Finite Group Databasebeta

← #13152715

Subgroups — #13152715

Order 1728.

1099 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88722C211yesmaximal, characteristic
2C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#64102⋊C3C213yesmaximal, complemented
5C4⋊C4×(C3⋊D3)⋊C3C213yesmaximal, complemented
6C4×C4⋊((C3⋊D3)⋊C3)C213yesmaximal, complemented
7#64081⋊C3C213yesmaximal, complemented
8#10596045C311yesmaximal, characteristic, complemented
9C3×#5574—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88722C418yescomplemented
12C2×C4⋊((C3⋊D3)⋊C3)C2213yes—
13C2×C4×(C3⋊D3)⋊C3C2216yes—
14C2×C4×(C3⋊D3)⋊C3C2213yes—
15C4⋊C4×H(3,𝔽3)C2213yes—
16C42×H(3,𝔽3)C2213yes—
17#3752⋊C3C2213yes—
18#3752⋊C3C2213yes—
19C4⋊((C3⋊Dic3)⋊C3)C2213yes—
20#3750⋊C3C2211yescharacteristic
21#3750⋊C3C2213yes—
22C4×(C3⋊Dic3)⋊C3C2213yes—
23C2×#3832C2×C311yescharacteristic
24C2×C3×#239—31no—
25C32×C4×Q8—21no—
26C32×C4×Q8D311yescharacteristic, complemented
27C32×C4×Q8C2×C311yescharacteristic, complemented
28Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
29#64102C2×C313yescomplemented
30C4⋊C4×C3⋊D3C2×C313yescomplemented
31C4×C4⋊(C3⋊D3)C2×C313yescomplemented
32#64081C2×C313yescomplemented
33C3×Q8×Dic3—31no—
34C3×#893—33no—
35C3×D3×C4⋊C4—33no—
36C3×C4×D12—33no—
37C3×#872—33no—
38C22×(C3⋊D3)⋊C3C2313yes—
39Q8×H(3,𝔽3)C2×C414yescomplemented
40C2×C4×H(3,𝔽3)C2313yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×C4×H(3,𝔽3)C2313yes—
43C2×(C3⋊Dic3)⋊C3C2313yes—
44C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
46C4⋊((C3⋊D3)⋊C3)C2×C4112yescomplemented
47C4×(C3⋊D3)⋊C3—46no—
48C4×(C3⋊D3)⋊C3C2×C4112yescomplemented
49C3×#520—91no—
50#5574—31no—
51#5574—91no—
52C2×C32×Q8—21no—
53C2×C32×Q8C2×D311yescharacteristic
54C2×C32×Q8C22×C311yescharacteristic
55#3832C3×C418yescomplemented
56C2×C4⋊(C3⋊D3)C22×C313yes—
57C2×C4×C3⋊D3C22×C316yes—
58C2×C4×C3⋊D3C22×C313yes—
59C3×#239—38no—
60C2×C3×D12—33no—
61C2×C3×C4×D3—33no—
62C2×C3×C4×D3—36no—
63C32×C4⋊C4—23no—
64C32×C4⋊C4C2×D313yes—
65C32×C4⋊C4C22×C313yes—
66C32×C42—23no—
67C32×C42C2×D313yes—
68C32×C42C22×C313yes—
69#3752C22×C313yes—
70#3752C22×C313yes—
71C4⋊(C3⋊Dic3)C22×C313yes—
72#3750C22×C311yescharacteristic
73#3750C22×C313yes—
74C4×C3⋊Dic3C22×C313yes—
75C3×(C2×D3)⋊C4—33no—
76C3×(C2×D3)⋊C4—33no—
77C3×C4⋊Dic3—33no—
78C3×Dic3⋊C4—33no—
79C3×Dic3⋊C4—31no—
80C3×C4×Dic3—33no—
81C22×H(3,𝔽3)C2411yescharacteristic
82C2×(C3⋊D3)⋊C3—23no—
83C2×(C3⋊D3)⋊C3—23no—
84C2×(C3⋊D3)⋊C3C22×C416yes—
85C4×H(3,𝔽3)—23no—
86C4×H(3,𝔽3)—21no—
87C4×H(3,𝔽3)C22×C416yes—
88(C3⋊Dic3)⋊C3—23no—
89(C3⋊Dic3)⋊C3—21no—
90(C3⋊Dic3)⋊C3C22×C412yes—
91C2×C3×#41—91no—
92C2×#239—31no—
93C2×#239—91no—
94C3×C4×Q8—31no—
95C3×C4×Q8—31no—
96C3×C4×Q8—61no—
97C3×C4×Q8—91no—
98C3×C4×Q8C3×D311yescharacteristic
99C3×C4×D4—93no—
100C3×C4×D4—93no—
101C3×#117—93no—
102C2×C3×C4⋊C4—93no—
103Q8×Dic3—31no—
104Q8×Dic3—91no—
105#893—33no—
106#893—93no—
107D3×C4⋊C4—33no—
108D3×C4⋊C4—93no—
109C4×D12—33no—
110C4×D12—93no—
111#872—33no—
112#872—93no—
113C22×C3⋊D3C23×C313yes—
114C22×C3×D3—33no—
115C32×Q8—24no—
116C32×Q8C4×D314yescomplemented
117C32×Q8C2×C3×C414yescomplemented
118C2×C32×C4—21no—
