Finite Group Databasebeta

← #10592725⋊C3 #13142797

Subgroups — #10592725⋊C3 #13142797

Order 1728.

490 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
2C2×#3755⋊C3C211yesmaximal, characteristic
3C4×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
4#63662⋊C3C211yesmaximal, characteristic, complemented
5#63636⋊C3C211yesmaximal, characteristic
6#63634⋊C3C211yesmaximal, characteristic, complemented
7#63630⋊C3C211yesmaximal, characteristic, complemented
8#10592725C311yesmaximal, characteristic, complemented
9C3×#5026—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
12C2×C4×(C3⋊D3)⋊C3—21no—
13C4⋊C4×H(3,𝔽3)—21no—
14C4⋊C4×H(3,𝔽3)C2211yescharacteristic
15C42×H(3,𝔽3)—21no—
16C42×H(3,𝔽3)C2211yescharacteristic
17#3755⋊C3C414yescomplemented
18#3752⋊C3—21no—
19C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
22C32×C4×Q8—21no—
23C32×C4×Q8D311yescharacteristic, complemented
24C32×C4×Q8C2×C311yescharacteristic, complemented
25C2×#3755C2×C311yescharacteristic
26C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
27C2×C3×#215—31no—
28C3×C4×D12—31no—
29#63662C2×C311yescharacteristic, complemented
30#63636C2×C311yescharacteristic
31#63634C2×C311yescharacteristic, complemented
32#63630C2×C311yescharacteristic, complemented
33C3×#835—31no—
34C3×#809—31no—
35C3×#807—31no—
36C3×#803—31no—
37C22×(C3⋊D3)⋊C3—21no—
38Q8×H(3,𝔽3)—21no—
39Q8×H(3,𝔽3)C2×C412yescomplemented
40C2×C4×H(3,𝔽3)—21no—
41C2×C4×H(3,𝔽3)—21no—
42C2×C4×H(3,𝔽3)D411yescharacteristic
43C2×C4×H(3,𝔽3)D411yescharacteristic
44C2×C4×H(3,𝔽3)C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3—21no—
46C4⋊((C3⋊D3)⋊C3)—21no—
47C4⋊((C3⋊D3)⋊C3)C2×C412yes—
48C4×(C3⋊D3)⋊C3—42no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3C2×C412yescomplemented
51C3×#440—91no—
52#5026—31no—
53#5026—91no—
54C2×C32×Q8—21no—
55C2×C32×Q8C2×D311yescharacteristic
56C2×C32×Q8C22×C311yescharacteristic
57C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
58C2×C4×C3⋊D3—21no—
59C2×C3×D12—31no—
60C2×C3×C4×D3—61no—
61C32×C4⋊C4—22no—
62C32×C4⋊C4—21no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C4⋊C4C22×C311yescharacteristic
67C32×C42—22no—
68C32×C42—21no—
69C32×C42—21no—
70C32×C42—21no—
71C32×C42C2×D311yescharacteristic
72C32×C42C22×C311yescharacteristic
73#3755C3×C414yescomplemented
74#3752—21no—
75C4⋊(C3⋊Dic3)C22×C311yescharacteristic
76C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
77C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
78C3×#215—34no—
79C3×(C2×D3)⋊C4—61no—
80C3×C4⋊Dic3—31no—
81C2×C3×C3⋊C8—31no—
82C2×C3×C3⋊C8—31no—
83C22×H(3,𝔽3)C2×D411yescharacteristic
84C2×(C3⋊D3)⋊C3—22no—
85C2×(C3⋊D3)⋊C3—41no—
86C2×(C3⋊D3)⋊C3—41no—
87C4×H(3,𝔽3)—22no—
88C4×H(3,𝔽3)—21no—
89C4×H(3,𝔽3)—21no—
90C4×H(3,𝔽3)—41no—
91C4×H(3,𝔽3)C22×C411yescharacteristic
92C4×H(3,𝔽3)#4111yescharacteristic
93(C3⋊Dic3)⋊C3—41no—
94C2×C3×SD16—91no—
95C3×C4×Q8—31no—
96C3×C4×Q8—31no—
97C3×C4×Q8—61no—
98C3×C4×Q8C3×D311yescharacteristic
99C3×C4×D4—91no—
100C2×#215—31no—
101C2×#215—91no—
102C4×D12—31no—
103C4×D12—91no—
104C3×#107—91no—
105C3×#103—91no—
106C3×#102—91no—
107C3×#97—91no—
108#835—31no—
109#835—91no—
110#809—31no—
111#809—91no—
112#807—31no—
113#807—91no—
114#803—31no—
115#803—91no—
116C22×C3⋊D3—21no—
117C22×C3×D3—61no—
118C32×Q8—22no—
119C32×Q8—21no—
120C32×Q8—21no—
121C32×Q8—22no—
122C32×Q8C4×D312yescomplemented
123C32×Q8C2×C3×C412yescomplemented
