Finite Group Databasebeta

← #10592538⋊C3 #13142060

Subgroups — #10592538⋊C3 #13142060

Order 1728.

437 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3743⋊C3C211yesmaximal, characteristic
2C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3#64081⋊C3C211yesmaximal, characteristic
4H(3,𝔽3)×C4⋊C8C211yesmaximal, characteristic, complemented
5#63641⋊C3C211yesmaximal, characteristic, complemented
6#63641⋊C3C211yesmaximal, characteristic, complemented
7C4⋊((C3⋊(C3⋊C8))⋊C3)C211yesmaximal, characteristic, complemented
8#10592538C311yesmaximal, characteristic, complemented
9C3×#4839—31nomaximal
10C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
11C2×C8×H(3,𝔽3)C2211yescharacteristic
12C2×C8×H(3,𝔽3)C2211yescharacteristic
13C42×H(3,𝔽3)C2211yescharacteristic
14#3752⋊C3C411yescharacteristic
15#3752⋊C3C411yescharacteristic
16#3750⋊C3C411yescharacteristic
17#3750⋊C3C411yescharacteristic
18C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
20C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
21#3743⋊C3—22no—
22C8×(C3⋊D3)⋊C3—22no—
23C2×#3743C2×C311yescharacteristic
24C2×C8×C3⋊D3C2×C311yescharacteristic
25#64081C2×C311yescharacteristic
26C2×C3×ΓL(1,𝔽25)—31no—
27C2×C3×D3×C8—31no—
28C3×#872—31no—
29C32×C4⋊C8—21no—
30C32×C4⋊C8D311yescharacteristic, complemented
31C32×C4⋊C8C2×C311yescharacteristic, complemented
32#63641C2×C311yescharacteristic, complemented
33#63641C2×C311yescharacteristic, complemented
34C4⋊(C3⋊(C3⋊C8))C2×C311yescharacteristic, complemented
35C3×Dic3⋊C8—31no—
36C3×Dic3⋊C8—31no—
37C3×C4⋊(C3⋊C8)—31no—
38C22×(C3⋊D3)⋊C3C2×C411yescharacteristic
39C2×C4×H(3,𝔽3)C2×C411yescharacteristic
40C2×C4×H(3,𝔽3)C2×C411yescharacteristic
41C2×C4×H(3,𝔽3)C2311yescharacteristic
42C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
43C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
44C2×(C3⋊Dic3)⋊C3C2×C411yescharacteristic
45C4×(C3⋊D3)⋊C3D412yes—
46C4×(C3⋊D3)⋊C3Q812yes—
47C8×H(3,𝔽3)—21no—
48C8×H(3,𝔽3)—21no—
49(C3⋊(C3⋊C8))⋊C3—21no—
50(C3⋊(C3⋊C8))⋊C3—21no—
51C3×#424—91no—
52#4839—31no—
53#4839—91no—
54C2×C4×C3⋊D3C22×C311yescharacteristic
55C2×C3×C4×D3—31no—
56C2×C32×C8—21no—
57C2×C32×C8—21no—
58C2×C32×C8C2×D311yescharacteristic
59C2×C32×C8C2×D311yescharacteristic
60C2×C32×C8C22×C311yescharacteristic
61C2×C32×C8C22×C311yescharacteristic
62C32×C42—21no—
63C32×C42C2×D311yescharacteristic
64C32×C42C22×C311yescharacteristic
65#3752C3×C411yescharacteristic
66#3752C3×C411yescharacteristic
67#3750C3×C411yescharacteristic
68#3750C3×C411yescharacteristic
69C4×C3⋊Dic3C22×C311yescharacteristic
