Finite Group Databasebeta

← #10592726⋊C3 #13142798

Subgroups — #10592726⋊C3 #13142798

Order 1728.

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

normal subgroups only

This group has 518 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×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C212yesmaximal, complemented
3#11721385C211yesmaximal, characteristic
4#63636⋊C3C212yesmaximal, complemented
5C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
6#10592726C311yesmaximal, characteristic, complemented
7C3×#5027—31nomaximal
8C2×Q8×H(3,𝔽3)C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)—21no—
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
12C4⋊C4×H(3,𝔽3)—21no—
13C4⋊C4×H(3,𝔽3)C2211yescharacteristic
14C42×H(3,𝔽3)—21no—
15C42×H(3,𝔽3)C2211yescharacteristic
16#3755⋊C3—24no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18C32×C4×Q8—21no—
19C32×C4×Q8D311yescharacteristic, complemented
20C32×C4×Q8C2×C311yescharacteristic, complemented
21C2×#3755C2×C312yescomplemented
22#64083C2×C311yescharacteristic
23C2×C3×#215—32no—
24C3×#874—31no—
25#63636C2×C312yescomplemented
26C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
27C3×#809—32no—
28C3×C4⋊(C3⋊C8)—31no—
29C22×(C3⋊D3)⋊C3—22no—
30Q8×H(3,𝔽3)—21no—
31Q8×H(3,𝔽3)D412yescomplemented
32C2×C4×H(3,𝔽3)—21no—
33C2×C4×H(3,𝔽3)—21no—
34C2×C4×H(3,𝔽3)D411yescharacteristic
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)C2311yescharacteristic
37C4⋊((C3⋊D3)⋊C3)—22no—
38C4⋊((C3⋊D3)⋊C3)—22no—
39C4⋊((C3⋊D3)⋊C3)—42no—
40C4⋊((C3⋊D3)⋊C3)—42no—
41(C3⋊(C3⋊C8))⋊C3—22no—
42C3×#460—91no—
43#5027—31no—
44#5027—91no—
45C2×C32×Q8—21no—
46C2×C32×Q8C2×D311yescharacteristic
47C2×C32×Q8C22×C311yescharacteristic
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4⋊(C3⋊D3)C22×C312yes—
51C2×C3×D12—32no—
52C2×C3×D12—61no—
53C2×C3×D12—61no—
54C32×C4⋊C4—22no—
55C32×C4⋊C4—21no—
56C32×C4⋊C4—21no—
57C32×C4⋊C4—21no—
58C32×C4⋊C4C2×D311yescharacteristic
59C32×C4⋊C4C22×C311yescharacteristic
60C32×C42—22no—
61C32×C42—21no—
62C32×C42—21no—
63C32×C42—21no—
64C32×C42C2×D311yescharacteristic
65C32×C42C22×C311yescharacteristic
66#3755—24no—
67C2×C3⋊(C3⋊C8)C22×C312yes—
68C3×#215—64no—
69C2×C3×C3⋊C8—32no—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C2×(C3⋊D3)⋊C3—42no—
72C2×(C3⋊D3)⋊C3—42no—
73C2×(C3⋊D3)⋊C3—42no—
74C4×H(3,𝔽3)—22no—
75C4×H(3,𝔽3)—21no—
76C4×H(3,𝔽3)—41no—
77C4×H(3,𝔽3)SD1612yescomplemented
78C4×H(3,𝔽3)C2×D411yescharacteristic
79C4×H(3,𝔽3)#4111yescharacteristic
80C2×C3×SD16—92no—
81C3×#127—91no—
82C3×C4×Q8—31no—
83C3×C4×Q8—31no—
84C3×C4×Q8—61no—
85C3×C4×Q8C3×D311yescharacteristic
86C2×#215—32no—
87C2×#215—92no—
88#874—31no—
89#874—91no—
90C3×C4⋊C8—91no—
91C3×#102—92no—
92#809—32no—
93#809—92no—
94C4⋊(C3⋊C8)—31no—
95C4⋊(C3⋊C8)—91no—
96C22×C3⋊D3—22no—
97C22×C3×D3—62no—
98C32×Q8—22no—
99C32×Q8—21no—
100C32×Q8—21no—
101C32×Q8—22no—
102C32×Q8D1212yescomplemented
103C32×Q8C3×D412yescomplemented
104C2×C32×C4—22no—
105C2×C32×C4—21no—
106C2×C32×C4—21no—
107C2×C32×C4—22no—
108C2×C32×C4—21no—
109C2×C32×C4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—21no—
