Finite Group Databasebeta

← #10592765⋊C3 #13142954

Subgroups — #10592765⋊C3 #13142954

Order 1728.

413 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 96 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
1H(3,𝔽3)×#125C211yesmaximal, characteristic, complemented
2C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
3#63636⋊C3C211yesmaximal, characteristic
4#63636⋊C3C211yesmaximal, characteristic
5#63635⋊C3C211yesmaximal, characteristic, complemented
6#63634⋊C3C211yesmaximal, characteristic, complemented
7C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
8#10592765C311yesmaximal, characteristic, complemented
9C3×#5066—31nomaximal
10C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
11C2×C4×(C3⋊D3)⋊C3—21no—
12C4⋊C4×H(3,𝔽3)—21no—
13C4⋊C4×H(3,𝔽3)—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C4⋊C4×H(3,𝔽3)C2211yescharacteristic
16C42×H(3,𝔽3)C2211yescharacteristic
17#3752⋊C3—21no—
18C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C32×#125—21no—
22C32×#125D311yescharacteristic, complemented
23C32×#125C2×C311yescharacteristic, complemented
24C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
25C3×C4×D12—31no—
26#63636C2×C311yescharacteristic
27#63636C2×C311yescharacteristic
28#63635C2×C311yescharacteristic, complemented
29#63634C2×C311yescharacteristic, complemented
30C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
31C3×#809—31no—
32C3×#809—31no—
33C3×#808—31no—
34C3×#807—31no—
35C3×C4⋊(C3⋊C8)—31no—
36C22×(C3⋊D3)⋊C3—21no—
37C2×C4×H(3,𝔽3)—21no—
38C2×C4×H(3,𝔽3)—21no—
39C2×C4×H(3,𝔽3)D411yescharacteristic
40C2×C4×H(3,𝔽3)D411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×(C3⋊Dic3)⋊C3—21no—
43C4⋊((C3⋊D3)⋊C3)—21no—
44C4⋊((C3⋊D3)⋊C3)Q812yes—
45C4×(C3⋊D3)⋊C3—42no—
46(C3⋊(C3⋊C8))⋊C3—21no—
47(C3⋊(C3⋊C8))⋊C3—21no—
48C3×#478—91no—
49#5066—31no—
50#5066—91no—
51C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
52C2×C4×C3⋊D3—21no—
53C2×C3×D12—31no—
54C2×C3×C4×D3—61no—
55C32×C4⋊C4—22no—
56C32×C4⋊C4—22no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4—21no—
59C32×C4⋊C4—21no—
60C32×C4⋊C4—21no—
61C32×C4⋊C4—21no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4C2×D311yescharacteristic
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C4⋊C4C22×C311yescharacteristic
66C32×C4⋊C4C22×C311yescharacteristic
67C32×C42—21no—
68C32×C42C2×D311yescharacteristic
69C32×C42C22×C311yescharacteristic
70#3752—21no—
71C4⋊(C3⋊Dic3)C22×C311yescharacteristic
72C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
73C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
74C3×(C2×D3)⋊C4—61no—
75C3×C4⋊Dic3—31no—
76C2×C3×C3⋊C8—31no—
77C2×C3×C3⋊C8—31no—
78C22×H(3,𝔽3)C2×D411yescharacteristic
79C2×(C3⋊D3)⋊C3—22no—
80C2×(C3⋊D3)⋊C3—41no—
81C2×(C3⋊D3)⋊C3—41no—
82C4×H(3,𝔽3)—21no—
83C4×H(3,𝔽3)—21no—
84C4×H(3,𝔽3)—41no—
85C4×H(3,𝔽3)—41no—
86C4×H(3,𝔽3)C2×Q811yescharacteristic
87C4×H(3,𝔽3)#4111yescharacteristic
88(C3⋊Dic3)⋊C3—41no—
89C3×#125—31no—
90C3×#125—31no—
91C3×#125—61no—
92C3×#125C3×D311yescharacteristic
93C3×C4×D4—91no—
94C4×D12—31no—
95C4×D12—91no—
96C3×#107—91no—
97C3×#106—91no—
98C3×C4⋊C8—91no—
99C3×#102—91no—
100C3×#102—91no—
101#809—31no—
102#809—31no—
103#809—91no—