70C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
71C2×C3⋊(C3⋊C8)C22×C311yescharacteristic
72#3743—22no—
73C8×C3⋊D3—22no—
74C3×(C2×D3)⋊C4—31no—
75C3×(C2×D3)⋊C4—31no—
76C3×Dic3⋊C4—31no—
77C3×Dic3⋊C4—31no—
78C3×C4×Dic3—31no—
79C2×C3×C3⋊C8—31no—
80C2×C3×C3⋊C8—31no—
81C3×ΓL(1,𝔽25)—62no—
82C3×D3×C8—62no—
83C22×H(3,𝔽3)C22×C411yescharacteristic
84C2×(C3⋊D3)⋊C3—21no—
85C2×(C3⋊D3)⋊C3—21no—
86C2×(C3⋊D3)⋊C3C4⋊C412yes—
87C4×H(3,𝔽3)—21no—
88C4×H(3,𝔽3)—21no—
89C4×H(3,𝔽3)C2×D411yescharacteristic
90C4×H(3,𝔽3)C2×Q811yescharacteristic
91(C3⋊Dic3)⋊C3—21no—
92(C3⋊Dic3)⋊C3—21no—
93(C3⋊Dic3)⋊C3C4⋊C412yes—
94C2×C3×M16—91no—
95C22×C3×C8—91no—
96C3×#117—91no—
97C2×ΓL(1,𝔽25)—31no—
98C2×ΓL(1,𝔽25)—91no—
99C2×D3×C8—31no—
100C2×D3×C8—91no—
101#872—31no—
102#872—91no—
103C3×C4⋊C8—31no—
104C3×C4⋊C8—31no—
105C3×C4⋊C8—61no—
106C3×C4⋊C8—91no—
107C3×C4⋊C8—91no—
108C3×C4⋊C8—91no—
109C3×C4⋊C8C3×D311yescharacteristic
110Dic3⋊C8—31no—
111Dic3⋊C8—31no—
112Dic3⋊C8—91no—
113Dic3⋊C8—91no—
114C4⋊(C3⋊C8)—31no—
115C4⋊(C3⋊C8)—91no—
116C22×C3⋊D3C2×C3×C411yescharacteristic
117C22×C3×D3—31no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4C4×D311yescharacteristic
122C2×C32×C4C4×D311yescharacteristic
123C2×C32×C4C2×C3×C411yescharacteristic
124C2×C32×C4C2×C3×C411yescharacteristic
125C2×C32×C4C22×D311yescharacteristic
126C2×C32×C4C23×C311yescharacteristic
127C2×C3⋊Dic3C2×C3×C411yescharacteristic
128C2×C3⋊Dic3C2×C3×C411yescharacteristic
129C2×C3⋊Dic3C2×C3×C411yescharacteristic
130C4×C3⋊D3C3×D412yes—
131C4×C3⋊D3C3×Q812yes—
132C2×C3×Dic3—31no—
133C2×C3×Dic3—31no—
134C2×C3×Dic3—31no—
135C3×C4×D3—32no—
136C3×C4×D3—32no—
137C32×C8—21no—
138C32×C8—21no—
139C32×C8—21no—
140C32×C8—21no—
141C32×C8—41no—
142C32×C8—41no—
143C3⋊(C3⋊C8)—21no—
144C3⋊(C3⋊C8)—21no—
145C3×C3⋊C8—61no—
146C3×C3⋊C8—61no—
147#424—91noSylow
148C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
149C2×H(3,𝔽3)#13111yescharacteristic
150C2×H(3,𝔽3)#13111yescharacteristic
151(C3⋊D3)⋊C3—22no—
152C22×C3×C4—91no—
153C2×C4×D3—31no—
154C2×C4×D3—91no—
155C3×M16—182no—
156C2×C3×C8—31no—
157C2×C3×C8—31no—
158C2×C3×C8—31no—
159C2×C3×C8—31no—
160C2×C3×C8—61no—
161C2×C3×C8—61no—
162C2×C3×C8—91no—
163C2×C3×C8—91no—
164C2×C3×C8—182no—
165C2×C3×C8C2×C3×D311yescharacteristic
166C2×C3×C8C2×C3×D311yescharacteristic
167C3×C4⋊C4—91no—
168C3×C4⋊C4—91no—
169C3×C22⋊C4—91no—
170C3×C22⋊C4—91no—
171C3×C42—31no—
172C3×C42—31no—
173C3×C42—61no—
174C3×C42—91no—
175C3×C42C2×C3×D311yescharacteristic
176(C2×D3)⋊C4—31no—
177(C2×D3)⋊C4—31no—
178(C2×D3)⋊C4—91no—
179(C2×D3)⋊C4—91no—
