Finite Group Databasebeta

← C4×Q8×(C3⋊D3)⋊C3 #13152712

Subgroups — C4×Q8×(C3⋊D3)⋊C3 #13152712

Order 1728.

1113 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
2C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C4⋊C4×(C3⋊D3)⋊C3C213yesmaximal
5#64096⋊C3C213yesmaximal, complemented
6C42×(C3⋊D3)⋊C3C213yesmaximal
7C4×#634⋊C3C213yesmaximal, complemented
8C4×Q8×C3⋊D3C311yesmaximal, characteristic, complemented
9C3×C4×D3×Q8—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×(C3⋊D3)⋊C3C418yesdirect product factor
12C2×C4×(C3⋊D3)⋊C3C2213yes—
13C2×C4×(C3⋊D3)⋊C3C2213yes—
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#634⋊C3C2213yes—
16C4⋊C4×H(3,𝔽3)C2213yes—
17C42×H(3,𝔽3)C2213yes—
18C4⋊((C3⋊Dic3)⋊C3)C2213yes—
19#3750⋊C3C2216yes—
20C4×(C3⋊Dic3)⋊C3C2213yes—
21C4×(C3⋊Dic3)⋊C3C2213yes—
22C4×(C3⋊Dic3)⋊C3C2213yes—
23C2×Q8×C3⋊D3C2×C311yescharacteristic
24C2×C3×D3×Q8—31no—
25C32×C4×Q8—21no—
26C32×C4×Q8D311yescharacteristic, complemented
27C32×C4×Q8C2×C311yescharacteristic, complemented
28Q8×C3⋊Dic3C2×C311yescharacteristic, complemented
29C4⋊C4×C3⋊D3C2×C313yes—
30#64096C2×C313yescomplemented
31C42×C3⋊D3C2×C313yes—
32C4×#634C2×C313yescomplemented
33C3×Q8×Dic3—31no—
34C3×D3×C4⋊C4—33no—
35C3×#887—33no—
36C3×C42×D3—33no—
37C3×C4×Dic6—33no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39Q8×H(3,𝔽3)C2×C414yescomplemented
40C2×C4×H(3,𝔽3)C2313yes—
41C2×C4×H(3,𝔽3)C2313yes—
42C2×C4×H(3,𝔽3)C2311yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2313yes—
44C2×(C3⋊Dic3)⋊C3C2313yes—
45C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
46C4×(C3⋊D3)⋊C3—26no—
47C4×(C3⋊D3)⋊C3C2×C4112yes—
48C4×(C3⋊D3)⋊C3Q814yesdirect product factor
49#634⋊C3C2×C4112yescomplemented
50C2×C3×C4×Q8—91no—
51C4×D3×Q8—31no—
52C4×D3×Q8—91no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C22×C311yescharacteristic
56Q8×C3⋊D3C3×C418yescomplemented
57C2×C4×C3⋊D3C22×C313yes—
58C2×C4×C3⋊D3C22×C313yes—
59C2×C4×C3⋊D3C22×C311yescharacteristic
60C2×#634C22×C313yes—
61C3×D3×Q8—38no—
62C2×C3×C4×D3—33no—
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—33no—
65C2×C3×Dic6—33no—
66C32×C4⋊C4—23no—
67C32×C4⋊C4C2×D313yes—
68C32×C4⋊C4C22×C313yes—
69C32×C42—23no—
70C32×C42C2×D313yes—
71C32×C42C22×C313yes—
72C4⋊(C3⋊Dic3)C22×C313yes—
73#3750C22×C316yes—
74C4×C3⋊Dic3C22×C313yes—
75C4×C3⋊Dic3C22×C313yes—
76C4×C3⋊Dic3C22×C313yes—
77C3×C4⋊Dic3—33no—
78C3×Dic3⋊C4—36no—
79C3×C4×Dic3—33no—
80C3×C4×Dic3—33no—
81C3×C4×Dic3—33no—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×(C3⋊D3)⋊C3C22×C412yes—
