Finite Group Databasebeta

← #10592706⋊C3 #13142741

Subgroups — #10592706⋊C3 #13142741

Order 1728.

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

normal subgroups only

This group has 472 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
1C4×D4×H(3,𝔽3)C211yesmaximal, characteristic
2#64078⋊C3C211yesmaximal, characteristic, complemented
3#63659⋊C3C212yesmaximal
4#63634⋊C3C212yesmaximal, complemented
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#10592706C311yesmaximal, characteristic, complemented
7C3×#5007—31nomaximal
8C2×D4×H(3,𝔽3)C2211yescharacteristic
9C22×C4×H(3,𝔽3)—21no—
10C2×#634⋊C3—21no—
11C4⋊C4×H(3,𝔽3)C2211yescharacteristic
12C22⋊C4×H(3,𝔽3)—21no—
13C42×H(3,𝔽3)C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
17C32×C4×D4—21no—
18C32×C4×D4D311yescharacteristic
19C32×C4×D4C2×C311yescharacteristic
20#64078C2×C311yescharacteristic, complemented
21C3×#869—31no—
22#63659C2×C312yes—
23#63634C2×C312yescomplemented
24C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
25C3×#832—32no—
26C3×#807—32no—
27C3×C4⋊(C3⋊C8)—31no—
28C23×H(3,𝔽3)—21no—
29D4×H(3,𝔽3)—21no—
30D4×H(3,𝔽3)Q812yescomplemented
31C2×C4×H(3,𝔽3)—21no—
32C2×C4×H(3,𝔽3)—42no—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×(C3⋊Dic3)⋊C3—22no—
37#634⋊C3—42no—
38(C3⋊(C3⋊C8))⋊C3—22no—
39C3×#474—91no—
40#5007—31no—
41#5007—91no—
42C2×C32×D4—21no—
43C2×C32×D4C2×D311yescharacteristic
44C2×C32×D4C22×C311yescharacteristic
45C22×C32×C4—22no—
46C22×C32×C4—21no—
47C22×C32×C4—21no—
48C2×#634—21no—
49C2×C3×Dic6—61no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4C2×D311yescharacteristic
52C32×C4⋊C4C22×C311yescharacteristic
53C32×C22⋊C4—22no—
54C32×C22⋊C4—21no—
55C32×C22⋊C4—21no—
56C32×C42—21no—
57C32×C42C2×D311yescharacteristic
58C32×C42C22×C311yescharacteristic
59C4⋊(C3⋊Dic3)—21no—
60C4⋊(C3⋊Dic3)C22×C312yes—
61C2×C3⋊(C3⋊C8)C22×C312yes—
62C3×C4⋊Dic3—32no—
63C3×C4⋊Dic3—61no—
64C2×C3×C3⋊C8—32no—
65C22×H(3,𝔽3)—22no—
66C22×H(3,𝔽3)—41no—
67C22×H(3,𝔽3)—41no—
68C22×H(3,𝔽3)C2×D411yescharacteristic
69C4×H(3,𝔽3)—21no—
70C4×H(3,𝔽3)—41no—
71C4×H(3,𝔽3)D812yes—
72C4×H(3,𝔽3)C2×Q811yescharacteristic
73C4×H(3,𝔽3)#4111yescharacteristic
74(C3⋊Dic3)⋊C3—42no—
75C3×C4⋊Q8—91no—
76C3×C4×D4—31no—
77C3×C4×D4—31no—
78C3×C4×D4—61no—
79C3×C4×D4C3×D311yescharacteristic
80#869—31no—
81#869—91no—
82C3×#107—92no—
83C3×C4⋊C8—91no—
84C3×#102—92no—
85#832—32no—
86#832—92no—
87#807—32no—
88#807—92no—
89C4⋊(C3⋊C8)—31no—
90C4⋊(C3⋊C8)—91no—
91C23×C32—22no—
92C23×C32—21no—
93C23×C32—21no—
94C32×D4—22no—
95C32×D4—21no—
96C32×D4—21no—
97C32×D4—22no—
98C32×D4Dic612yescomplemented
99C32×D4C3×Q812yescomplemented
100C2×C32×C4—22no—
101C2×C32×C4—21no—
102C2×C32×C4—21no—
103C2×C32×C4—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—44no—
107C2×C32×C4—42no—
108C2×C32×C4—42no—
109C2×C32×C4#6711yescharacteristic