180Dic3⋊C4—31no—
181Dic3⋊C4—31no—
182Dic3⋊C4—91no—
183Dic3⋊C4—91no—
184C4×Dic3—31no—
185C4×Dic3—91no—
186C2×C3⋊C8—31no—
187C2×C3⋊C8—31no—
188C2×C3⋊C8—91no—
189C2×C3⋊C8—91no—
190ΓL(1,𝔽25)—62no—
191ΓL(1,𝔽25)—182no—
192D3×C8—62no—
193D3×C8—182no—
194C22×C32—21no—
195C22×C32C2×C4×D311yescharacteristic
196C22×C32C22×C3×C411yescharacteristic
197C2×C3⋊D3—21no—
198C2×C3⋊D3—21no—
199C2×C3⋊D3C3×C4⋊C412yes—
200C2×C3×D3—32no—
201C2×C3×D3—61no—
202C2×C3×D3—61no—
203C32×C4—22no—
204C32×C4—21no—
205C32×C4—21no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4D3×D411yescharacteristic
212C32×C4D3×Q811yescharacteristic
213C32×C4C2×C3×D411yescharacteristic
214C32×C4C2×C3×Q811yescharacteristic
215C3⋊Dic3—21no—
216C3⋊Dic3—21no—
217C3⋊Dic3C3×C4⋊C412yes—
218C3×Dic3—32no—
219C3×Dic3—61no—
220C3×Dic3—61no—
221C2×M16—91no—
222C22×C8—91no—
223#117—91no—
224C4⋊C8—91no—
225C4⋊C8—91no—
226C4⋊C8—91no—
227C4⋊C8(C3⋊D3)⋊C311yescharacteristic, complemented
228H(3,𝔽3)#42411yescharacteristic, complemented, Sylow
229C23×C3—91no—
230C22×D3—31no—
231C22×D3—91no—
232C2×C3×C4—31no—
233C2×C3×C4—31no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—31no—
237C2×C3×C4—31no—
238C2×C3×C4—61no—
239C2×C3×C4—61no—
240C2×C3×C4—61no—
241C2×C3×C4—92no—
242C2×C3×C4—92no—
243C2×C3×C4—91no—
244C2×C3×C4—91no—
245C2×C3×C4—91no—
246C2×C3×C4C3×C4×D311yescharacteristic
247C2×C3×C4C3×C4×D311yescharacteristic
248C2×C3×C4C22×C3×D311yescharacteristic
249C2×Dic3—31no—
250C2×Dic3—31no—
251C2×Dic3—31no—
252C2×Dic3—91no—
253C2×Dic3—91no—
254C2×Dic3—91no—
255C4×D3—32no—
256C4×D3—32no—
257C4×D3—92no—
258C4×D3—92no—
259C3×C8—21no—
260C3×C8—21no—
261C3×C8—61no—
262C3×C8—61no—
263C3×C8—61no—
264C3×C8—61no—
265C3×C8—121no—
266C3×C8—121no—
267C3×C8—181no—
268C3×C8—181no—
269C3⋊C8—61no—
270C3⋊C8—61no—
271C3⋊C8—181no—
272C3⋊C8—181no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—21no—
276C2×C32D3×C4⋊C411yescharacteristic
277C2×C32#90111yescharacteristic
278C2×C32#90711yescharacteristic
279C2×C32C2×C3×C4⋊C411yescharacteristic
280C2×C32C3×#13111yescharacteristic
281C2×C32C3×#13111yescharacteristic
282C3⋊D3—22no—
283C3×D3—62no—
284C22×C4—91no—
285M16—182no—
286C2×C8—91no—
287C2×C8—91no—
288C2×C8—182no—
289C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
290C2×C8C2×(C3⋊D3)⋊C311yescharacteristic
291C4⋊C4—91no—
292C4⋊C4—91no—
293C22⋊C4—91no—
294C22⋊C4—91no—
295C42—91no—
296C42C2×(C3⋊D3)⋊C311yescharacteristic
297C22×C3—31no—
298C22×C3—31no—
299C22×C3—61no—
300C22×C3—92no—
301C22×C3—181no—
302C22×C3—181no—
303C22×C3C2×C3×C4×D311yescharacteristic