124C2×C32×C4—22no—
125C2×C32×C4—21no—
126C2×C32×C4—21no—
127C2×C32×C4—22no—
128C2×C32×C4—21no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4#6711yescharacteristic
134C2×C32×C4#6711yescharacteristic
135C2×C32×C4C3×D411yescharacteristic
136C2×C32×C4C3×D411yescharacteristic
137C2×C32×C4C22×D311yescharacteristic
138C2×C32×C4C23×C311yescharacteristic
139C2×C3⋊Dic3—21no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)C2×C3×C412yes—
142C4×C3⋊D3—42no—
143C2×C3×Dic3—61no—
144C3×D12—32no—
145C3×D12—61no—
146C3×C4×D3—122no—
147C3⋊(C3⋊C8)—21no—
148C3⋊(C3⋊C8)C2×C3×C412yescomplemented
149C3×C3⋊C8—32no—
150C3×C3⋊C8—61no—
151#440—91noSylow
152C2×H(3,𝔽3)C4×D411yescharacteristic
153C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
154C2×H(3,𝔽3)#13711yescharacteristic
155(C3⋊D3)⋊C3—42no—
156C2×C3×Q8—31no—
157C2×C3×Q8—31no—
158C2×C3×Q8—61no—
159C2×C3×Q8C2×C3×D311yescharacteristic
160C2×C3×D4—91no—
161C22×C3×C4—181no—
162C2×D12—31no—
163C2×D12—91no—
164C2×C4×D3—61no—
165C2×C4×D3—181no—
166C3×SD16—94no—
167C2×C3×C8—91no—
168C2×C3×C8—91no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—62no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—91no—
177C3×C4⋊C4C2×C3×D311yescharacteristic
178C3×C22⋊C4—181no—
179C3×C42—21no—
180C3×C42—31no—
181C3×C42—31no—
182C3×C42—61no—
183C3×C42—61no—
184C3×C42—62no—
185C3×C42—61no—
186C3×C42C2×C3×D311yescharacteristic
187#215—34no—
188#215—94no—
189(C2×D3)⋊C4—61no—
190(C2×D3)⋊C4—181no—
191C4⋊Dic3—31no—
192C4⋊Dic3—91no—
193C2×C3⋊C8—31no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—91no—
196C2×C3⋊C8—91no—
197C22×C32—21no—
198C22×C32C2×#6711yescharacteristic
199C22×C32C2×C3×D411yescharacteristic
200C2×C3⋊D3—22no—
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3×D3—62no—
204C2×C3×D3—121no—
205C2×C3×D3—121no—
206C32×C4—22no—
207C32×C4—24no—
208C32×C4—22no—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—22no—
212C32×C4—21no—
213C32×C4—21no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—41no—
217C32×C4—42no—
218C32×C4—41no—
219C32×C4—41no—
220C32×C4C2×C4×D311yescharacteristic
221C32×C4#23511yescharacteristic
222C32×C4C22×C3×C411yescharacteristic
223C32×C4C3×#4111yescharacteristic
224C3⋊Dic3—41no—
225C3×Dic3—121no—
226C2×SD16—91no—
227C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
228C4×D4—91no—
229#107—91no—
230#103—91no—
231#102—91no—
232#97—91no—
233H(3,𝔽3)#44011yescharacteristic, complemented, Sylow
234C23×C3—181no—
235C22×D3—61no—
236C22×D3—181no—
237C3×Q8—21no—
238C3×Q8—32no—
239C3×Q8—32no—
240C3×Q8—61no—
241C3×Q8—61no—
242C3×Q8—62no—
243C3×Q8—62no—
244C3×Q8C3×C4×D312yes—
245C3×D4—92no—
246C3×D4—181no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—31no—
250C2×C3×C4—31no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—31no—
254C2×C3×C4—31no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—62no—
258C2×C3×C4—61no—
259C2×C3×C4—61no—
260C2×C3×C4—62no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—181no—
265C2×C3×C4—362no—
266C2×C3×C4C3×#6711yescharacteristic
267C2×C3×C4C3×#6711yescharacteristic
268C2×C3×C4C22×C3×D311yescharacteristic
269C2×Dic3—61no—
270C2×Dic3—181no—
271D12—32no—
272D12—61no—
273D12—92no—
274D12—181no—
275C4×D3—122no—
276C4×D3—362no—
277C3×C8—92no—
278C3×C8—181no—
279C3⋊C8—32no—
280C3⋊C8—61no—
281C3⋊C8—92no—
282C3⋊C8—181no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32C4×#6711yescharacteristic