119C2×C32×C4—23no—
120C2×C32×C4—23no—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D313yes—
123C2×C32×C4C22×D313yes—
124C2×C32×C4C23×C311yescharacteristic
125C2×C32×C4C23×C313yes—
126C2×C32×C4C23×C313yes—
127C2×C3⋊Dic3C23×C311yescharacteristic
128C2×C3⋊Dic3C23×C313yes—
129C2×C3⋊Dic3C23×C311yescharacteristic
130C4⋊(C3⋊D3)C2×C3×C4112yescomplemented
131C4×C3⋊D3—46no—
132C4×C3⋊D3C2×C3×C4112yescomplemented
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C2×C3×Dic3—33no—
136C3×D12—312no—
137C3×C4×D3—312no—
138C3×C4×D3—126no—
139#520—91noSylow
140C2×H(3,𝔽3)C23×C411yescharacteristic
141C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
142C2×H(3,𝔽3)2−1+411yescharacteristic
143(C3⋊D3)⋊C3—26no—
144C3×#41—98no—
145C2×C3×Q8—31no—
146C2×C3×Q8—31no—
147C2×C3×Q8—61no—
148C2×C3×Q8C2×C3×D311yescharacteristic
149C2×C3×D4—93no—
150C22×C3×C4—93no—
151C22×C3×C4—96no—
152#239—38no—
153#239—98no—
154C2×D12—33no—
155C2×D12—93no—
156C2×C4×D3—33no—
157C2×C4×D3—36no—
158C2×C4×D3—93no—
159C2×C4×D3—96no—
160C3×C4⋊C4—33no—
161C3×C4⋊C4—33no—
162C3×C4⋊C4—63no—
163C3×C4⋊C4—93no—
164C3×C4⋊C4—91no—
165C3×C4⋊C4—93no—
166C3×C4⋊C4C2×C3×D313yes—
167C3×C22⋊C4—93no—
168C3×C22⋊C4—93no—
169C3×C42—33no—
170C3×C42—33no—
171C3×C42—63no—
172C3×C42—93no—
173C3×C42C2×C3×D313yes—
174(C2×D3)⋊C4—33no—
175(C2×D3)⋊C4—33no—
176(C2×D3)⋊C4—93no—
177(C2×D3)⋊C4—93no—
178C4⋊Dic3—33no—
179C4⋊Dic3—93no—
180Dic3⋊C4—31no—
181Dic3⋊C4—33no—
182Dic3⋊C4—91no—
183Dic3⋊C4—93no—
184C4×Dic3—33no—
185C4×Dic3—93no—
186C22×C32—21no—
187C22×C32C23×D311yescharacteristic
188C22×C32C24×C311yescharacteristic
189C2×C3⋊D3—23no—
190C2×C3⋊D3—23no—
191C2×C3⋊D3C22×C3×C416yes—
192C2×C3×D3—36no—
193C2×C3×D3—63no—
194C2×C3×D3—63no—
195C32×C4—22no—
196C32×C4—21no—
197C32×C4—23no—
198C32×C4—23no—
199C32×C4—21no—
200C32×C4—26no—
201C32×C4—43no—
202C32×C4C2×C4×D316yes—
203C32×C4C22×C3×C416yes—
204C3⋊Dic3—23no—
205C3⋊Dic3—21no—
206C3⋊Dic3C22×C3×C412yes—
207C3×Dic3—32no—
208C3×Dic3—61no—
209C3×Dic3—63no—
210C2×#41—91no—
211C4×Q8—91no—
212C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
213C4×D4—93no—
214C4×D4—93no—
215#117—93no—
216C2×C4⋊C4—93no—
217H(3,𝔽3)#52011yescharacteristic, complemented, Sylow
218C23×C3—93no—
219C22×D3—33no—
220C22×D3—93no—
221C3×Q8—34no—
222C3×Q8—34no—
223C3×Q8—64no—
224C3×Q8C3×C4×D314yes—
225C3×D4—912no—
226C2×C3×C4—33no—
227C2×C3×C4—33no—
228C2×C3×C4—31no—
229C2×C3×C4—33no—
230C2×C3×C4—31no—
231C2×C3×C4—33no—
232C2×C3×C4—61no—
233C2×C3×C4—63no—
234C2×C3×C4—63no—
235C2×C3×C4—912no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—93no—
239C2×C3×C4—366no—
240C2×C3×C4C22×C3×D311yescharacteristic
241C2×C3×C4C22×C3×D313yes—
242C2×C3×C4C22×C3×D313yes—
243C2×Dic3—31no—
244C2×Dic3—31no—
245C2×Dic3—33no—
246C2×Dic3—91no—
247C2×Dic3—91no—
248C2×Dic3—93no—
249D12—312no—
250D12—912no—
251C4×D3—312no—
252C4×D3—912no—
253C4×D3—126no—
254C4×D3—366no—
255C2×C32—21no—
256C2×C32—21no—
257C2×C32—21no—
258C2×C32C22×C4×D311yescharacteristic
259C2×C32#100711yescharacteristic
260C2×C32#100911yescharacteristic
261C2×C32C23×C3×C411yescharacteristic
262C2×C32C3×H(5,𝔽2)11yescharacteristic
263C2×C32C3×2−1+411yescharacteristic