112C2×C32×C4—21no—
113C2×C32×C4#6711yescharacteristic
114C2×C32×C4#6711yescharacteristic
115C2×C32×C4C3×D411yescharacteristic
116C2×C32×C4C3×D411yescharacteristic
117C2×C32×C4C22×D311yescharacteristic
118C2×C32×C4C23×C311yescharacteristic
119C4⋊(C3⋊D3)—22no—
120C4⋊(C3⋊D3)—22no—
121C4⋊(C3⋊D3)—42no—
122C4⋊(C3⋊D3)—42no—
123C3×D12—62no—
124C3×D12—62no—
125C3×D12—122no—
126C3×D12—122no—
127C3⋊(C3⋊C8)—22no—
128C3×C3⋊C8—62no—
129#460—91noSylow
130C2×H(3,𝔽3)#12111yescharacteristic
131C2×H(3,𝔽3)C2×SD1611yescharacteristic
132C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
133(C3⋊D3)⋊C3—82no—
134C2×C3×Q8—31no—
135C2×C3×Q8—31no—
136C2×C3×Q8—61no—
137C2×C3×Q8C2×C3×D311yescharacteristic
138C2×C3×D4—92no—
139C2×C3×D4—181no—
140C2×C3×D4—181no—
141C2×D12—32no—
142C2×D12—61no—
143C2×D12—61no—
144C2×D12—92no—
145C2×D12—181no—
146C2×D12—181no—
147C3×SD16—184no—
148C2×C3×C8—92no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—31no—
151C3×C4⋊C4—31no—
152C3×C4⋊C4—61no—
153C3×C4⋊C4—61no—
154C3×C4⋊C4—62no—
155C3×C4⋊C4—61no—
156C3×C4⋊C4C2×C3×D311yescharacteristic
157C3×C42—21no—
158C3×C42—31no—
159C3×C42—31no—
160C3×C42—61no—
161C3×C42—61no—
162C3×C42—62no—
163C3×C42—61no—
164C3×C42C2×C3×D311yescharacteristic
165#215—64no—
166#215—184no—
167C2×C3⋊C8—32no—
168C2×C3⋊C8—92no—
169C22×C32—21no—
170C22×C32C2×#6711yescharacteristic
171C22×C32C2×C3×D411yescharacteristic
172C2×C3⋊D3—42no—
173C2×C3⋊D3—42no—
174C2×C3⋊D3—42no—
175C2×C3×D3—122no—
176C2×C3×D3—122no—
177C2×C3×D3—122no—
178C32×C4—24no—
179C32×C4—22no—
180C32×C4—21no—
181C32×C4—22no—
182C32×C4—21no—
183C32×C4—22no—
184C32×C4—21no—
185C32×C4—21no—
186C32×C4—41no—
187C32×C4—42no—
188C32×C4—41no—
189C32×C4—41no—
190C32×C4#21512yescomplemented
191C32×C4C3×SD1612yescomplemented
192C32×C4C2×D1211yescharacteristic
193C32×C4#23511yescharacteristic
194C32×C4C2×C3×D411yescharacteristic
195C32×C4C3×#4111yescharacteristic
196C2×SD16—92no—
197#127—91no—
198C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
199C4⋊C8—91no—
200#102—92no—
201H(3,𝔽3)#46011yescharacteristic, complemented, Sylow
202C23×C3—182no—
203C22×D3—62no—
204C22×D3—182no—
205C3×Q8—21no—
206C3×Q8—32no—
207C3×Q8—32no—
208C3×Q8—61no—
209C3×Q8—61no—
210C3×Q8—62no—
211C3×Q8—62no—
212C3×Q8C3×D1212yes—
213C3×D4—182no—
214C3×D4—182no—
215C3×D4—362no—
216C3×D4—362no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—31no—
220C2×C3×C4—31no—
221C2×C3×C4—31no—
222C2×C3×C4—31no—
223C2×C3×C4—31no—
224C2×C3×C4—31no—
225C2×C3×C4—61no—
226C2×C3×C4—61no—
227C2×C3×C4—62no—
228C2×C3×C4—61no—
229C2×C3×C4—61no—
230C2×C3×C4—62no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4C3×#6711yescharacteristic
235C2×C3×C4C3×#6711yescharacteristic
236C2×C3×C4C22×C3×D311yescharacteristic
237D12—62no—
238D12—62no—
239D12—122no—
240D12—122no—
241D12—182no—
242D12—182no—
243D12—362no—
244D12—362no—
245C3×C8—182no—
246C3⋊C8—62no—
247C3⋊C8—182no—
248C2×C32—21no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32#93011yescharacteristic
252C2×C32C2×#21511yescharacteristic
253C2×C32#94911yescharacteristic
254C2×C32C3×#12111yescharacteristic
