Finite Group Databasebeta

← #10592828⋊C3 #13143212

Subgroups — #10592828⋊C3 #13143212

Order 1728.

621 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic
2C2×M16×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#63647⋊C3C214yesmaximal, complemented
5#10592828C311yesmaximal, characteristic, complemented
6C3×#5129—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#638⋊C3C412yes—
9#3828⋊C3—22no—
10#3828⋊C3—22no—
11C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic
12C2×C4×(C3⋊D3)⋊C3C2212yes—
13C2×#634⋊C3C411yescharacteristic
14M16×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2212yes—
16C2×(C3⋊(C3⋊C8))⋊C3—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18C2×#3828C2×C311yescharacteristic
19C2×C3×#235—31no—
20C2×C32×M16—21no—
21C2×C32×M16D311yescharacteristic, complemented
22C2×C32×M16C2×C311yescharacteristic, complemented
23C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
24C22×C3×C3⋊C8—31no—
25#63647C2×C314yescomplemented
26C3×(C2×D3)⋊C8—34no—
27C23×H(3,𝔽3)C2×C411yescharacteristic
28C22×(C3⋊D3)⋊C3C2×C412yes—
29C2×C4×H(3,𝔽3)C2×C411yescharacteristic
30C2×C4×H(3,𝔽3)D412yes—
31C2×C4×H(3,𝔽3)D412yes—
32C2×C4×H(3,𝔽3)C2311yescharacteristic
33#638⋊C3—24no—
34C2×(C3⋊Dic3)⋊C3C2×C412yes—
35C4⋊((C3⋊D3)⋊C3)—22no—
36C4×(C3⋊D3)⋊C3—22no—
37C4×(C3⋊D3)⋊C3—22no—
38#634⋊C3—22no—
39C8×H(3,𝔽3)—22no—
40(C3⋊(C3⋊C8))⋊C3—22no—
41C3×#408—91no—
42#5129—31no—
43#5129—91no—
44C22×C32×C4—21no—
45C22×C32×C4C2×D311yescharacteristic
46C22×C32×C4C22×C311yescharacteristic
47C2×#638C3×C412yes—
48#3828—22no—
49#3828—22no—
50C2×C4⋊(C3⋊D3)C3×C411yescharacteristic
51C2×C4×C3⋊D3C22×C312yes—
52C2×#634C3×C411yescharacteristic
53C2×C3×#67—32no—
54C3×#235—62no—
55C3×#235—62no—
56C2×C3×D12—31no—
57C2×C3×C4×D3—32no—
58C2×C3×Dic6—31no—
59C32×M16—24no—
60C32×M16—22no—
61C32×M16—22no—
62C2×C32×C8—22no—
63C2×C32×C8C2×D312yes—
64C2×C32×C8C22×C312yes—
65C2×C3⋊(C3⋊C8)—22no—
66C2×C3⋊(C3⋊C8)C22×C312yes—
67C2×C3×C3⋊C8—32no—
68C2×C3×C3⋊C8—62no—
69C22×H(3,𝔽3)—22no—
70C22×H(3,𝔽3)C22⋊C412yes—
71C22×H(3,𝔽3)C22×C411yescharacteristic
72C2×(C3⋊D3)⋊C3—22no—
73C2×(C3⋊D3)⋊C3—24no—
74C4×H(3,𝔽3)C22⋊C412yes—
75C4×H(3,𝔽3)C2×D411yescharacteristic
76C4×H(3,𝔽3)C2×D411yescharacteristic
77(C3⋊Dic3)⋊C3—22no—
78C2×C3×#41—91no—
79C2×#235—31no—
80C2×#235—91no—
81C2×C3×M16—31no—
82C2×C3×M16—31no—
83C2×C3×M16—61no—
84C2×C3×M16C3×D311yescharacteristic
85C22×C3×C8—91no—
86C22×C3⋊C8—31no—
87C22×C3⋊C8—91no—
88C3×C22⋊C8—94no—
89(C2×D3)⋊C8—34no—
90(C2×D3)⋊C8—94no—
91C23×C32—21no—
92C23×C32C4×D311yescharacteristic
93C23×C32C2×C3×C411yescharacteristic
94C22×C3⋊D3C2×C3×C412yes—
95C22×C3×D3—32no—
96C2×C32×C4—21no—
97C2×C32×C4—22no—
98C2×C32×C4—22no—
99C2×C32×C4—21no—
100C2×C32×C4C4×D311yescharacteristic
101C2×C32×C4D1212yes—
102C2×C32×C4#6712yes—
103C2×C32×C4C2×C3×C411yescharacteristic
104C2×C32×C4C3×D412yes—
105C2×C32×C4C3×D412yes—