84C2×(C3⋊D3)⋊C3C2×Q812yes—
85C2×(C3⋊D3)⋊C3#4112yes—
86C4×H(3,𝔽3)—23no—
87C4×H(3,𝔽3)C22×C416yes—
88C4×H(3,𝔽3)C2×Q812yescomplemented
89(C3⋊Dic3)⋊C3—23no—
90(C3⋊Dic3)⋊C3C22×C416yes—
91(C3⋊Dic3)⋊C3C2×Q812yescomplemented
92C22×C3×Q8—91no—
93C2×D3×Q8—31no—
94C2×D3×Q8—91no—
95C3×C4×Q8—31no—
96C3×C4×Q8—31no—
97C3×C4×Q8—61no—
98C3×C4×Q8—93no—
99C3×C4×Q8—93no—
100C3×C4×Q8—91no—
101C3×C4×Q8C3×D311yescharacteristic
102C2×C3×C4⋊C4—93no—
103C2×C3×C42—93no—
104Q8×Dic3—31no—
105Q8×Dic3—91no—
106D3×C4⋊C4—33no—
107D3×C4⋊C4—93no—
108#887—33no—
109#887—93no—
110C42×D3—33no—
111C42×D3—93no—
112C4×Dic6—33no—
113C4×Dic6—93no—
114C22×C3⋊D3C23×C311yescharacteristic
115C22×C3×D3—31no—
116C32×Q8—24no—
117C32×Q8C4×D314yescomplemented
118C32×Q8C2×C3×C414yescomplemented
119C2×C32×C4—21no—
120C2×C32×C4—23no—
121C2×C32×C4—23no—
122C2×C32×C4C22×D313yes—
123C2×C32×C4C22×D313yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C32×C4C23×C313yes—
126C2×C32×C4C23×C313yes—
127C2×C32×C4C23×C311yescharacteristic
128C2×C3⋊Dic3C23×C313yes—
129C2×C3⋊Dic3C23×C313yes—
130C2×C3⋊Dic3C23×C311yescharacteristic
131C4×C3⋊D3—26no—
132C4×C3⋊D3C2×C3×C4112yes—
133C4×C3⋊D3C3×Q814yescomplemented
134#634C2×C3×C4112yescomplemented
135C2×C3×Dic3—31no—
136C2×C3×Dic3—33no—
137C2×C3×Dic3—33no—
138C3×C4×D3—312no—
139C3×C4×D3—34no—
140C3×C4×D3—66no—
141C3×Dic6—312no—
142C2×C4×Q8—91noSylow
143C2×H(3,𝔽3)C23×C411yescharacteristic
144C2×H(3,𝔽3)C22×Q811yescharacteristic
145C2×H(3,𝔽3)C2×#4111yescharacteristic
146(C3⋊D3)⋊C3C4×Q814yesdirect product factor
147C2×C3×Q8—31no—
148C2×C3×Q8—31no—
149C2×C3×Q8—61no—
150C2×C3×Q8—93no—
151C2×C3×Q8—98no—
152C2×C3×Q8C2×C3×D311yescharacteristic
153C22×C3×C4—91no—
154C22×C3×C4—93no—
155C22×C3×C4—93no—
156D3×Q8—38no—
157D3×Q8—98no—
158C2×C4×D3—31no—
159C2×C4×D3—33no—
160C2×C4×D3—33no—
161C2×C4×D3—91no—
162C2×C4×D3—93no—
163C2×C4×D3—93no—
164C2×Dic6—33no—
165C2×Dic6—93no—
166C3×C4⋊C4—33no—
167C3×C4⋊C4—33no—
168C3×C4⋊C4—63no—
169C3×C4⋊C4—96no—
170C3×C4⋊C4—93no—
171C3×C4⋊C4C2×C3×D313yes—
172C3×C42—33no—
173C3×C42—33no—
174C3×C42—63no—
175C3×C42—93no—
176C3×C42—93no—
177C3×C42—93no—
178C3×C42C2×C3×D313yes—
179C4⋊Dic3—33no—
180C4⋊Dic3—93no—
181Dic3⋊C4—36no—
182Dic3⋊C4—96no—
183C4×Dic3—33no—
184C4×Dic3—33no—
185C4×Dic3—33no—
186C4×Dic3—93no—
187C4×Dic3—93no—
188C4×Dic3—93no—
189C22×C32—21no—
190C22×C32C23×D311yescharacteristic
191C22×C32C24×C311yescharacteristic
192C2×C3⋊D3C22×C3×C412yes—
193C2×C3⋊D3C2×C3×Q812yes—
194C2×C3⋊D3C3×#4112yes—
