Finite Group Databasebeta

← #13152713

Subgroups — #13152713

Order 1728.

1001 conjugacy classes of subgroups.

The subgroup lattice has 354 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
4#64101⋊C3C213yesmaximal
5#64096⋊C3C213yesmaximal, complemented
6#64081⋊C3C213yesmaximal
7C4×#634⋊C3C213yesmaximal, complemented
8#10596043C311yesmaximal, characteristic, complemented
9C3×#5572—31nomaximal
10C2×Q8×H(3,𝔽3)C2211yescharacteristic
11Q8×(C3⋊D3)⋊C3C418yescomplemented
12C2×C4×(C3⋊D3)⋊C3C2213yes—
13C2×#634⋊C3C2213yes—
14C4⋊C4×H(3,𝔽3)C2213yes—
15C42×H(3,𝔽3)C2213yes—
16#3752⋊C3C2211yescharacteristic
17#3752⋊C3C2213yes—
18C4⋊((C3⋊Dic3)⋊C3)C2213yes—
19#3750⋊C3C2213yes—
20#3750⋊C3C2213yes—
21C4×(C3⋊Dic3)⋊C3C2216yes—
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
29#64101C2×C313yes—
30#64096C2×C313yescomplemented
31#64081C2×C313yes—
32C4×#634C2×C313yescomplemented
33C3×Q8×Dic3—31no—
34C3×#892—33no—
35C3×#887—33no—
36C3×#872—33no—
37C3×C4×Dic6—33no—
38C22×(C3⋊D3)⋊C3C2311yescharacteristic
39Q8×H(3,𝔽3)C2×C414yes—
40C2×C4×H(3,𝔽3)C2313yes—
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×C4×H(3,𝔽3)C2313yes—
43C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2313yes—
45C2×(C3⋊Dic3)⋊C3C2313yes—
46C4×(C3⋊D3)⋊C3C2×C4112yes—
47#634⋊C3C2×C4112yes—
48C3×#519—91no—
49#5572—31no—
50#5572—91no—
51C2×C32×Q8—21no—
52C2×C32×Q8C2×D311yescharacteristic
53C2×C32×Q8C22×C311yescharacteristic
54Q8×C3⋊D3C3×C418yescomplemented
55C2×C4×C3⋊D3C22×C313yes—
56C2×#634C22×C313yes—
57C3×D3×Q8—38no—
58C2×C3×C4×D3—33no—
59C2×C3×Dic6—33no—
60C32×C4⋊C4—23no—
61C32×C4⋊C4C2×D313yes—
62C32×C4⋊C4C22×C313yes—
63C32×C42—23no—
64C32×C42C2×D313yes—
65C32×C42C22×C313yes—
66#3752C22×C311yescharacteristic
67#3752C22×C313yes—
68C4⋊(C3⋊Dic3)C22×C313yes—
69#3750C22×C313yes—
70#3750C22×C313yes—
71C4×C3⋊Dic3C22×C316yes—
72C4×C3⋊Dic3C22×C313yes—
73C3×(C2×D3)⋊C4—33no—
74C3×(C2×D3)⋊C4—31no—
75C3×C4⋊Dic3—33no—
76C3×Dic3⋊C4—33no—
77C3×Dic3⋊C4—33no—
78C3×C4×Dic3—36no—
79C3×C4×Dic3—33no—
80C22×H(3,𝔽3)C2411yescharacteristic
81C2×(C3⋊D3)⋊C3—21no—
82C2×(C3⋊D3)⋊C3—21no—
83C2×(C3⋊D3)⋊C3C22×C412yes—
84C4×H(3,𝔽3)—21no—
85C4×H(3,𝔽3)—23no—
86C4×H(3,𝔽3)C22×C416yes—
87(C3⋊Dic3)⋊C3—23no—
88(C3⋊Dic3)⋊C3—21no—
89(C3⋊Dic3)⋊C3C22×C416yes—
90C22×C3×Q8—91no—
91C2×D3×Q8—31no—
92C2×D3×Q8—91no—
93C3×C4×Q8—31no—
94C3×C4×Q8—31no—
95C3×C4×Q8—61no—
96C3×C4×Q8—93no—
97C3×C4×Q8—93no—
98C3×C4×Q8—91no—
99C3×C4×Q8C3×D311yescharacteristic
100C3×#117—93no—
101C3×#117—93no—
102Q8×Dic3—31no—
103Q8×Dic3—91no—
104#892—33no—
105#892—93no—
106#887—33no—
107#887—93no—
108#872—33no—
109#872—93no—
110C4×Dic6—33no—
111C4×Dic6—93no—
112C22×C3⋊D3C23×C311yescharacteristic
113C22×C3×D3—31no—
114C32×Q8—24no—
115C32×Q8C4×D314yes—
116C32×Q8C2×C3×C414yes—
117C2×C32×C4—21no—
