Finite Group Databasebeta

← #13152717

Subgroups — #13152717

Order 1728.

1085 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88722C211yesmaximal, characteristic, complemented
2C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
3C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
4#64106⋊C3C213yesmaximal, complemented
5#64104⋊C3C213yesmaximal, complemented
6#64084⋊C3C213yesmaximal, complemented
7C4×C4⋊((C3⋊D3)⋊C3)C213yesmaximal, complemented
8#10596047C311yesmaximal, characteristic, complemented
9C3×#5576—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88722—24no—
12Q8×(C3⋊D3)⋊C3—24no—
13C2×C4⋊((C3⋊D3)⋊C3)C2213yes—
14C2×C4⋊((C3⋊D3)⋊C3)C2213yes—
15C2×C4×(C3⋊D3)⋊C3C2213yes—
16C2×C4×(C3⋊D3)⋊C3C2213yes—
17C2×#634⋊C3C2213yes—
18C4⋊C4×H(3,𝔽3)C2213yes—
19C42×H(3,𝔽3)C2213yes—
20#3752⋊C3C2211yescharacteristic
21#3752⋊C3C2213yes—
22#3752⋊C3C2216yes—
23C4⋊((C3⋊Dic3)⋊C3)C2213yes—
24C2×#3832C2×C311yescharacteristic, complemented
25C2×Q8×C3⋊D3C2×C311yescharacteristic, complemented
26C2×C3×#239—31no—
27C2×C3×D3×Q8—31no—
28C32×C4×Q8—21no—
29C32×C4×Q8D311yescharacteristic, complemented
30C32×C4×Q8C2×C311yescharacteristic, complemented
31#64106C2×C313yescomplemented
32#64104C2×C313yescomplemented
33#64084C2×C313yescomplemented
34C4×C4⋊(C3⋊D3)C2×C313yescomplemented
35C3×#897—33no—
36C3×#895—33no—
37C3×#875—33no—
38C3×C4×D12—33no—
39C22×(C3⋊D3)⋊C3C2313yes—
40C22×(C3⋊D3)⋊C3C2311yescharacteristic
41Q8×H(3,𝔽3)D414yescomplemented
42C2×C4×H(3,𝔽3)C2313yes—
43C2×C4×H(3,𝔽3)C2311yescharacteristic
44C2×C4×H(3,𝔽3)C2313yes—
45C2×(C3⋊Dic3)⋊C3C2313yes—
46C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
47C4⋊((C3⋊D3)⋊C3)—26no—
48C4⋊((C3⋊D3)⋊C3)—26no—
49C4×(C3⋊D3)⋊C3—26no—
50C4×(C3⋊D3)⋊C3—26no—
51#634⋊C3—26no—
52C3×#548—91no—
53#5576—31no—
54#5576—91no—
55C2×C32×Q8—21no—
56C2×C32×Q8C2×D311yescharacteristic
57C2×C32×Q8C22×C311yescharacteristic
58#3832—24no—
59Q8×C3⋊D3—24no—
60C2×C4⋊(C3⋊D3)C22×C313yes—
61C2×C4⋊(C3⋊D3)C22×C313yes—
62C2×C4×C3⋊D3C22×C313yes—
63C2×C4×C3⋊D3C22×C313yes—
64C2×#634C22×C313yes—
65C3×#239—64no—
66C3×D3×Q8—64no—
67C2×C3×D12—33no—
68C2×C3×D12—33no—
69C2×C3×C4×D3—33no—
70C2×C3×C4×D3—33no—
71C2×C3×Dic6—33no—
72C32×C4⋊C4—23no—
73C32×C4⋊C4C2×D313yes—
74C32×C4⋊C4C22×C313yes—
75C32×C42—23no—
76C32×C42C2×D313yes—
77C32×C42C22×C313yes—
78#3752C22×C311yescharacteristic
79#3752C22×C313yes—
80#3752C22×C316yes—
81C4⋊(C3⋊Dic3)C22×C313yes—
82C3×(C2×D3)⋊C4—36no—