255C2×C32C2×C3×SD1611yescharacteristic
256C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
257C3⋊D3—82no—
258C3×D3—242no—
259C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
260C2×D4—92no—
261C2×D4—181no—
262C2×D4—181no—
263SD16—184no—
264C2×C8—92no—
265C4⋊C4—21no—
266C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
267C42—21no—
268C42C2×(C3⋊D3)⋊C311yescharacteristic
269C22×C3—31no—
270C22×C3—31no—
271C22×C3—61no—
272C22×C3—362no—
273C22×C3—362no—
274C22×C3—362no—
275C22×C3C2×C3×#6711yescharacteristic
276C2×D3—122no—
277C2×D3—122no—
278C2×D3—122no—
279C2×D3—362no—
280C2×D3—362no—
281C2×D3—362no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—31no—
288C3×C4—32no—
289C3×C4—32no—
290C3×C4—41no—
291C3×C4—61no—
292C3×C4—61no—
293C3×C4—62no—
294C3×C4—62no—
295C3×C4—64no—
296C3×C4—61no—
297C3×C4—61no—
298C3×C4—62no—
299C3×C4—121no—
300C3×C4—121no—
301C3×C4—121no—
302C3×C4—122no—
303C3×C4C3×#21512yes—
304C3×C4C2×C3×D1211yescharacteristic
305C3×C4C3×#23511yescharacteristic
306C32—21no—
307C32#502711yescharacteristic, complemented
308C32C3×#46011yescharacteristic, complemented
309C23—182no—
310Q8—21no—
311Q8C4⋊((C3⋊D3)⋊C3)12yescomplemented
312D4—182no—
313D4—182no—
314D4—362no—
315D4—362no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4#638⋊C311yescharacteristic
319C2×C4#638⋊C311yescharacteristic
320C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
321C8—182no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—31no—
327C2×C3—31no—
328C2×C3—61no—
329C2×C3—61no—
330C2×C3—61no—
331C2×C3—722no—
332C2×C3C3×#93011yescharacteristic
333C2×C3C2×C3×#21511yescharacteristic
334C2×C3C3×#94911yescharacteristic
335D3—242no—
336D3—722no—
337C22—362no—
338C22—362no—
339C22—362no—
340C22C2×#638⋊C311yescharacteristic
341C4—22no—
342C4—21no—
343C4—41no—
344C4#3755⋊C312yescomplemented
345C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
346C4#3828⋊C311yescharacteristic
347C3—31no—
348C3—31no—
349C3—61no—
350C3C3×#502711yescharacteristic
351C2—722no—
352C2#64139⋊C311yescharacteristic
353C2C2×#3755⋊C311yescharacteristic
354C2#64158⋊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 radical#11721385C2
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#460
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
1. C4×Q8×H(3,𝔽3)C2semidirect
2. C2×#3755⋊C3C2semidirect
4. #63636⋊C3C2semidirect
5. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
6. #10592726C3semidirect
19. C32×C4×Q8D3semidirect
20. C32×C4×Q8C2×C3semidirect
21. C2×#3755C2×C3semidirect
25. #63636C2×C3semidirect
26. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
31. Q8×H(3,𝔽3)D4semidirect
77. C4×H(3,𝔽3)SD16semidirect
102. C32×Q8D12semidirect
103. C32×Q8C3×D4semidirect
190. C32×C4#215semidirect
191. C32×C4C3×SD16semidirect
198. C4×Q8(C3⋊D3)⋊C3semidirect
201. H(3,𝔽3)#460semidirect
307. C32#5027semidirect
308. C32C3×#460semidirect
311. Q8C4⋊((C3⋊D3)⋊C3)semidirect
344. C4#3755⋊C3semidirect

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
C2×#3755⋊C32
#117213852
#63636⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105927263
C3×#50273