104#809—91no—
105#808—31no—
106#808—91no—
107#807—31no—
108#807—91no—
109C4⋊(C3⋊C8)—31no—
110C4⋊(C3⋊C8)—91no—
111C22×C3⋊D3—21no—
112C22×C3×D3—61no—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4#6711yescharacteristic
123C2×C32×C4#6711yescharacteristic
124C2×C32×C4C3×D411yescharacteristic
125C2×C32×C4C3×D411yescharacteristic
126C2×C32×C4C22×D311yescharacteristic
127C2×C32×C4C23×C311yescharacteristic
128C2×C3⋊Dic3—21no—
129C4⋊(C3⋊D3)—21no—
130C4⋊(C3⋊D3)C3×Q812yes—
131C4×C3⋊D3—42no—
132C2×C3×Dic3—61no—
133C3×D12—32no—
134C3×D12—61no—
135C3×C4×D3—122no—
136C3⋊(C3⋊C8)—21no—
137C3⋊(C3⋊C8)—21no—
138C3×C3⋊C8—61no—
139C3×C3⋊C8—61no—
140#478—91noSylow
141C2×H(3,𝔽3)C22⋊Q811yescharacteristic
142C2×H(3,𝔽3)#13511yescharacteristic
143C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
144(C3⋊D3)⋊C3—42no—
145C2×C3×D4—91no—
146C22×C3×C4—181no—
147C2×D12—31no—
148C2×D12—91no—
149C2×C4×D3—61no—
150C2×C4×D3—181no—
151C2×C3×C8—91no—
152C2×C3×C8—91no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—31no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—61no—
160C3×C4⋊C4—61no—
161C3×C4⋊C4—61no—
162C3×C4⋊C4—61no—
163C3×C4⋊C4—62no—
164C3×C4⋊C4—62no—
165C3×C4⋊C4—61no—
166C3×C4⋊C4—61no—
167C3×C4⋊C4—91no—
168C3×C4⋊C4C2×C3×D311yescharacteristic
169C3×C4⋊C4C2×C3×D311yescharacteristic
170C3×C22⋊C4—181no—
171C3×C42—31no—
172C3×C42—31no—
173C3×C42—61no—
174C3×C42C2×C3×D311yescharacteristic
175(C2×D3)⋊C4—61no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—31no—
178C4⋊Dic3—91no—
179C2×C3⋊C8—31no—
180C2×C3⋊C8—31no—
181C2×C3⋊C8—91no—
182C2×C3⋊C8—91no—
183C22×C32—21no—
184C22×C32C2×#6711yescharacteristic
185C22×C32C2×C3×D411yescharacteristic
186C2×C3⋊D3—22no—
187C2×C3⋊D3—41no—
188C2×C3⋊D3—41no—
189C2×C3×D3—62no—
190C2×C3×D3—121no—
191C2×C3×D3—121no—
192C32×C4—21no—
193C32×C4—21no—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4—21no—
197C32×C4—21no—
198C32×C4—41no—
199C32×C4—41no—
200C32×C4—42no—
201C32×C4—42no—
202C32×C4—41no—
203C32×C4—41no—
204C32×C4—41no—
205C32×C4—41no—
206C32×C4D3×Q811yescharacteristic
207C32×C4#23911yescharacteristic
208C32×C4C2×C3×Q811yescharacteristic
209C32×C4C3×#4111yescharacteristic
210C3⋊Dic3—41no—
211C3×Dic3—121no—
212#125(C3⋊D3)⋊C311yescharacteristic, complemented
213C4×D4—91no—
214#107—91no—
215#106—91no—
216C4⋊C8—91no—
217#102—91no—
218#102—91no—
219H(3,𝔽3)#47811yescharacteristic, complemented, Sylow
220C23×C3—181no—
221C22×D3—61no—
222C22×D3—181no—
223C3×D4—92no—
224C3×D4—181no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—31no—
228C2×C3×C4—31no—
229C2×C3×C4—31no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—31no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—61no—
237C2×C3×C4—62no—
238C2×C3×C4—62no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—61no—
242C2×C3×C4—181no—
243C2×C3×C4—362no—
244C2×C3×C4C3×#6711yescharacteristic
245C2×C3×C4C3×#6711yescharacteristic
246C2×C3×C4C22×C3×D311yescharacteristic
247C2×Dic3—61no—
248C2×Dic3—181no—
249D12—32no—
250D12—61no—
251D12—92no—
252D12—181no—
253C4×D3—122no—
254C4×D3—362no—
255C3×C8—181no—
256C3×C8—181no—
257C3⋊C8—61no—
258C3⋊C8—61no—
259C3⋊C8—181no—
