C2×C4×H(3,𝔽3)—21no—
40C2×C4×H(3,𝔽3)—42no—
41C2×C4×H(3,𝔽3)D411yescharacteristic
42C2×C4×H(3,𝔽3)D411yescharacteristic
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3—21no—
45C2×(C3⋊Dic3)⋊C3—21no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48C3×#478—91no—
49#5009—31no—
50#5009—91no—
51C2×C32×D4—21no—
52C2×C32×D4C2×D311yescharacteristic
53C2×C32×D4C22×C311yescharacteristic
54C22×C32×C4—22no—
55C22×C32×C4—21no—
56C22×C32×C4—21no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C4⋊C4C22×C311yescharacteristic
60C32×C22⋊C4—22no—
61C32×C22⋊C4—21no—
62C32×C22⋊C4—21no—
63C32×C42—21no—
64C32×C42C2×D311yescharacteristic
65C32×C42C22×C311yescharacteristic
66C4⋊(C3⋊Dic3)C22×C311yescharacteristic
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic
68#3750—21no—
69#3750—21no—
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72C3×C4⋊Dic3—31no—
73C3×C4⋊Dic3—31no—
74C3×Dic3⋊C4—61no—
75C3×Dic3⋊C4—61no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C22×H(3,𝔽3)—22no—
79C22×H(3,𝔽3)—41no—
80C22×H(3,𝔽3)—41no—
81C22×H(3,𝔽3)C2×D411yescharacteristic
82C4×H(3,𝔽3)—21no—
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)C2×Q811yescharacteristic
86C4×H(3,𝔽3)#4111yescharacteristic
87(C3⋊Dic3)⋊C3—41no—
88(C3⋊Dic3)⋊C3—41no—
89C3×#125—91no—
90C3×C4×D4—31no—
91C3×C4×D4—31no—
92C3×C4×D4—61no—
93C3×C4×D4C3×D311yescharacteristic
94#870—31no—
95#870—91no—
96C3×#107—91no—
97C3×#106—91no—
98C3×C4⋊C8—91no—
99C3×#102—91no—
100C3×#102—91no—
101#832—31no—
102#832—31no—
103#832—91no—
104#832—91no—
105#808—31no—
106#808—91no—
107#807—31no—
108#807—91no—
109C4⋊(C3⋊C8)—31no—
110C4⋊(C3⋊C8)—91no—
111C23×C32—22no—
112C23×C32—21no—
113C23×C32—21no—
114C32×D4—22no—
115C32×D4—21no—
116C32×D4—21no—
117C32×D4—22no—
118C32×D4Dic612yes—
119C32×D4C3×Q812yes—
120C2×C32×C4—22no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4—21no—
125C2×C32×C4—21no—
126C2×C32×C4—44no—
127C2×C32×C4—42no—
128C2×C32×C4—42no—
129C2×C32×C4#6711yescharacteristic
130C2×C32×C4#6711yescharacteristic
131C2×C32×C4C3×D411yescharacteristic
132C2×C32×C4C3×D411yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C23×C311yescharacteristic
135C2×C3⋊Dic3—21no—
136C2×C3⋊Dic3—21no—
137C2×C3×Dic3—61no—
138C2×C3×Dic3—61no—
139C3⋊(C3⋊C8)—21no—
140C3⋊(C3⋊C8)—21no—
141C3×C3⋊C8—61no—
142C3×C3⋊C8—61no—
143#478—91noSylow
144C2×H(3,𝔽3)—42no—
145C2×H(3,𝔽3)C22⋊Q811yescharacteristic
146C2×H(3,𝔽3)#13511yescharacteristic
147C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
148C2×C3×D4—31no—
149C2×C3×D4—31no—
150C2×C3×D4—61no—
151C2×C3×D4C2×C3×D311yescharacteristic
152C22×C3×C4—21no—
153C22×C3×C4—61no—
154C22×C3×C4—61no—
155C22×C3×C4—62no—
156C2×C3×C8—91no—
157C2×C3×C8—91no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—61no—
161C3×C4⋊C4—91no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4—181no—
164C3×C4⋊C4—181no—
165C3×C4⋊C4C2×C3×D311yescharacteristic
166C3×C22⋊C4—21no—
167C3×C22⋊C4—61no—
168C3×C22⋊C4—61no—
169C3×C22⋊C4—62no—
170C3×C42—31no—
171C3×C42—31no—
172C3×C42—61no—
173C3×C42C2×C3×D311yescharacteristic
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—91no—
177C4⋊Dic3—91no—
178Dic3⋊C4—61no—
179Dic3⋊C4—61no—
180Dic3⋊C4—181no—
181Dic3⋊C4—181no—
182C2×C3⋊C8—31no—
183C2×C3⋊C8—31no—
184C2×C3⋊C8—91no—
185C2×C3⋊C8—91no—
186C22×C32—24no—
187C22×C32—22no—
188C22×C32—22no—
189C22×C32—21no—
190C22×C32—42no—
191C22×C32—42no—
192C22×C32—41no—
193C22×C32—41no—
194C22×C32—41no—
195C22×C32—41no—
196C22×C32C2×#6711yescharacteristic
197C22×C32C2×C3×D411yescharacteristic
198C32×C4—22no—
199C32×C4—21no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—21no—
205C32×C4—41no—
206C32×C4—42no—
207C32×C4—41no—
208C32×C4—41no—
209C32×C4C2×Dic611yescharacteristic
210C32×C4#23511yescharacteristic
211C32×C4C2×C3×Q811yescharacteristic
212C32×C4C3×#4111yescharacteristic
213C3⋊Dic3—41no—
214C3⋊Dic3—41no—
215C3×Dic3—121no—
216C3×Dic3—121no—
217#125—91no—
218C4×D4(C3⋊D3)⋊C311yescharacteristic
219#107—91no—
220#106—91no—
221C4⋊C8—91no—
222#102—91no—
223#102—91no—
224H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