264C3⋊D3—26no—
265C3×D3—66no—
266#41—98no—
267C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
268C2×D4—93no—
269C22×C4—93no—
270C22×C4—96no—
271C4⋊C4—93no—
272C4⋊C4—93no—
273C4⋊C4—91no—
274C4⋊C4C2×(C3⋊D3)⋊C313yes—
275C22⋊C4—93no—
276C22⋊C4—93no—
277C42—93no—
278C42C2×(C3⋊D3)⋊C313yes—
279C22×C3—31no—
280C22×C3—31no—
281C22×C3—61no—
282C22×C3—96no—
283C22×C3—183no—
284C22×C3—183no—
285C22×C3C23×C3×D311yescharacteristic
286C2×D3—36no—
287C2×D3—63no—
288C2×D3—63no—
289C2×D3—96no—
290C2×D3—183no—
291C2×D3—183no—
292C3×C4—21no—
293C3×C4—23no—
294C3×C4—36no—
295C3×C4—36no—
296C3×C4—63no—
297C3×C4—63no—
298C3×C4—61no—
299C3×C4—61no—
300C3×C4—62no—
301C3×C4—66no—
302C3×C4—92no—
303C3×C4—123no—
304C3×C4—183no—
305C3×C4—181no—
306C3×C4C2×C3×C4×D316yes—
307Dic3—32no—
308Dic3—61no—
309Dic3—63no—
310Dic3—92no—
311Dic3—183no—
312Dic3—181no—
313C32—21no—
314C32#557411yescharacteristic, complemented
315C32C3×#52011yescharacteristic, complemented
316C23—93no—
317Q8C4×(C3⋊D3)⋊C314yescomplemented
318D4—912no—
319C2×C4—91no—
320C2×C4—912no—
321C2×C4—91no—
322C2×C4—93no—
323C2×C4—366no—
324C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
325C2×C4C22×(C3⋊D3)⋊C313yes—
326C2×C4C22×(C3⋊D3)⋊C313yes—
327C2×C3—31no—
328C2×C3—31no—
329C2×C3—31no—
330C2×C3—31no—
331C2×C3—31no—
332C2×C3—31no—
333C2×C3—61no—
334C2×C3—61no—
335C2×C3—61no—
336C2×C3—186no—
337C2×C3C22×C3×C4×D311yescharacteristic
338C2×C3C3×#100711yescharacteristic
339C2×C3C3×#100911yescharacteristic
340D3—66no—
341D3—186no—
342C22—96no—
343C22—183no—
344C22—183no—
345C22C23×(C3⋊D3)⋊C311yescharacteristic
346C4—21no—
347C4—23no—
348C4—92no—
349C4—181no—
350C4—183no—
351C4C2×C4×(C3⋊D3)⋊C316yes—
352C3—31no—
353C3—31no—
354C3—61no—
355C3C3×#557411yescharacteristic
356C2—186no—
357C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
358C2#1172262711yescharacteristic
359C2#1172263811yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC2×#88722C2
cosocleC22×C32C24×C3
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#520
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
2. C4×Q8×H(3,𝔽3)C2semidirect
3. Q8×(C3⋊Dic3)⋊C3C2semidirect
4. #64102⋊C3C2semidirect
5. C4⋊C4×(C3⋊D3)⋊C3C2semidirect
6. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
7. #64081⋊C3C2semidirect
8. #10596045C3semidirect
11. #88722C4semidirect
26. C32×C4×Q8D3semidirect
27. C32×C4×Q8C2×C3semidirect
28. Q8×C3⋊Dic3C2×C3semidirect
29. #64102C2×C3semidirect
30. C4⋊C4×C3⋊D3C2×C3semidirect
31. C4×C4⋊(C3⋊D3)C2×C3semidirect
32. #64081C2×C3semidirect
39. Q8×H(3,𝔽3)C2×C4semidirect
46. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
48. C4×(C3⋊D3)⋊C3C2×C4semidirect
55. #3832C3×C4semidirect
116. C32×Q8C4×D3semidirect
117. C32×Q8C2×C3×C4semidirect
130. C4⋊(C3⋊D3)C2×C3×C4semidirect
132. C4×C3⋊D3C2×C3×C4semidirect
212. C4×Q8(C3⋊D3)⋊C3semidirect
217. H(3,𝔽3)#520semidirect
314. C32#5574semidirect
315. C32C3×#520semidirect
317. Q8C4×(C3⋊D3)⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C4×Q8×H(3,𝔽3)2
Q8×(C3⋊Dic3)⋊C32
#64102⋊C32
C4⋊C4×(C3⋊D3)⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#64081⋊C32
#105960453
C3×#55743