304C2×D3—32no—
305C2×D3—61no—
306C2×D3—61no—
307C2×D3—92no—
308C2×D3—181no—
309C2×D3—181no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—31no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—61no—
317C3×C4—61no—
318C3×C4—61no—
319C3×C4—61no—
320C3×C4—62no—
321C3×C4—61no—
322C3×C4—61no—
323C3×C4—92no—
324C3×C4—121no—
325C3×C4—181no—
326C3×C4—181no—
327C3×C4C3×D3×D411yescharacteristic
328C3×C4C3×D3×Q811yescharacteristic
329Dic3—32no—
330Dic3—61no—
331Dic3—61no—
332Dic3—92no—
333Dic3—181no—
334Dic3—181no—
335C32—21no—
336C32#483911yescharacteristic, complemented
337C32C3×#42411yescharacteristic, complemented
338C23—91no—
339C2×C4—91no—
340C2×C4—92no—
341C2×C4—92no—
342C2×C4—91no—
343C2×C4—91no—
344C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
345C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
346C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
347C8—21no—
348C8—21no—
349C8—181no—
350C8—181no—
351C2×C3—31no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—61no—
358C2×C3—61no—
359C2×C3—61no—
360C2×C3—182no—
361C2×C3C3×D3×C4⋊C411yescharacteristic
362C2×C3C3×#90111yescharacteristic
363C2×C3C3×#90711yescharacteristic
364D3—62no—
365D3—182no—
366C22—92no—
367C22—181no—
368C22—181no—
369C22C2×C4×(C3⋊D3)⋊C311yescharacteristic
370C4—21no—
371C4—21no—
372C4—92no—
373C4—181no—
374C4—181no—
375C4D4×(C3⋊D3)⋊C311yescharacteristic
376C4Q8×(C3⋊D3)⋊C311yescharacteristic
377C3—31no—
378C3—31no—
379C3—61no—
380C3C3×#483911yescharacteristic
381C2—182no—
382C2C4⋊C4×(C3⋊D3)⋊C311yescharacteristic
383C2#64110⋊C311yescharacteristic
384C2#64116⋊C311yescharacteristic
subgroupG/H
centerC2×C4C22×(C3⋊D3)⋊C3
commutatorC22×C32C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingH(3,𝔽3)×C4⋊C8C2
socleC22×C3C2×C3×C4×D3
prime radicalC22×(C3⋊D3)⋊C3C2×C4
cosocleC2×C32×C4C23×C3
hypercenterC4⋊C8(C3⋊D3)⋊C3
hypocenterC32C3×#424
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
4. H(3,𝔽3)×C4⋊C8C2semidirect
5. #63641⋊C3C2semidirect
6. #63641⋊C3C2semidirect
7. C4⋊((C3⋊(C3⋊C8))⋊C3)C2semidirect
8. #10592538C3semidirect
30. C32×C4⋊C8D3semidirect
31. C32×C4⋊C8C2×C3semidirect
32. #63641C2×C3semidirect
33. #63641C2×C3semidirect
34. C4⋊(C3⋊(C3⋊C8))C2×C3semidirect
227. C4⋊C8(C3⋊D3)⋊C3semidirect
228. H(3,𝔽3)#424semidirect
336. C32#4839semidirect
337. C32C3×#424semidirect

Maximal subgroups

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

subgroupindex
C2×#3743⋊C32
C2×C8×(C3⋊D3)⋊C32
#64081⋊C32
H(3,𝔽3)×C4⋊C82
#63641⋊C32
C4⋊((C3⋊(C3⋊C8))⋊C3)2
#105925383
C3×#48393