Finite Group Databasebeta

← C4×#88722 #13152714

Subgroups — C4×#88722 #13152714

Order 1728.

1155 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×#88722C211yesmaximal, characteristic
2C4×Q8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3Q8×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#64102⋊C3C213yesmaximal, complemented
5#64101⋊C3C213yesmaximal, complemented
6C4×C4⋊((C3⋊D3)⋊C3)C213yesmaximal, complemented
7C42×(C3⋊D3)⋊C3C213yesmaximal, complemented
8C4×#3832C311yesmaximal, characteristic, complemented
9C3×C4×#239—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11#88722C418yesdirect product factor
12C2×C4⋊((C3⋊D3)⋊C3)C2213yes—
13C2×C4×(C3⋊D3)⋊C3C2213yes—
14C2×C4×(C3⋊D3)⋊C3C2213yes—
15C2×C4×(C3⋊D3)⋊C3C2213yes—
16C4⋊C4×H(3,𝔽3)C2213yes—
17C42×H(3,𝔽3)C2213yes—
18#3752⋊C3C2216yes—
19C4⋊((C3⋊Dic3)⋊C3)C2213yes—
20C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
21C4×(C3⋊Dic3)⋊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
30#64101C2×C313yescomplemented
31C4×C4⋊(C3⋊D3)C2×C313yescomplemented
32C42×C3⋊D3C2×C313yescomplemented
33C3×Q8×Dic3—31no—
34C3×#893—33no—
35C3×#892—33no—
36C3×C4×D12—33no—
37C3×C42×D3—33no—
38C22×(C3⋊D3)⋊C3C2313yes—
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)⋊C3C2311yescharacteristic
45C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
46C4⋊((C3⋊D3)⋊C3)C2×C4112yescomplemented
47C4×(C3⋊D3)⋊C3—26no—
48C4×(C3⋊D3)⋊C3—26no—
49C4×(C3⋊D3)⋊C3C2×C4112yescomplemented
50C3×C4×#41—91no—
51C4×#239—31no—
52C4×#239—91no—
53C2×C32×Q8—21no—
54C2×C32×Q8C2×D311yescharacteristic
55C2×C32×Q8C22×C311yescharacteristic
56#3832C3×C418yescomplemented
57C2×C4⋊(C3⋊D3)C22×C313yes—
58C2×C4×C3⋊D3C22×C313yes—
59C2×C4×C3⋊D3C22×C313yes—
60C2×C4×C3⋊D3C22×C313yes—
61C3×#239—38no—
62C2×C3×D12—33no—
63C2×C3×C4×D3—33no—
64C2×C3×C4×D3—33no—
65C2×C3×C4×D3—33no—
66C32×C4⋊C4—23no—
67C32×C4⋊C4C2×D313yes—
68C32×C4⋊C4C22×C313yes—
69C32×C42—23no—
70C32×C42C2×D313yes—
71C32×C42C22×C313yes—
72#3752C22×C316yes—
73C4⋊(C3⋊Dic3)C22×C313yes—
74C4×C3⋊Dic3C22×C311yescharacteristic
75C4×C3⋊Dic3C22×C313yes—
76C4×C3⋊Dic3C22×C313yes—
77C3×(C2×D3)⋊C4—36no—
78C3×C4⋊Dic3—33no—
79C3×C4×Dic3—33no—
80C3×C4×Dic3—33no—
81C3×C4×Dic3—31no—
82C22×H(3,𝔽3)C2411yescharacteristic
83C2×(C3⋊D3)⋊C3—23no—
84C2×(C3⋊D3)⋊C3—23no—
85C2×(C3⋊D3)⋊C3C22×C416yes—
86C4×H(3,𝔽3)—23no—
87C4×H(3,𝔽3)C22×C416yes—
88C4×H(3,𝔽3)#4112yescomplemented
89(C3⋊Dic3)⋊C3—23no—
90(C3⋊Dic3)⋊C3C22×C412yes—
91(C3⋊Dic3)⋊C3#4112yescomplemented
92C2×C3×#41—91no—
93C2×#239—31no—
94C2×#239—91no—
95C3×C4×Q8—31no—
96C3×C4×Q8—31no—
97C3×C4×Q8—61no—
98C3×C4×Q8—91no—
99C3×C4×Q8C3×D311yescharacteristic
100C3×C4×D4—93no—
101C3×C4×D4—93no—
102C3×#117—93no—
103C2×C3×C42—93no—
104Q8×Dic3—31no—
105Q8×Dic3—91no—
106#893—33no—
107#893—93no—
108#892—33no—
109#892—93no—
110C4×D12—33no—
111C4×D12—93no—
112C42×D3—33no—
113C42×D3—93no—
114C22×C3⋊D3C23×C313yes—
115C22×C3×D3—33no—