195C2×C3×D3—32no—
196C2×C3×D3—32no—
197C2×C3×D3—32no—
198C32×C4—23no—
199C32×C4—23no—
200C32×C4—26no—
201C32×C4—22no—
202C32×C4—43no—
203C32×C4C2×C4×D316yes—
204C32×C4D3×Q812yescomplemented
205C32×C4C22×C3×C416yes—
206C32×C4C2×C3×Q812yescomplemented
207C3⋊Dic3—23no—
208C3⋊Dic3C22×C3×C416yes—
209C3⋊Dic3C2×C3×Q812yescomplemented
210C3×Dic3—36no—
211C3×Dic3—32no—
212C3×Dic3—63no—
213C22×Q8—91no—
214C4×Q8—91no—
215C4×Q8—93no—
216C4×Q8—93no—
217C4×Q8(C3⋊D3)⋊C311yescharacteristic, direct product factor
218C2×C4⋊C4—93no—
219C2×C42—93no—
220H(3,𝔽3)C2×C4×Q811yescharacteristic, complemented, Sylow
221C23×C3—91no—
222C22×D3—31no—
223C22×D3—91no—
224C3×Q8—34no—
225C3×Q8—34no—
226C3×Q8—64no—
227C3×Q8—912no—
228C3×Q8C3×C4×D314yes—
229C2×C3×C4—33no—
230C2×C3×C4—33no—
231C2×C3×C4—33no—
232C2×C3×C4—33no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—63no—
236C2×C3×C4—63no—
237C2×C3×C4—61no—
238C2×C3×C4—94no—
239C2×C3×C4—912no—
240C2×C3×C4—91no—
241C2×C3×C4—93no—
242C2×C3×C4—93no—
243C2×C3×C4—186no—
244C2×C3×C4C22×C3×D313yes—
245C2×C3×C4C22×C3×D313yes—
246C2×C3×C4C22×C3×D311yescharacteristic
247C2×Dic3—31no—
248C2×Dic3—33no—
249C2×Dic3—33no—
250C2×Dic3—91no—
251C2×Dic3—93no—
252C2×Dic3—93no—
253C4×D3—312no—
254C4×D3—34no—
255C4×D3—66no—
256C4×D3—912no—
257C4×D3—94no—
258C4×D3—186no—
259Dic6—312no—
260Dic6—912no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32C22×C4×D311yescharacteristic
265C2×C32C2×D3×Q811yescharacteristic
266C2×C32D3×#4111yescharacteristic
267C2×C32C23×C3×C411yescharacteristic
268C2×C32C22×C3×Q811yescharacteristic
269C2×C32C2×C3×#4111yescharacteristic
270C3⋊D3C3×C4×Q814yescomplemented
271C3×D3—34no—
272C2×Q8—98no—
273C2×Q8—93no—
274C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
275C22×C4—93no—
276C22×C4—91no—
277C22×C4—93no—
278C4⋊C4—96no—
279C4⋊C4—93no—
280C4⋊C4C2×(C3⋊D3)⋊C313yes—
281C42—93no—
282C42—93no—
283C42—93no—
284C42C2×(C3⋊D3)⋊C313yes—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—61no—
288C22×C3—92no—
289C22×C3—92no—
290C22×C3—92no—
291C22×C3C23×C3×D311yescharacteristic
292C2×D3—32no—
293C2×D3—32no—
294C2×D3—32no—
295C2×D3—92no—
296C2×D3—92no—
297C2×D3—92no—
298C3×C4—23no—
299C3×C4—36no—
300C3×C4—36no—
301C3×C4—32no—
302C3×C4—32no—
303C3×C4—63no—
304C3×C4—63no—
305C3×C4—66no—
306C3×C4—62no—
307C3×C4—96no—
308C3×C4—92no—
309C3×C4—123no—
310C3×C4—183no—
311C3×C4C2×C3×C4×D316yes—
312C3×C4C3×D3×Q812yes—
313Dic3—36no—
314Dic3—32no—
315Dic3—63no—
316Dic3—96no—
317Dic3—92no—
318Dic3—183no—
319C32—21no—