110C2×C32×C4#6711yescharacteristic
111C2×C32×C4C3×D411yescharacteristic
112C2×C32×C4C3×D411yescharacteristic
113C2×C32×C4C22×D311yescharacteristic
114C2×C32×C4C23×C311yescharacteristic
115C2×C3⋊Dic3—22no—
116#634—42no—
117C2×C3×Dic3—62no—
118C3×Dic6—122no—
119C3⋊(C3⋊C8)—22no—
120C3×C3⋊C8—62no—
121#474—91noSylow
122C2×H(3,𝔽3)—42no—
123C2×H(3,𝔽3)C22⋊Q811yescharacteristic
124C2×H(3,𝔽3)C2×D811yescharacteristic
125C2×H(3,𝔽3)#13711yescharacteristic
126C2×C3×Q8—181no—
127C2×C3×D4—31no—
128C2×C3×D4—31no—
129C2×C3×D4—61no—
130C2×C3×D4C2×C3×D311yescharacteristic
131C22×C3×C4—21no—
132C22×C3×C4—61no—
133C22×C3×C4—61no—
134C22×C3×C4—62no—
135C2×Dic6—61no—
136C2×Dic6—181no—
137C2×C3×C8—92no—
138C3×C4⋊C4—31no—
139C3×C4⋊C4—31no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—92no—
142C3×C4⋊C4—181no—
143C3×C4⋊C4C2×C3×D311yescharacteristic
144C3×C22⋊C4—21no—
145C3×C22⋊C4—61no—
146C3×C22⋊C4—61no—
147C3×C22⋊C4—62no—
148C3×C42—31no—
149C3×C42—31no—
150C3×C42—61no—
151C3×C42C2×C3×D311yescharacteristic
152C4⋊Dic3—32no—
153C4⋊Dic3—61no—
154C4⋊Dic3—92no—
155C4⋊Dic3—181no—
156C2×C3⋊C8—32no—
157C2×C3⋊C8—92no—
158C22×C32—24no—
159C22×C32—22no—
160C22×C32—22no—
161C22×C32—21no—
162C22×C32—42no—
163C22×C32—42no—
164C22×C32—41no—
165C22×C32—41no—
166C22×C32—41no—
167C22×C32—41no—
168C22×C32C2×#6711yescharacteristic
169C22×C32C2×C3×D411yescharacteristic
170C32×C4—21no—
171C32×C4—21no—
172C32×C4—22no—
173C32×C4—21no—
174C32×C4—21no—
175C32×C4—41no—
176C32×C4—42no—
177C32×C4—41no—
178C32×C4—41no—
179C32×C4#21312yes—
180C32×C4C3×D812yes—
181C32×C4C2×Dic611yescharacteristic
182C32×C4#23511yescharacteristic
183C32×C4C2×C3×Q811yescharacteristic
184C32×C4C3×#4111yescharacteristic
185C3⋊Dic3—42no—
186C3×Dic3—122no—
187C4⋊Q8—91no—
188C4×D4(C3⋊D3)⋊C311yescharacteristic
189#107—92no—
190C4⋊C8—91no—
191#102—92no—
192H(3,𝔽3)#47411yescharacteristic, complemented, Sylow
193C23×C3—21no—
194C23×C3—61no—
195C23×C3—61no—
196C23×C3—62no—
197C3×Q8—362no—
198C3×D4—21no—
199C3×D4—32no—
200C3×D4—32no—
201C3×D4—61no—
202C3×D4—61no—
203C3×D4—62no—
204C3×D4—62no—
205C3×D4C3×Dic612yes—
206C2×C3×C4—21no—
207C2×C3×C4—31no—
208C2×C3×C4—31no—
209C2×C3×C4—31no—
210C2×C3×C4—31no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—42no—
214C2×C3×C4—61no—
215C2×C3×C4—61no—
216C2×C3×C4—62no—
217C2×C3×C4—61no—
218C2×C3×C4—61no—
219C2×C3×C4—61no—
220C2×C3×C4—122no—
221C2×C3×C4—122no—
222C2×C3×C4—124no—
223C2×C3×C4—182no—
224C2×C3×C4C3×#6711yescharacteristic
225C2×C3×C4C3×#6711yescharacteristic
226C2×C3×C4C22×C3×D311yescharacteristic
227C2×Dic3—62no—
228C2×Dic3—182no—
229Dic6—122no—
230Dic6—362no—
231C3×C8—182no—
232C3⋊C8—62no—
233C3⋊C8—182no—
234C2×C32—21no—
235C2×C32—21no—
236C2×C32—21no—
237C2×C32—44no—
238C2×C32—42no—
239C2×C32—42no—
240C2×C32#92411yescharacteristic
241C2×C32C2×#21311yescharacteristic
