Finite Group Databasebeta

← C2×#63648⋊C3 #13143127

Subgroups — C2×#63648⋊C3 #13143127

Order 1728.

783 conjugacy classes of subgroups.

The subgroup lattice has 387 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C22×C8×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C2×C4⋊((C3⋊Dic3)⋊C3)C211yesmaximal, characteristic, complemented
4#63648⋊C3C214yesmaximal, direct product factor
5C2×#63648C311yesmaximal, characteristic, complemented
6C2×C3×#821—31nomaximal
7C23×(C3⋊D3)⋊C3—21no—
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9C22×(C3⋊Dic3)⋊C3—21no—
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)—22no—
12C2×C4⋊((C3⋊D3)⋊C3)C414yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C2×C8×H(3,𝔽3)—22no—
15C2×C8×H(3,𝔽3)C2212yescomplemented
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
18C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
19C22×C3×D12—31no—
20C22×C32×C8—21no—
21C22×C32×C8D311yescharacteristic, complemented
22C22×C32×C8C2×C311yescharacteristic, complemented
23C2×C4⋊(C3⋊Dic3)C2×C311yescharacteristic, complemented
24C2×C3×C4⋊Dic3—31no—
25#63648C2×C314yescomplemented
26C3×#821—34no—
27C23×H(3,𝔽3)D411yescharacteristic
28C22×(C3⋊D3)⋊C3—24no—
29C22×(C3⋊D3)⋊C3—22no—
30C22×(C3⋊D3)⋊C3—42no—
31C22×(C3⋊D3)⋊C3—42no—
32C2×C4×H(3,𝔽3)C2×C412yes—
33C2×C4×H(3,𝔽3)D411yescharacteristic
34C2×C4×H(3,𝔽3)D412yes—
35C2×C4×H(3,𝔽3)C2311yescharacteristic
36C2×(C3⋊Dic3)⋊C3—22no—
37C2×(C3⋊Dic3)⋊C3—42no—
38C4⋊((C3⋊D3)⋊C3)—24no—
39C4⋊((C3⋊D3)⋊C3)—22no—
40C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
41C8×H(3,𝔽3)—22no—
42C2×C3×#102—91no—
43C2×#821—31no—
44C2×#821—91no—
45C23×C3⋊D3—21no—
46C23×C3×D3—61no—
47C22×C32×C4—21no—
48C22×C32×C4C2×D311yescharacteristic
49C22×C32×C4C22×C311yescharacteristic
50C22×C3⋊Dic3—21no—
51C2×C4⋊(C3⋊D3)—21no—
52C2×C4⋊(C3⋊D3)—22no—
53C2×C4⋊(C3⋊D3)C3×C414yescomplemented
54C2×C4⋊(C3⋊D3)C22×C312yes—
55C22×C3×Dic3—61no—
56C2×C3×D12—34no—
57C2×C3×D12—32no—
58C2×C3×D12—61no—
59C2×C3×D12—62no—
60C2×C32×C8—24no—
61C2×C32×C8—22no—
62C2×C32×C8—22no—
63C2×C32×C8—22no—
64C2×C32×C8C2×D312yescomplemented
65C2×C32×C8C22×C312yescomplemented
66C4⋊(C3⋊Dic3)—21no—
67C4⋊(C3⋊Dic3)C22×C312yescomplemented
68C3×C4⋊Dic3—32no—
69C3×C4⋊Dic3—61no—
70C22×H(3,𝔽3)C22⋊C412yes—
71C22×H(3,𝔽3)D812yes—
72C22×H(3,𝔽3)SD1612yes—
73C22×H(3,𝔽3)C2×D411yescharacteristic
74C2×(C3⋊D3)⋊C3—24no—
75C2×(C3⋊D3)⋊C3—48no—
76C2×(C3⋊D3)⋊C3—42no—
77C2×(C3⋊D3)⋊C3—42no—
78C4×H(3,𝔽3)C22⋊C412yes—
79C4×H(3,𝔽3)C22×C411yescharacteristic
80C4×H(3,𝔽3)C2×D411yescharacteristic
81(C3⋊Dic3)⋊C3—42no—
82C22×C3×D4—91no—
83C22×D12—31no—
84C22×D12—91no—