118C2×C32×C4—23no—
119C2×C32×C4—23no—
120C2×C32×C4C22×D313yes—
121C2×C32×C4C22×D311yescharacteristic
122C2×C32×C4C22×D313yes—
123C2×C32×C4C23×C313yes—
124C2×C32×C4C23×C311yescharacteristic
125C2×C32×C4C23×C313yes—
126C2×C3⋊Dic3C23×C311yescharacteristic
127C2×C3⋊Dic3C23×C313yes—
128C2×C3⋊Dic3C23×C313yes—
129C4×C3⋊D3C2×C3×C4112yes—
130#634C2×C3×C4112yes—
131C2×C3×Dic3—33no—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—33no—
134C3×C4×D3—312no—
135C3×Dic6—312no—
136#519—91noSylow
137C2×H(3,𝔽3)C23×C411yescharacteristic
138C2×H(3,𝔽3)2−1+411yescharacteristic
139C2×H(3,𝔽3)2−1+411yescharacteristic
140(C3⋊D3)⋊C3—22no—
141C2×C3×Q8—31no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×Q8—93no—
145C2×C3×Q8—98no—
146C2×C3×Q8C2×C3×D311yescharacteristic
147C22×C3×C4—93no—
148D3×Q8—38no—
149D3×Q8—98no—
150C2×C4×D3—33no—
151C2×C4×D3—93no—
152C2×Dic6—33no—
153C2×Dic6—93no—
154C3×C4⋊C4—33no—
155C3×C4⋊C4—33no—
156C3×C4⋊C4—63no—
157C3×C4⋊C4—93no—
158C3×C4⋊C4—93no—
159C3×C4⋊C4—93no—
160C3×C4⋊C4C2×C3×D313yes—
161C3×C22⋊C4—91no—
162C3×C22⋊C4—93no—
163C3×C42—33no—
164C3×C42—33no—
165C3×C42—63no—
166C3×C42—93no—
167C3×C42—96no—
168C3×C42C2×C3×D313yes—
169(C2×D3)⋊C4—31no—
170(C2×D3)⋊C4—33no—
171(C2×D3)⋊C4—91no—
172(C2×D3)⋊C4—93no—
173C4⋊Dic3—33no—
174C4⋊Dic3—93no—
175Dic3⋊C4—33no—
176Dic3⋊C4—33no—
177Dic3⋊C4—93no—
178Dic3⋊C4—93no—
179C4×Dic3—36no—
180C4×Dic3—33no—
181C4×Dic3—96no—
182C4×Dic3—93no—
183C22×C32—21no—
184C22×C32C23×D311yescharacteristic
185C22×C32C24×C311yescharacteristic
186C2×C3⋊D3—21no—
187C2×C3⋊D3—21no—
188C2×C3⋊D3C22×C3×C412yes—
189C2×C3×D3—32no—
190C2×C3×D3—61no—
191C2×C3×D3—61no—
192C32×C4—22no—
193C32×C4—21no—
194C32×C4—23no—
195C32×C4—21no—
196C32×C4—23no—
197C32×C4—26no—
198C32×C4—43no—
199C32×C4C2×C4×D316yes—
200C32×C4C22×C3×C416yes—
201C3⋊Dic3—23no—
202C3⋊Dic3—21no—
203C3⋊Dic3C22×C3×C416yes—
204C3×Dic3—36no—
205C3×Dic3—63no—
206C3×Dic3—61no—
207C22×Q8—91no—
208C4×Q8—91no—
209C4×Q8—93no—
210C4×Q8—93no—
211C4×Q8(C3⋊D3)⋊C311yescharacteristic, complemented
212#117—93no—
213#117—93no—
214H(3,𝔽3)#51911yescharacteristic, complemented, Sylow
215C23×C3—91no—
216C22×D3—31no—
217C22×D3—91no—
218C3×Q8—34no—
219C3×Q8—34no—
220C3×Q8—64no—
221C3×Q8—912no—
222C3×Q8C3×C4×D314yes—
223C2×C3×C4—33no—
224C2×C3×C4—33no—
225C2×C3×C4—31no—
226C2×C3×C4—33no—
227C2×C3×C4—33no—
228C2×C3×C4—31no—
229C2×C3×C4—61no—
230C2×C3×C4—63no—
231C2×C3×C4—63no—
232C2×C3×C4—912no—
233C2×C3×C4—93no—
234C2×C3×C4—91no—
235C2×C3×C4—93no—
236C2×C3×C4C22×C3×D313yes—
237C2×C3×C4C22×C3×D311yescharacteristic
238C2×C3×C4C22×C3×D313yes—
239C2×Dic3—33no—
240C2×Dic3—31no—
241C2×Dic3—33no—
242C2×Dic3—93no—
243C2×Dic3—91no—
244C2×Dic3—93no—
245C4×D3—312no—
246C4×D3—912no—
247Dic6—312no—
248Dic6—912no—
249C2×C32—21no—