242C2×C32#95111yescharacteristic
243C2×C32C3×C22⋊Q811yescharacteristic
244C2×C32C2×C3×D811yescharacteristic
245C2×C32C3×#13711yescharacteristic
246C2×Q8—181no—
247C2×D4C2×(C3⋊D3)⋊C311yescharacteristic
248C22×C4—21no—
249C2×C8—92no—
250C4⋊C4—92no—
251C4⋊C4—181no—
252C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
253C22⋊C4—21no—
254C42C2×(C3⋊D3)⋊C311yescharacteristic
255C22×C3—22no—
256C22×C3—31no—
257C22×C3—31no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—62no—
261C22×C3—62no—
262C22×C3—64no—
263C22×C3—61no—
264C22×C3—121no—
265C22×C3—121no—
266C22×C3—121no—
267C22×C3—121no—
268C22×C3—122no—
269C22×C3—122no—
270C22×C3C2×C3×#6711yescharacteristic
271C3×C4—21no—
272C3×C4—31no—
273C3×C4—31no—
274C3×C4—31no—
275C3×C4—31no—
276C3×C4—32no—
277C3×C4—32no—
278C3×C4—41no—
279C3×C4—61no—
280C3×C4—61no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4—62no—
284C3×C4—121no—
285C3×C4—121no—
286C3×C4—121no—
287C3×C4—122no—
288C3×C4—362no—
289C3×C4C3×#21312yes—
290C3×C4C2×C3×Dic611yescharacteristic
291C3×C4C3×#23511yescharacteristic
292Dic3—122no—
293Dic3—362no—
294C32—21no—
295C32#500711yescharacteristic, complemented
296C32C3×#47411yescharacteristic, complemented
297C23—21no—
298Q8—362no—
299D4—21no—
300D4#634⋊C312yescomplemented
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—182no—
304C2×C4#638⋊C311yescharacteristic
305C2×C4#638⋊C311yescharacteristic
306C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
307C8—182no—
308C2×C3—31no—
309C2×C3—31no—
310C2×C3—31no—
311C2×C3—31no—
312C2×C3—31no—
313C2×C3—31no—
314C2×C3—42no—
315C2×C3—61no—
316C2×C3—61no—
317C2×C3—61no—
318C2×C3—122no—
319C2×C3—122no—
320C2×C3—124no—
321C2×C3C3×#92411yescharacteristic
322C2×C3C2×C3×#21311yescharacteristic
323C2×C3C3×#95111yescharacteristic
324C22—22no—
325C22—41no—
326C22—41no—
327C22C2×#638⋊C311yescharacteristic
328C4—21no—
329C4—41no—
330C4—362no—
331C4#3753⋊C312yes—
332C4C2×#634⋊C311yescharacteristic
333C4#3828⋊C311yescharacteristic
334C3—31no—
335C3—31no—
336C3—61no—
337C3C3×#500711yescharacteristic
338C2—42no—
339C2#64133⋊C311yescharacteristic
340C2C2×#3753⋊C311yescharacteristic
341C2#64160⋊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×#474
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. #64078⋊C3C2semidirect
4. #63634⋊C3C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #10592706C3semidirect
20. #64078C2×C3semidirect
23. #63634C2×C3semidirect
24. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
30. D4×H(3,𝔽3)Q8semidirect
98. C32×D4Dic6semidirect
99. C32×D4C3×Q8semidirect
192. H(3,𝔽3)#474semidirect
295. C32#5007semidirect
296. C32C3×#474semidirect
300. D4#634⋊C3semidirect

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
#64078⋊C32
#63659⋊C32
#63634⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927063
C3×#50073