106C2×C32×C4C22×D311yescharacteristic
107C2×C32×C4C23×C311yescharacteristic
108#638—24no—
109C2×C3⋊Dic3C2×C3×C412yes—
110C4⋊(C3⋊D3)—22no—
111C4×C3⋊D3—22no—
112C4×C3⋊D3—22no—
113#634—22no—
114C3×#67—64no—
115C2×C3×Dic3—32no—
116C3×D12—62no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119C3×Dic6—62no—
120C32×C8—24no—
121C32×C8—22no—
122C32×C8—22no—
123C3⋊(C3⋊C8)—22no—
124C3×C3⋊C8—62no—
125#408—91noSylow
126C2×H(3,𝔽3)—22no—
127C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
128C2×H(3,𝔽3)#13112yes—
129(C3⋊D3)⋊C3—42no—
130C3×#41—182no—
131C3×#41—182no—
132C2×C3×Q8—91no—
133C2×C3×D4—91no—
134C2×C3×D4—92no—
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—61no—
138C22×C3×C4—92no—
139C22×C3×C4C2×C3×D311yescharacteristic
140C2×#67—32no—
141C2×#67—92no—
142#235—62no—
143#235—62no—
144#235—182no—
145#235—182no—
146C2×D12—31no—
147C2×D12—91no—
148C2×C4×D3—32no—
149C2×C4×D3—92no—
150C2×Dic6—31no—
151C2×Dic6—91no—
152C3×M16—22no—
153C3×M16—62no—
154C3×M16—62no—
155C3×M16—64no—
156C2×C3×C8—32no—
157C2×C3×C8—32no—
158C2×C3×C8—62no—
159C2×C3×C8—92no—
160C2×C3×C8—182no—
161C2×C3×C8C2×C3×D312yes—
162C2×C3⋊C8—32no—
163C2×C3⋊C8—62no—
164C2×C3⋊C8—92no—
165C2×C3⋊C8—182no—
166C22×C32—22no—
167C22×C32—22no—
168C22×C32—22no—
169C22×C32—21no—
170C22×C32—42no—
171C22×C32(C2×D3)⋊C412yes—
172C22×C32C3×C22⋊C412yes—
173C22×C32C2×C4×D311yescharacteristic
174C22×C32C22×C3×C411yescharacteristic
175C2×C3⋊D3—22no—
176C2×C3⋊D3—24no—
177C2×C3×D3—64no—
178C2×C3×D3—62no—
179C32×C4—22no—
180C32×C4—21no—
181C32×C4—21no—
182C32×C4(C2×D3)⋊C412yes—
183C32×C4C3×C22⋊C412yes—
184C32×C4C2×D1211yescharacteristic
185C32×C4C2×#6711yescharacteristic
186C32×C4C2×C3×D411yescharacteristic
187C32×C4C2×C3×D411yescharacteristic
188C3⋊Dic3—22no—
189C3×Dic3—62no—
190C2×#41—91no—
191C2×M16(C3⋊D3)⋊C311yescharacteristic, complemented
192C22×C8—91no—
193C22⋊C8—94no—
194H(3,𝔽3)#40811yescharacteristic, complemented, Sylow
195C23×C3—31no—
196C23×C3—31no—
197C23×C3—61no—
198C23×C3—92no—
199C23×C3C3×C4×D311yescharacteristic
200C22×D3—32no—
201C22×D3—92no—
202C3×Q8—182no—
203C3×D4—182no—
204C3×D4—184no—
205C2×C3×C4—31no—
206C2×C3×C4—31no—
207C2×C3×C4—32no—
208C2×C3×C4—32no—
209C2×C3×C4—32no—
210C2×C3×C4—32no—
211C2×C3×C4—31no—
212C2×C3×C4—31no—
213C2×C3×C4—61no—
214C2×C3×C4—62no—
215C2×C3×C4—62no—
216C2×C3×C4—61no—
217C2×C3×C4—92no—
218C2×C3×C4—182no—
219C2×C3×C4—182no—
220C2×C3×C4C3×C4×D311yescharacteristic
221C2×C3×C4C3×D1212yes—
222C2×C3×C4C3×#6712yes—
223C2×C3×C4C22×C3×D311yescharacteristic
224#67—64no—
225#67—184no—
226C2×Dic3—32no—
227C2×Dic3—92no—
228D12—62no—
229D12—182no—
230C4×D3—62no—
231C4×D3—62no—
232C4×D3—182no—
233C4×D3—182no—
234Dic6—62no—
235Dic6—182no—
236C3×C8—22no—
237C3×C8—62no—
238C3×C8—62no—
239C3×C8—64no—
240C3×C8—182no—
241C3⋊C8—62no—
242C3⋊C8—182no—
243C2×C32—22no—
244C2×C32—22no—
245C2×C32—22no—