250C2×C32—21no—
251C2×C32—21no—
252C2×C32C22×C4×D311yescharacteristic
253C2×C32#100711yescharacteristic
254C2×C32#101011yescharacteristic
255C2×C32C23×C3×C411yescharacteristic
256C2×C32C3×2−1+411yescharacteristic
257C2×C32C3×2−1+411yescharacteristic
258C3⋊D3—22no—
259C3×D3—62no—
260C2×Q8—98no—
261C2×Q8—93no—
262C2×Q8C2×(C3⋊D3)⋊C311yescharacteristic
263C22×C4—93no—
264C4⋊C4—93no—
265C4⋊C4—93no—
266C4⋊C4—93no—
267C4⋊C4C2×(C3⋊D3)⋊C313yes—
268C22⋊C4—93no—
269C22⋊C4—91no—
270C42—96no—
271C42—93no—
272C42C2×(C3⋊D3)⋊C313yes—
273C22×C3—31no—
274C22×C3—31no—
275C22×C3—61no—
276C22×C3—92no—
277C22×C3—181no—
278C22×C3—181no—
279C22×C3C23×C3×D311yescharacteristic
280C2×D3—32no—
281C2×D3—61no—
282C2×D3—61no—
283C2×D3—92no—
284C2×D3—181no—
285C2×D3—181no—
286C3×C4—21no—
287C3×C4—23no—
288C3×C4—36no—
289C3×C4—36no—
290C3×C4—61no—
291C3×C4—61no—
292C3×C4—63no—
293C3×C4—63no—
294C3×C4—62no—
295C3×C4—66no—
296C3×C4—96no—
297C3×C4—123no—
298C3×C4—183no—
299C3×C4—181no—
300C3×C4C2×C3×C4×D316yes—
301Dic3—36no—
302Dic3—63no—
303Dic3—61no—
304Dic3—96no—
305Dic3—183no—
306Dic3—181no—
307C32—21no—
308C32#557211yescharacteristic, complemented
309C32C3×#51911yescharacteristic, complemented
310C23—91no—
311Q8—912no—
312Q8C4×(C3⋊D3)⋊C314yes—
313C2×C4—93no—
314C2×C4—912no—
315C2×C4—93no—
316C2×C4—91no—
317C2×C4C22×(C3⋊D3)⋊C313yes—
318C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
319C2×C4C22×(C3⋊D3)⋊C313yes—
320C2×C3—31no—
321C2×C3—31no—
322C2×C3—31no—
323C2×C3—31no—
324C2×C3—31no—
325C2×C3—31no—
326C2×C3—61no—
327C2×C3—61no—
328C2×C3—61no—
329C2×C3—182no—
330C2×C3C22×C3×C4×D311yescharacteristic
331C2×C3C3×#100711yescharacteristic
332C2×C3C3×#101011yescharacteristic
333D3—62no—
334D3—182no—
335C22—92no—
336C22—181no—
337C22—181no—
338C22C23×(C3⋊D3)⋊C311yescharacteristic
339C4—21no—
340C4—23no—
341C4—96no—
342C4—183no—
343C4—181no—
344C4C2×C4×(C3⋊D3)⋊C316yes—
345C3—31no—
346C3—31no—
347C3—61no—
348C3C3×#557211yescharacteristic
349C2—182no—
350C2C22×C4×(C3⋊D3)⋊C311yescharacteristic
351C2#1172262711yescharacteristic
352C2#1172263911yescharacteristic
subgroupG/H
centerC22C23×(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
hypocenterC32C3×#519
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
5. #64096⋊C3C2semidirect
7. C4×#634⋊C3C2semidirect
8. #10596043C3semidirect
11. Q8×(C3⋊D3)⋊C3C4semidirect
26. C32×C4×Q8D3semidirect
27. C32×C4×Q8C2×C3semidirect
28. Q8×C3⋊Dic3C2×C3semidirect
30. #64096C2×C3semidirect
32. C4×#634C2×C3semidirect
54. Q8×C3⋊D3C3×C4semidirect
211. C4×Q8(C3⋊D3)⋊C3semidirect
214. H(3,𝔽3)#519semidirect
308. C32#5572semidirect
309. C32C3×#519semidirect

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
#64101⋊C32
#64096⋊C32
#64081⋊C32
C4×#634⋊C32
#105960433
C3×#55723