85C22×C3×C8—31no—
86C22×C3×C8—31no—
87C22×C3×C8—61no—
88C22×C3×C8C3×D311yescharacteristic
89C2×C3×C4⋊C4—91no—
90C2×C4⋊Dic3—31no—
91C2×C4⋊Dic3—91no—
92C3×#102—94no—
93#821—34no—
94#821—94no—
95C23×C32—21no—
96C23×C32D1211yescharacteristic
97C23×C32C3×D411yescharacteristic
98C22×C3⋊D3—24no—
99C22×C3⋊D3—22no—
100C22×C3⋊D3—42no—
101C22×C3⋊D3—42no—
102C22×C3×D3—64no—
103C22×C3×D3—62no—
104C22×C3×D3—122no—
105C22×C3×D3—122no—
106C2×C32×C4—22no—
107C2×C32×C4—21no—
108C2×C32×C4—22no—
109C2×C32×C4—21no—
110C2×C32×C4C4×D312yes—
111C2×C32×C4D1211yescharacteristic
112C2×C32×C4#6712yes—
113C2×C32×C4C2×C3×C412yes—
114C2×C32×C4C3×D411yescharacteristic
115C2×C32×C4C3×D412yes—
116C2×C32×C4C22×D311yescharacteristic
117C2×C32×C4C23×C311yescharacteristic
118C2×C3⋊Dic3—22no—
119C2×C3⋊Dic3—42no—
120C4⋊(C3⋊D3)—24no—
121C4⋊(C3⋊D3)—22no—
122C4⋊(C3⋊D3)C2×C3×C414yescomplemented
123C2×C3×Dic3—62no—
124C2×C3×Dic3—122no—
125C3×D12—34no—
126C3×D12—64no—
127C3×D12—62no—
128C32×C8—24no—
129C32×C8—22no—
130C32×C8—22no—
131C2×#102—91noSylow
132C2×H(3,𝔽3)#10214yescomplemented
133C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
134C2×H(3,𝔽3)C2×D811yescharacteristic
135C2×H(3,𝔽3)C2×SD1611yescharacteristic
136(C3⋊D3)⋊C3—44no—
137C24×C3—181no—
138C23×D3—61no—
139C23×D3—181no—
140C2×C3×D4—92no—
141C2×C3×D4—94no—
142C2×C3×D4—181no—
143C2×C3×D4—182no—
144C22×C3×C4—31no—
145C22×C3×C4—31no—
146C22×C3×C4—61no—
147C22×C3×C4—181no—
148C22×C3×C4C2×C3×D311yescharacteristic
149C22×Dic3—61no—
150C22×Dic3—181no—
151C2×D12—34no—
152C2×D12—32no—
153C2×D12—61no—
154C2×D12—62no—
155C2×D12—92no—
156C2×D12—94no—
157C2×D12—181no—
158C2×D12—182no—
159C2×C3×C8—22no—
160C2×C3×C8—32no—
161C2×C3×C8—32no—
162C2×C3×C8—62no—
163C2×C3×C8—62no—
164C2×C3×C8—64no—
165C2×C3×C8—62no—
166C2×C3×C8C2×C3×D312yes—
167C3×C4⋊C4—92no—
168C3×C4⋊C4—181no—
169C4⋊Dic3—32no—
170C4⋊Dic3—61no—
171C4⋊Dic3—92no—
172C4⋊Dic3—181no—
173C22×C32—22no—
174C22×C32—22no—
175C22×C32—22no—
176C22×C32—21no—
177C22×C32#20412yes—
178C22×C32D2412yes—
179C22×C32(C2×D3)⋊C412yes—
180C22×C32C3×C22⋊C412yes—
181C22×C32C3×D812yes—
182C22×C32C3×SD1612yes—
183C22×C32C2×D1211yescharacteristic
184C22×C32C2×C3×D411yescharacteristic
185C2×C3⋊D3—24no—
186C2×C3⋊D3—48no—
187C2×C3⋊D3—42no—
188C2×C3⋊D3—42no—
189C2×C3×D3—64no—
190C2×C3×D3—128no—
191C2×C3×D3—122no—
192C2×C3×D3—122no—
193C32×C4—22no—
194C32×C4—21no—
195C32×C4—21no—
196C32×C4(C2×D3)⋊C412yes—
197C32×C4C3×C22⋊C412yes—
198C32×C4C2×C4×D311yescharacteristic
199C32×C4C2×#6711yescharacteristic
200C32×C4C22×C3×C411yescharacteristic
201C32×C4C2×C3×D411yescharacteristic
202C3⋊Dic3—42no—