320C32C4×D3×Q811yescharacteristic, complemented
321C32C2×C3×C4×Q811yescharacteristic, complemented
322C23—91no—
323Q8—912no—
324Q8C4×(C3⋊D3)⋊C314yesdirect product factor
325C2×C4—91no—
326C2×C4—93no—
327C2×C4—912no—
328C2×C4—94no—
329C2×C4—93no—
330C2×C4—186no—
331C2×C4C22×(C3⋊D3)⋊C313yes—
332C2×C4C22×(C3⋊D3)⋊C313yes—
333C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
334C2×C3—31no—
335C2×C3—31no—
336C2×C3—31no—
337C2×C3—31no—
338C2×C3—31no—
339C2×C3—31no—
340C2×C3—61no—
341C2×C3—61no—
342C2×C3—61no—
343C2×C3—94no—
344C2×C3C22×C3×C4×D311yescharacteristic
345C2×C3C2×C3×D3×Q811yescharacteristic
346C2×C3C3×D3×#4111yescharacteristic
347D3—34no—
348D3—94no—
349C22—92no—
350C22—92no—
351C22—92no—
352C22C23×(C3⋊D3)⋊C311yescharacteristic
353C4—23no—
354C4—96no—
355C4—92no—
356C4—183no—
357C4C2×C4×(C3⋊D3)⋊C316yes—
358C4Q8×(C3⋊D3)⋊C312yesdirect product factor
359C3—31no—
360C3—31no—
361C3—61no—
362C3C3×C4×D3×Q811yescharacteristic
363C2—94no—
364C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
365C2C2×Q8×(C3⋊D3)⋊C311yescharacteristic
366C2#41×(C3⋊D3)⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC2×C32C23×C3×C4
Frattini, socleC22×C3C23×C3×D3
FittingC4×Q8×H(3,𝔽3)C2
prime radicalC22×(C3⋊D3)⋊C3C23
cosocleC22×C32C24×C3
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C2×C3×C4×Q8
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ 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
5. #64096⋊C3C2semidirect
7. C4×#634⋊C3C2semidirect
8. C4×Q8×C3⋊D3C3semidirect
11. Q8×(C3⋊D3)⋊C3C4direct (N × H)
26. C32×C4×Q8D3semidirect
27. C32×C4×Q8C2×C3semidirect
28. Q8×C3⋊Dic3C2×C3semidirect
30. #64096C2×C3semidirect
32. C4×#634C2×C3semidirect
39. Q8×H(3,𝔽3)C2×C4semidirect
48. C4×(C3⋊D3)⋊C3Q8direct (N × H)
49. #634⋊C3C2×C4semidirect
56. Q8×C3⋊D3C3×C4semidirect
88. C4×H(3,𝔽3)C2×Q8semidirect
91. (C3⋊Dic3)⋊C3C2×Q8semidirect
117. C32×Q8C4×D3semidirect
118. C32×Q8C2×C3×C4semidirect
133. C4×C3⋊D3C3×Q8semidirect
134. #634C2×C3×C4semidirect
146. (C3⋊D3)⋊C3C4×Q8direct (N × H)
204. C32×C4D3×Q8semidirect
206. C32×C4C2×C3×Q8semidirect
209. C3⋊Dic3C2×C3×Q8semidirect
217. C4×Q8(C3⋊D3)⋊C3direct (N × H)
220. H(3,𝔽3)C2×C4×Q8semidirect
270. C3⋊D3C3×C4×Q8semidirect
320. C32C4×D3×Q8semidirect
321. C32C2×C3×C4×Q8semidirect
324. Q8C4×(C3⋊D3)⋊C3direct (N × H)
358. C4Q8×(C3⋊D3)⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C2×Q8×(C3⋊D3)⋊C32
C4×Q8×H(3,𝔽3)2
Q8×(C3⋊Dic3)⋊C32
C4⋊C4×(C3⋊D3)⋊C32
#64096⋊C32
C42×(C3⋊D3)⋊C32
C4×#634⋊C32
C4×Q8×C3⋊D33
C3×C4×D3×Q83