287C2×C32#94211yescharacteristic
288C2×C32#94911yescharacteristic
289C2×C32C3×C4×D411yescharacteristic
290C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
291C2×C32C3×#13711yescharacteristic
292C3⋊D3—42no—
293C3×D3—122no—
294C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
295C2×D4—91no—
296C22×C4—181no—
297SD16—94no—
298C2×C8—91no—
299C2×C8—91no—
300C4⋊C4—21no—
301C4⋊C4—91no—
302C4⋊C4C2×(C3⋊D3)⋊C311yescharacteristic
303C22⋊C4—181no—
304C42—21no—
305C42C2×(C3⋊D3)⋊C311yescharacteristic
306C22×C3—31no—
307C22×C3—31no—
308C22×C3—61no—
309C22×C3—182no—
310C22×C3—361no—
311C22×C3—361no—
312C22×C3C2×C3×#6711yescharacteristic
313C2×D3—62no—
314C2×D3—121no—
315C2×D3—121no—
316C2×D3—182no—
317C2×D3—361no—
318C2×D3—361no—
319C3×C4—22no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—31no—
323C3×C4—31no—
324C3×C4—31no—
325C3×C4—31no—
326C3×C4—41no—
327C3×C4—61no—
328C3×C4—61no—
329C3×C4—61no—
330C3×C4—61no—
331C3×C4—62no—
332C3×C4—62no—
333C3×C4—64no—
334C3×C4—62no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—121no—
338C3×C4—121no—
339C3×C4—121no—
340C3×C4—122no—
341C3×C4—361no—
342C3×C4C2×C3×C4×D311yescharacteristic
343C3×C4C3×#23511yescharacteristic
344Dic3—121no—
345Dic3—361no—
346C32—21no—
347C32#502611yescharacteristic, complemented
348C32C3×#44011yescharacteristic, complemented
349C23—181no—
350Q8—21no—
351Q8C4×(C3⋊D3)⋊C312yescomplemented
352D4—92no—
353D4—181no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—181no—
357C2×C4—362no—
358C2×C4#638⋊C311yescharacteristic
359C2×C4#638⋊C311yescharacteristic
360C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
361C8—92no—
362C8—181no—
363C2×C3—31no—
364C2×C3—31no—
365C2×C3—31no—
366C2×C3—31no—
367C2×C3—31no—
368C2×C3—31no—
369C2×C3—61no—
370C2×C3—61no—
371C2×C3—61no—
372C2×C3—362no—
373C2×C3C3×C4×#6711yescharacteristic
374C2×C3C3×#94211yescharacteristic
375C2×C3C3×#94911yescharacteristic
376D3—122no—
377D3—362no—
378C22—182no—
379C22—361no—
380C22—361no—
381C22C2×#638⋊C311yescharacteristic
382C4—22no—
383C4—21no—
384C4—21no—
385C4—41no—
386C4—361no—
387C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
388C4#3828⋊C311yescharacteristic
389C3—31no—
390C3—31no—
391C3—61no—
392C3C3×#502611yescharacteristic
393C2—362no—
394C2C4×#638⋊C311yescharacteristic
395C2#64151⋊C311yescharacteristic
396C2#64158⋊C311yescharacteristic
subgroupG/H
centerC22C2×#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
socleC22×C3C2×C3×#67
prime radicalC2×C4⋊((C3⋊D3)⋊C3)C22
cosocleC2×C32×C4C23×C3
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#440
LCSG ▷ 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
4. #63662⋊C3C2semidirect
6. #63634⋊C3C2semidirect
7. #63630⋊C3C2semidirect
8. #10592725C3semidirect
17. #3755⋊C3C4semidirect
23. C32×C4×Q8D3semidirect
24. C32×C4×Q8C2×C3semidirect
29. #63662C2×C3semidirect
31. #63634C2×C3semidirect
32. #63630C2×C3semidirect
39. Q8×H(3,𝔽3)C2×C4semidirect
50. (C3⋊(C3⋊C8))⋊C3C2×C4semidirect
73. #3755C3×C4semidirect
122. C32×Q8C4×D3semidirect
123. C32×Q8C2×C3×C4semidirect
148. C3⋊(C3⋊C8)C2×C3×C4semidirect
227. C4×Q8(C3⋊D3)⋊C3semidirect
233. H(3,𝔽3)#440semidirect
347. C32#5026semidirect
348. C32C3×#440semidirect
351. Q8C4×(C3⋊D3)⋊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
C4×C4⋊((C3⋊D3)⋊C3)2
#63662⋊C32
#63636⋊C32
#63634⋊C32
#63630⋊C32
#105927253
C3×#50263