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×C311yescharacteristic
130C2×C3⋊Dic3C23×C311yescharacteristic
131C4⋊(C3⋊D3)C2×C3×C4112yescomplemented
132C4×C3⋊D3—26no—
133C4×C3⋊D3—26no—
134C4×C3⋊D3C2×C3×C4112yescomplemented
135C2×C3×Dic3—31no—
136C2×C3×Dic3—31no—
137C2×C3×Dic3—33no—
138C3×D12—312no—
139C3×C4×D3—312no—
140C3×C4×D3—66no—
141C3×C4×D3—66no—
142C4×#41—91noSylow
143C2×H(3,𝔽3)C23×C411yescharacteristic
144C2×H(3,𝔽3)C2×#4111yescharacteristic
145C2×H(3,𝔽3)C2×#4111yescharacteristic
146(C3⋊D3)⋊C3—26no—
147C3×#41—98no—
148C2×C3×Q8—31no—
149C2×C3×Q8—31no—
150C2×C3×Q8—61no—
151C2×C3×Q8C2×C3×D311yescharacteristic
152C2×C3×D4—93no—
153C22×C3×C4—93no—
154C22×C3×C4—93no—
155C22×C3×C4—93no—
156#239—38no—
157#239—98no—
158C2×D12—33no—
159C2×D12—93no—
160C2×C4×D3—33no—
161C2×C4×D3—33no—
162C2×C4×D3—33no—
163C2×C4×D3—93no—
164C2×C4×D3—93no—
165C2×C4×D3—93no—
166C3×C4⋊C4—33no—
167C3×C4⋊C4—33no—
168C3×C4⋊C4—63no—
169C3×C4⋊C4—93no—
170C3×C4⋊C4C2×C3×D313yes—
171C3×C22⋊C4—96no—
172C3×C42—33no—
173C3×C42—33no—
174C3×C42—63no—
175C3×C42—91no—
176C3×C42—93no—
177C3×C42—93no—
178C3×C42C2×C3×D313yes—
179(C2×D3)⋊C4—36no—
180(C2×D3)⋊C4—96no—
181C4⋊Dic3—33no—
182C4⋊Dic3—93no—
183C4×Dic3—31no—
184C4×Dic3—33no—
185C4×Dic3—33no—
186C4×Dic3—91no—
187C4×Dic3—93no—
188C4×Dic3—93no—
189C22×C32—21no—
190C22×C32C23×D311yescharacteristic
191C22×C32C24×C311yescharacteristic
192C2×C3⋊D3—23no—
193C2×C3⋊D3—23no—
194C2×C3⋊D3C22×C3×C416yes—
195C2×C3×D3—36no—
196C2×C3×D3—63no—
197C2×C3×D3—63no—
198C32×C4—23no—
199C32×C4—23no—
200C32×C4—26no—
201C32×C4—22no—
202C32×C4—43no—
203C32×C4C2×C4×D316yes—
204C32×C4#23912yescomplemented
205C32×C4C22×C3×C416yes—
206C32×C4C3×#4112yescomplemented
207C3⋊Dic3—23no—
208C3⋊Dic3C22×C3×C412yes—
209C3⋊Dic3C3×#4112yescomplemented
210C3×Dic3—32no—
211C3×Dic3—32no—
212C3×Dic3—63no—
213C2×#41—91no—
214C4×Q8—91no—
215C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
216C4×D4—93no—
217C4×D4—93no—
218#117—93no—
219C2×C42—93no—
220H(3,𝔽3)C4×#4111yescharacteristic, complemented, Sylow
221C23×C3—93no—
222C22×D3—33no—
223C22×D3—93no—
224C3×Q8—34no—
225C3×Q8—34no—
226C3×Q8—64no—
227C3×Q8C3×C4×D314yes—
228C3×D4—912no—
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—912no—
239C2×C3×C4—91no—
240C2×C3×C4—91no—
241C2×C3×C4—93no—
242C2×C3×C4—186no—
243C2×C3×C4—186no—
244C2×C3×C4C22×C3×D313yes—
245C2×C3×C4C22×C3×D313yes—
246C2×C3×C4C22×C3×D311yescharacteristic
247C2×Dic3—31no—
248C2×Dic3—31no—
249C2×Dic3—33no—
250C2×Dic3—91no—
251C2×Dic3—91no—
252C2×Dic3—93no—
253D12—312no—
254D12—912no—
255C4×D3—312no—
256C4×D3—66no—
257C4×D3—66no—
258C4×D3—912no—
259C4×D3—186no—
260C4×D3—186no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32C22×C4×D311yescharacteristic
265C2×C32C2×#23911yescharacteristic
266C2×C32D3×#4111yescharacteristic
267C2×C32C23×C3×C411yescharacteristic
268C2×C32C2×C3×#4111yescharacteristic
269C2×C32C2×C3×#4111yescharacteristic