83C3×(C2×D3)⋊C4—31no—
84C3×(C2×D3)⋊C4—33no—
85C3×C4⋊Dic3—33no—
86C22×H(3,𝔽3)C2411yescharacteristic
87C2×(C3⋊D3)⋊C3—21no—
88C2×(C3⋊D3)⋊C3—21no—
89C2×(C3⋊D3)⋊C3—23no—
90C2×(C3⋊D3)⋊C3—23no—
91C2×(C3⋊D3)⋊C3—23no—
92C2×(C3⋊D3)⋊C3#4112yescomplemented
93C4×H(3,𝔽3)—23no—
94C4×H(3,𝔽3)—21no—
95C4×H(3,𝔽3)C2×D416yes—
96(C3⋊Dic3)⋊C3—23no—
97(C3⋊Dic3)⋊C3—21no—
98C2×C3×#41—91no—
99C22×C3×Q8—91no—
100C2×#239—31no—
101C2×#239—91no—
102C2×D3×Q8—31no—
103C2×D3×Q8—91no—
104C3×#124—93no—
105C3×C22⋊Q8—93no—
106C3×#121—93no—
107C3×C4×Q8—31no—
108C3×C4×Q8—31no—
109C3×C4×Q8—61no—
110C3×C4×Q8C3×D311yescharacteristic
111C3×C4×D4—93no—
112#897—33no—
113#897—93no—
114#895—33no—
115#895—93no—
116#875—33no—
117#875—93no—
118C4×D12—33no—
119C4×D12—93no—
120C22×C3⋊D3C23×C313yes—
121C22×C3⋊D3C23×C311yescharacteristic
122C22×C3×D3—31no—
123C22×C3×D3—33no—
124C32×Q8—24no—
125C32×Q8D1214yescomplemented
126C32×Q8C3×D414yescomplemented
127C2×C32×C4—21no—
128C2×C32×C4—23no—
129C2×C32×C4—23no—
130C2×C32×C4C22×D313yes—
131C2×C32×C4C22×D311yescharacteristic
132C2×C32×C4C22×D313yes—
133C2×C32×C4C23×C313yes—
134C2×C32×C4C23×C311yescharacteristic
135C2×C32×C4C23×C313yes—
136C2×C3⋊Dic3C23×C311yescharacteristic
137C2×C3⋊Dic3C23×C313yes—
138C4⋊(C3⋊D3)—26no—
139C4⋊(C3⋊D3)—26no—
140C4×C3⋊D3—26no—
141C4×C3⋊D3—26no—
142#634—26no—
143C2×C3×Dic3—31no—
144C2×C3×Dic3—33no—
145C3×D12—66no—
146C3×D12—66no—
147C3×C4×D3—66no—
148C3×C4×D3—66no—
149C3×Dic6—66no—
150#548—91noSylow
151C2×H(3,𝔽3)C22×D411yescharacteristic
152C2×H(3,𝔽3)C2×#4111yescharacteristic
153C2×H(3,𝔽3)2−1+411yescharacteristic
154(C3⋊D3)⋊C3—22no—
155(C3⋊D3)⋊C3—43no—
156C3×#41—184no—
157C2×C3×Q8—31no—
158C2×C3×Q8—31no—
159C2×C3×Q8—61no—
160C2×C3×Q8—93no—
161C2×C3×Q8—184no—
162C2×C3×Q8C2×C3×D311yescharacteristic
163C2×C3×D4—93no—
164C2×C3×D4—93no—
165C22×C3×C4—93no—
166C22×C3×C4—93no—
167#239—64no—
168#239—184no—
169D3×Q8—64no—
170D3×Q8—184no—
171C2×D12—33no—
172C2×D12—33no—
173C2×D12—93no—
174C2×D12—93no—
175C2×C4×D3—33no—
176C2×C4×D3—33no—
177C2×C4×D3—93no—
178C2×C4×D3—93no—
179C2×Dic6—33no—
180C2×Dic6—93no—
181C3×C4⋊C4—33no—
182C3×C4⋊C4—33no—
183C3×C4⋊C4—63no—
184C3×C4⋊C4—93no—
185C3×C4⋊C4C2×C3×D313yes—
186C3×C22⋊C4—91no—
187C3×C22⋊C4—93no—
188C3×C22⋊C4—96no—
189C3×C42—33no—
190C3×C42—33no—
191C3×C42—63no—
192C3×C42C2×C3×D313yes—
193(C2×D3)⋊C4—31no—
194(C2×D3)⋊C4—33no—
195(C2×D3)⋊C4—36no—
196(C2×D3)⋊C4—91no—
197(C2×D3)⋊C4—93no—
198(C2×D3)⋊C4—96no—