260C3⋊C8—181no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32#94611yescharacteristic
265C2×C32#94911yescharacteristic
266C2×C32#95011yescharacteristic
267C2×C32C3×C22⋊Q811yescharacteristic
268C2×C32C3×#13511yescharacteristic
269C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
270C3⋊D3—42no—
271C3×D3—122no—
272C2×D4—91no—
273C22×C4—181no—
274C2×C8—91no—
275C2×C8—91no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—91no—
279C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
280C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
281C22⋊C4—181no—
282C42C2×(C3⋊D3)⋊C311yescharacteristic
283C22×C3—31no—
284C22×C3—31no—
285C22×C3—61no—
286C22×C3—182no—
287C22×C3—361no—
288C22×C3—361no—
289C22×C3C2×C3×#6711yescharacteristic
290C2×D3—62no—
291C2×D3—121no—
292C2×D3—121no—
293C2×D3—182no—
294C2×D3—361no—
295C2×D3—361no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—31no—
299C3×C4—31no—
300C3×C4—31no—
301C3×C4—31no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—61no—
305C3×C4—61no—
306C3×C4—61no—
307C3×C4—61no—
308C3×C4—61no—
309C3×C4—61no—
310C3×C4—121no—
311C3×C4—121no—
312C3×C4—121no—
313C3×C4—121no—
314C3×C4—121no—
315C3×C4—121no—
316C3×C4—122no—
317C3×C4—122no—
318C3×C4—361no—
319C3×C4C3×D3×Q811yescharacteristic
320C3×C4C3×#23911yescharacteristic
321Dic3—121no—
322Dic3—361no—
323C32—21no—
324C32#506611yescharacteristic, complemented
325C32C3×#47811yescharacteristic, complemented
326C23—181no—
327D4—92no—
328D4—181no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—181no—
332C2×C4—362no—
333C2×C4#638⋊C311yescharacteristic
334C2×C4#638⋊C311yescharacteristic
335C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
336C8—181no—
337C8—181no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—31no—
341C2×C3—31no—
342C2×C3—31no—
343C2×C3—31no—
344C2×C3—61no—
345C2×C3—61no—
346C2×C3—61no—
347C2×C3—362no—
348C2×C3C3×#94611yescharacteristic
349C2×C3C3×#94911yescharacteristic
350C2×C3C3×#95011yescharacteristic
351D3—122no—
352D3—362no—
353C22—182no—
354C22—361no—
355C22—361no—
356C22C2×#638⋊C311yescharacteristic
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—361no—
362C4Q8×(C3⋊D3)⋊C311yescharacteristic
363C4#8872211yescharacteristic
364C3—31no—
365C3—31no—
366C3—61no—
367C3C3×#506611yescharacteristic
368C2—362no—
369C2#64155⋊C311yescharacteristic
370C2#64158⋊C311yescharacteristic
371C2#64159⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×#125C2
socleC22×C3C2×C3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
hypercenter#125(C3⋊D3)⋊C3
hypocenterC32C3×#478
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×#125C2semidirect
5. #63635⋊C3C2semidirect
6. #63634⋊C3C2semidirect
7. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
8. #10592765C3semidirect
22. C32×#125D3semidirect
23. C32×#125C2×C3semidirect
28. #63635C2×C3semidirect
29. #63634C2×C3semidirect
30. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
212. #125(C3⋊D3)⋊C3semidirect
219. H(3,𝔽3)#478semidirect
324. C32#5066semidirect
325. C32C3×#478semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×#1252
C4×C4⋊((C3⋊D3)⋊C3)2
#63636⋊C32
#63635⋊C32
#63634⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927653
C3×#50663