270C3⋊D3—26no—
271C3×D3—66no—
272#41—98no—
273C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
274C2×D4—93no—
275C22×C4—93no—
276C22×C4—93no—
277C22×C4—93no—
278C4⋊C4—93no—
279C4⋊C4C2×(C3⋊D3)⋊C313yes—
280C22⋊C4—96no—
281C42—93no—
282C42—93no—
283C42—91no—
284C42C2×(C3⋊D3)⋊C313yes—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—61no—
288C22×C3—96no—
289C22×C3—183no—
290C22×C3—183no—
291C22×C3C23×C3×D311yescharacteristic
292C2×D3—36no—
293C2×D3—63no—
294C2×D3—63no—
295C2×D3—96no—
296C2×D3—183no—
297C2×D3—183no—
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—92no—
308C3×C4—92no—
309C3×C4—123no—
310C3×C4—183no—
311C3×C4C2×C3×C4×D316yes—
312C3×C4C3×#23912yes—
313Dic3—32no—
314Dic3—32no—
315Dic3—63no—
316Dic3—92no—
317Dic3—92no—
318Dic3—183no—
319C32—21no—
320C32C4×#23911yescharacteristic, complemented
321C32C3×C4×#4111yescharacteristic, complemented
322C23—93no—
323Q8C4×(C3⋊D3)⋊C314yescomplemented
324D4—912no—
325C2×C4—91no—
326C2×C4—91no—
327C2×C4—912no—
328C2×C4—93no—
329C2×C4—186no—
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—186no—
344C2×C3C22×C3×C4×D311yescharacteristic
345C2×C3C2×C3×#23911yescharacteristic
346C2×C3C3×D3×#4111yescharacteristic
347D3—66no—
348D3—186no—
349C22—96no—
350C22—183no—
351C22—183no—
352C22C23×(C3⋊D3)⋊C311yescharacteristic
353C4—23no—
354C4—92no—
355C4—92no—
356C4—183no—
357C4C2×C4×(C3⋊D3)⋊C316yes—
358C4#8872212yesdirect product factor
359C3—31no—
360C3—31no—
361C3—61no—
362C3C3×C4×#23911yescharacteristic
363C2—186no—
364C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
365C2C2×#8872211yescharacteristic
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 radicalC2×#88722C2
cosocleC22×C32C24×C3
hypercenterC4×Q8(C3⋊D3)⋊C3
hypocenterC32C3×C4×#41
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
4. #64102⋊C3C2semidirect
5. #64101⋊C3C2semidirect
6. C4×C4⋊((C3⋊D3)⋊C3)C2semidirect
7. C42×(C3⋊D3)⋊C3C2semidirect
8. C4×#3832C3semidirect
11. #88722C4direct (N × H)
26. C32×C4×Q8D3semidirect
27. C32×C4×Q8C2×C3semidirect
28. Q8×C3⋊Dic3C2×C3semidirect
29. #64102C2×C3semidirect
30. #64101C2×C3semidirect
31. C4×C4⋊(C3⋊D3)C2×C3semidirect
32. C42×C3⋊D3C2×C3semidirect
39. Q8×H(3,𝔽3)C2×C4semidirect
46. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
49. C4×(C3⋊D3)⋊C3C2×C4semidirect
56. #3832C3×C4semidirect
88. C4×H(3,𝔽3)#41semidirect
91. (C3⋊Dic3)⋊C3#41semidirect
117. C32×Q8C4×D3semidirect
118. C32×Q8C2×C3×C4semidirect
131. C4⋊(C3⋊D3)C2×C3×C4semidirect
134. C4×C3⋊D3C2×C3×C4semidirect
204. C32×C4#239semidirect
206. C32×C4C3×#41semidirect
209. C3⋊Dic3C3×#41semidirect
215. C4×Q8(C3⋊D3)⋊C3semidirect
220. H(3,𝔽3)C4×#41semidirect
320. C32C4×#239semidirect
321. C32C3×C4×#41semidirect
323. Q8C4×(C3⋊D3)⋊C3semidirect
358. C4#88722direct (N × H)

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
#64101⋊C32
C4×C4⋊((C3⋊D3)⋊C3)2
C42×(C3⋊D3)⋊C32
C4×#38323
C3×C4×#2393