199C4⋊Dic3—33no—
200C4⋊Dic3—93no—
201C22×C32—21no—
202C22×C32C23×D311yescharacteristic
203C22×C32C24×C311yescharacteristic
204C2×C3⋊D3—21no—
205C2×C3⋊D3—21no—
206C2×C3⋊D3—23no—
207C2×C3⋊D3—23no—
208C2×C3⋊D3—23no—
209C2×C3⋊D3C3×#4112yescomplemented
210C2×C3×D3—32no—
211C2×C3×D3—61no—
212C2×C3×D3—61no—
213C2×C3×D3—63no—
214C2×C3×D3—63no—
215C2×C3×D3—63no—
216C32×C4—22no—
217C32×C4—21no—
218C32×C4—23no—
219C32×C4—23no—
220C32×C4—21no—
221C32×C4—26no—
222C32×C4—43no—
223C32×C4C2×D1216yes—
224C32×C4C2×C3×D416yes—
225C3⋊Dic3—23no—
226C3⋊Dic3—21no—
227C3×Dic3—63no—
228C3×Dic3—61no—
229C2×#41—91no—
230C22×Q8—91no—
231#124—93no—
232C22⋊Q8—93no—
233#121—93no—
234C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
235C4×D4—93no—
236H(3,𝔽3)#54811yescharacteristic, complemented, Sylow
237C23×C3—91no—
238C23×C3—93no—
239C22×D3—31no—
240C22×D3—33no—
241C22×D3—91no—
242C22×D3—93no—
243C3×Q8—34no—
244C3×Q8—34no—
245C3×Q8—64no—
246C3×Q8—186no—
247C3×Q8C3×D1214yes—
248C3×D4—186no—
249C3×D4—186no—
250C2×C3×C4—33no—
251C2×C3×C4—33no—
252C2×C3×C4—31no—
253C2×C3×C4—33no—
254C2×C3×C4—33no—
255C2×C3×C4—31no—
256C2×C3×C4—61no—
257C2×C3×C4—63no—
258C2×C3×C4—63no—
259C2×C3×C4—93no—
260C2×C3×C4—91no—
261C2×C3×C4—186no—
262C2×C3×C4—186no—
263C2×C3×C4C22×C3×D313yes—
264C2×C3×C4C22×C3×D311yescharacteristic
265C2×C3×C4C22×C3×D313yes—
266C2×Dic3—31no—
267C2×Dic3—33no—
268C2×Dic3—91no—
269C2×Dic3—93no—
270D12—66no—
271D12—66no—
272D12—186no—
273D12—186no—
274C4×D3—66no—
275C4×D3—66no—
276C4×D3—186no—
277C4×D3—186no—
278Dic6—66no—
279Dic6—186no—
280C2×C32—21no—
281C2×C32—21no—
282C2×C32—21no—
283C2×C32C22×D1211yescharacteristic
284C2×C32#100711yescharacteristic
285C2×C32D3×#4111yescharacteristic
286C2×C32C22×C3×D411yescharacteristic
287C2×C32C2×C3×#4111yescharacteristic
288C2×C32C3×2−1+411yescharacteristic
289C3⋊D3—22no—
290C3⋊D3—43no—
291C3×D3—62no—
292C3×D3—123no—
293#41—184no—
294C2×Q8—93no—
295C2×Q8—184no—
296C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
297C2×D4—93no—
298C2×D4—93no—
299C22×C4—93no—
300C22×C4—93no—
301C4⋊C4—93no—
302C4⋊C4C2×(C3⋊D3)⋊C313yes—
303C22⋊C4—91no—
304C22⋊C4—93no—
305C22⋊C4—96no—
306C42C2×(C3⋊D3)⋊C313yes—
307C22×C3—31no—
308C22×C3—31no—
309C22×C3—61no—
310C22×C3—92no—
311C22×C3—181no—
312C22×C3—181no—
313C22×C3—183no—
314C22×C3—183no—
315C22×C3—183no—
316C22×C3C23×C3×D311yescharacteristic
317C2×D3—32no—
318C2×D3—61no—
319C2×D3—61no—
320C2×D3—63no—