246C2×C32—21no—
247C2×C32—42no—
248C2×C32#90712yes—
249C2×C32C2×(C2×D3)⋊C411yescharacteristic
250C2×C32C2×C3×C22⋊C411yescharacteristic
251C2×C32C3×#13112yes—
252C3⋊D3—42no—
253C3×D3—122no—
254#41—182no—
255#41—182no—
256C2×Q8—91no—
257C2×D4—92no—
258C2×D4—91no—
259C22×C4—92no—
260C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
261M16—22no—
262C2×C8—92no—
263C2×C8—182no—
264C2×C8C2×(C3⋊D3)⋊C312yes—
265C22×C3—22no—
266C22×C3—31no—
267C22×C3—31no—
268C22×C3—32no—
269C22×C3—32no—
270C22×C3—62no—
271C22×C3—62no—
272C22×C3—61no—
273C22×C3—62no—
274C22×C3—122no—
275C22×C3—182no—
276C22×C3—184no—
277C22×C3C3×(C2×D3)⋊C412yes—
278C22×C3C2×C3×C4×D311yescharacteristic
279C2×D3—64no—
280C2×D3—62no—
281C2×D3—182no—
282C2×D3—184no—
283C3×C4—31no—
284C3×C4—31no—
285C3×C4—31no—
286C3×C4—31no—
287C3×C4—32no—
288C3×C4—32no—
289C3×C4—61no—
290C3×C4—61no—
291C3×C4—62no—
292C3×C4—182no—
293C3×C4C3×(C2×D3)⋊C412yes—
294C3×C4C2×C3×D1211yescharacteristic
295C3×C4C2×C3×#6711yescharacteristic
296Dic3—62no—
297Dic3—182no—
298C32—21no—
299C32#512911yescharacteristic, complemented
300C32C3×#40811yescharacteristic, complemented
301C23—92no—
302C23C4×(C3⋊D3)⋊C311yescharacteristic
303Q8—182no—
304D4—184no—
305D4—182no—
306C2×C4—92no—
307C2×C4—182no—
308C2×C4—182no—
309C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
310C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
311C2×C4#638⋊C312yes—
312C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
313C8—22no—
314C8—182no—
315C2×C3—22no—
316C2×C3—32no—
317C2×C3—32no—
318C2×C3—31no—
319C2×C3—31no—
320C2×C3—62no—
321C2×C3—62no—
322C2×C3—62no—
323C2×C3—61no—
324C2×C3—122no—
325C2×C3—362no—
326C2×C3C3×#90712yes—
327C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
328D3—122no—
329D3—362no—
330C22—22no—
331C22—184no—
332C22—182no—
333C22#3752⋊C312yes—
334C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
335C4—182no—
336C4#3752⋊C312yes—
337C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
338C4C2×#638⋊C311yescharacteristic
339C3—31no—
340C3—31no—
341C3—61no—
342C3C3×#512911yescharacteristic
343C2—22no—
344C2—362no—
345C2#64116⋊C312yes—
346C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×H(3,𝔽3)C2
socleC22×C3C2×C3×C4×D3
prime radicalC2×#3828⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenterC2×M16(C3⋊D3)⋊C3
hypocenterC32C3×#408
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×M16×H(3,𝔽3)C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. #63647⋊C3C2semidirect
5. #10592828C3semidirect
21. C2×C32×M16D3semidirect
22. C2×C32×M16C2×C3semidirect
23. C22×C3⋊(C3⋊C8)C2×C3semidirect
25. #63647C2×C3semidirect
191. C2×M16(C3⋊D3)⋊C3semidirect
194. H(3,𝔽3)#408semidirect
299. C32#5129semidirect
300. C32C3×#408semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C2×M16×H(3,𝔽3)2
C22×(C3⋊(C3⋊C8))⋊C32
#63647⋊C32
#105928283
C3×#51293