225C23×C3—21no—
226C23×C3—61no—
227C23×C3—61no—
228C23×C3—62no—
229C3×D4—21no—
230C3×D4—32no—
231C3×D4—32no—
232C3×D4—61no—
233C3×D4—61no—
234C3×D4—62no—
235C3×D4—62no—
236C3×D4C3×Dic612yes—
237C2×C3×C4—21no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—31no—
242C2×C3×C4—31no—
243C2×C3×C4—31no—
244C2×C3×C4—42no—
245C2×C3×C4—61no—
246C2×C3×C4—61no—
247C2×C3×C4—62no—
248C2×C3×C4—61no—
249C2×C3×C4—61no—
250C2×C3×C4—61no—
251C2×C3×C4—122no—
252C2×C3×C4—122no—
253C2×C3×C4—124no—
254C2×C3×C4—181no—
255C2×C3×C4—181no—
256C2×C3×C4C3×#6711yescharacteristic
257C2×C3×C4C3×#6711yescharacteristic
258C2×C3×C4C22×C3×D311yescharacteristic
259C2×Dic3—61no—
260C2×Dic3—61no—
261C2×Dic3—181no—
262C2×Dic3—181no—
263C3×C8—181no—
264C3×C8—181no—
265C3⋊C8—61no—
266C3⋊C8—61no—
267C3⋊C8—181no—
268C3⋊C8—181no—
269C2×C32—21no—
270C2×C32—21no—
271C2×C32—21no—
272C2×C32—44no—
273C2×C32—42no—
274C2×C32—42no—
275C2×C32#92411yescharacteristic
276C2×C32#93211yescharacteristic
277C2×C32#95011yescharacteristic
278C2×C32C3×C22⋊Q811yescharacteristic
279C2×C32C3×#13511yescharacteristic
280C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
281C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
282C22×C4—21no—
283C2×C8—91no—
284C2×C8—91no—
285C4⋊C4—91no—
286C4⋊C4—91no—
287C4⋊C4—181no—
288C4⋊C4—181no—
289C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
290C22⋊C4—21no—
291C42C2×(C3⋊D3)⋊C311yescharacteristic
292C22×C3—22no—
293C22×C3—31no—
294C22×C3—31no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—62no—
298C22×C3—62no—
299C22×C3—64no—
300C22×C3—61no—
301C22×C3—121no—
302C22×C3—121no—
303C22×C3—121no—
304C22×C3—121no—
305C22×C3—122no—
306C22×C3—122no—
307C22×C3C2×C3×#6711yescharacteristic
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—31no—
311C3×C4—31no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—41no—
315C3×C4—61no—
316C3×C4—61no—
317C3×C4—61no—
318C3×C4—61no—
319C3×C4—62no—
320C3×C4—61no—
321C3×C4—61no—
322C3×C4—121no—
323C3×C4—121no—
324C3×C4—121no—
325C3×C4—122no—
326C3×C4—361no—
327C3×C4—361no—
328C3×C4C2×C3×Dic611yescharacteristic
329C3×C4C3×#23511yescharacteristic
330Dic3—121no—
331Dic3—121no—
332Dic3—361no—
333Dic3—361no—
334C32—21no—
335C32#500911yescharacteristic, complemented
336C32C3×#47811yescharacteristic, complemented
337C23—21no—
338D4—21no—
339D4#634⋊C312yes—
340C2×C4—21no—
341C2×C4—42no—
342C2×C4—181no—
343C2×C4—181no—
344C2×C4#638⋊C311yescharacteristic
345C2×C4#638⋊C311yescharacteristic
346C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
347C8—181no—
348C8—181no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—42no—
356C2×C3—61no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—122no—
360C2×C3—122no—
361C2×C3—124no—
362C2×C3C3×#92411yescharacteristic
363C2×C3C3×#93211yescharacteristic
364C2×C3C3×#95011yescharacteristic
365C22—22no—
366C22—41no—
367C22—41no—
368C22C2×#638⋊C311yescharacteristic
369C4—21no—
370C4—21no—
371C4—41no—
372C4—361no—
373C4—361no—
374C4C2×#634⋊C311yescharacteristic
375C4#3828⋊C311yescharacteristic
376C3—31no—
377C3—31no—
378C3—61no—
379C3C3×#500911yescharacteristic
380C2—42no—
381C2#64133⋊C311yescharacteristic
382C2#64141⋊C311yescharacteristic
383C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingC4×D4×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC2×D4×H(3,𝔽3)C22
hypercenterC4×D4(C3⋊D3)⋊C3
hypocenterC32C3×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #64079⋊C3C2semidirect
5. #63635⋊C3C2semidirect
6. #63634⋊C3C2semidirect
7. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
8. #10592708C3semidirect
24. #64079C2×C3semidirect
28. #63635C2×C3semidirect
29. #63634C2×C3semidirect
30. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
224. H(3,𝔽3)#478semidirect
335. C32#5009semidirect
336. C32C3×#478semidirect

Maximal subgroups

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

subgroupindex
C4×D4×H(3,𝔽3)2
#64079⋊C32
#63659⋊C32
#63635⋊C32
#63634⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927083
C3×#50093