321C2×D3—63no—
322C2×D3—63no—
323C2×D3—92no—
324C2×D3—181no—
325C2×D3—181no—
326C2×D3—183no—
327C2×D3—183no—
328C2×D3—183no—
329C3×C4—21no—
330C3×C4—23no—
331C3×C4—36no—
332C3×C4—36no—
333C3×C4—63no—
334C3×C4—63no—
335C3×C4—61no—
336C3×C4—61no—
337C3×C4—62no—
338C3×C4—66no—
339C3×C4—123no—
340C3×C4—181no—
341C3×C4—183no—
342C3×C4C2×C3×D1216yes—
343Dic3—63no—
344Dic3—61no—
345Dic3—183no—
346Dic3—181no—
347C32—21no—
348C32#557611yescharacteristic, complemented
349C32C3×#54811yescharacteristic, complemented
350C23—91no—
351C23—93no—
352Q8—186no—
353Q8C4⋊((C3⋊D3)⋊C3)14yescomplemented
354D4—186no—
355D4—186no—
356C2×C4—93no—
357C2×C4—91no—
358C2×C4—186no—
359C2×C4—186no—
360C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
361C2×C4C22×(C3⋊D3)⋊C313yes—
362C2×C4C22×(C3⋊D3)⋊C313yes—
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—182no—
373C2×C3—363no—
374C2×C3C22×C3×D1211yescharacteristic
375C2×C3C3×#100711yescharacteristic
376C2×C3C3×D3×#4111yescharacteristic
377D3—62no—
378D3—123no—
379D3—182no—
380D3—363no—
381C22—92no—
382C22—181no—
383C22—181no—
384C22—183no—
385C22—183no—
386C22—183no—
387C22C23×(C3⋊D3)⋊C311yescharacteristic
388C4—21no—
389C4—23no—
390C4—181no—
391C4—183no—
392C4C2×C4⋊((C3⋊D3)⋊C3)16yes—
393C3—31no—
394C3—31no—
395C3—61no—
396C3C3×#557611yescharacteristic
397C2—182no—
398C2—363no—
399C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
400C2#1172262711yescharacteristic
401C2#41×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#88722C2semidirect
2. C2×Q8×(C3⋊D3)⋊C3C2semidirect
3. C4×Q8×H(3,𝔽3)C2semidirect
4. #64106⋊C3C2semidirect
5. #64104⋊C3C2semidirect
6. #64084⋊C3C2semidirect
7. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
8. #10596047C3semidirect
24. C2×#3832C2×C3semidirect
25. C2×Q8×C3⋊D3C2×C3semidirect
29. C32×C4×Q8D3semidirect
30. C32×C4×Q8C2×C3semidirect
31. #64106C2×C3semidirect
32. #64104C2×C3semidirect
33. #64084C2×C3semidirect
34. C4×C4⋊(C3⋊D3)C2×C3semidirect
41. Q8×H(3,𝔽3)D4semidirect
92. C2×(C3⋊D3)⋊C3#41semidirect
125. C32×Q8D12semidirect
126. C32×Q8C3×D4semidirect
209. C2×C3⋊D3C3×#41semidirect
234. C4×Q8(C3⋊D3)⋊C3semidirect
236. H(3,𝔽3)#548semidirect
348. C32#5576semidirect
349. C32C3×#548semidirect
353. Q8C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#887222
C2×Q8×(C3⋊D3)⋊C32
C4×Q8×H(3,𝔽3)2
#64106⋊C32
#64104⋊C32
#64084⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
#105960473
C3×#55763