203C3×Dic3—122no—
204C22×D4—91no—
205C22×C8(C3⋊D3)⋊C311yescharacteristic, complemented
206C2×C4⋊C4—91no—
207#102—94no—
208H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
209C23×C3—31no—
210C23×C3—31no—
211C23×C3—61no—
212C23×C3—182no—
213C23×C3—184no—
214C23×C3—362no—
215C23×C3—362no—
216C23×C3C3×D1211yescharacteristic
217C22×D3—64no—
218C22×D3—62no—
219C22×D3—122no—
220C22×D3—122no—
221C22×D3—182no—
222C22×D3—184no—
223C22×D3—362no—
224C22×D3—362no—
225C3×D4—94no—
226C3×D4—182no—
227C3×D4—184no—
228C2×C3×C4—32no—
229C2×C3×C4—32no—
230C2×C3×C4—31no—
231C2×C3×C4—31no—
232C2×C3×C4—32no—
233C2×C3×C4—32no—
234C2×C3×C4—31no—
235C2×C3×C4—31no—
236C2×C3×C4—62no—
237C2×C3×C4—61no—
238C2×C3×C4—62no—
239C2×C3×C4—61no—
240C2×C3×C4—182no—
241C2×C3×C4—362no—
242C2×C3×C4C3×C4×D312yes—
243C2×C3×C4C3×D1211yescharacteristic
244C2×C3×C4C3×#6712yes—
245C2×C3×C4C22×C3×D311yescharacteristic
246C2×Dic3—62no—
247C2×Dic3—122no—
248C2×Dic3—182no—
249C2×Dic3—362no—
250D12—34no—
251D12—64no—
252D12—62no—
253D12—94no—
254D12—182no—
255D12—184no—
256C3×C8—22no—
257C3×C8—62no—
258C3×C8—62no—
259C3×C8—64no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—24no—
264C2×C32#82114yescomplemented
265C2×C32C3×#10214yescomplemented
266C2×C32C2×#20411yescharacteristic
267C2×C32C2×D2411yescharacteristic
268C2×C32C2×(C2×D3)⋊C411yescharacteristic
269C2×C32C2×C3×C22⋊C411yescharacteristic
270C2×C32C2×C3×D811yescharacteristic
271C2×C32C2×C3×SD1611yescharacteristic
272C3⋊D3—44no—
273C3×D3—124no—
274C24—181no—
275C2×D4—94no—
276C2×D4—92no—
277C2×D4—181no—
278C2×D4—182no—
279C22×C4—181no—
280C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
281C2×C8—22no—
282C2×C8C2×(C3⋊D3)⋊C312yescomplemented
283C4⋊C4—92no—
284C4⋊C4—181no—
285C22×C3—31no—
286C22×C3—31no—
287C22×C3—32no—
288C22×C3—32no—
289C22×C3—32no—
290C22×C3—32no—
291C22×C3—32no—
292C22×C3—32no—
293C22×C3—61no—
294C22×C3—62no—
295C22×C3—62no—
296C22×C3—62no—
297C22×C3—184no—
298C22×C3—362no—
299C22×C3—362no—
300C22×C3—368no—
301C22×C3C3×#20412yes—
302C22×C3C3×D2412yes—
303C22×C3C3×(C2×D3)⋊C412yes—
304C22×C3C2×C3×D1211yescharacteristic
305C2×D3—64no—
306C2×D3—122no—
307C2×D3—122no—
308C2×D3—128no—
309C2×D3—184no—
310C2×D3—362no—
311C2×D3—362no—
312C2×D3—368no—
313C3×C4—31no—
314C3×C4—31no—
315C3×C4—31no—
316C3×C4—31no—
317C3×C4—32no—
318C3×C4—32no—
319C3×C4—61no—
320C3×C4—61no—
321C3×C4—62no—
322C3×C4—362no—
323C3×C4C3×(C2×D3)⋊C412yes—
324C3×C4C2×C3×C4×D311yescharacteristic
325C3×C4C2×C3×#6711yescharacteristic
326Dic3—122no—
327Dic3—362no—
328C32—21no—
329C32C2×#82111yescharacteristic, complemented
330C32C2×C3×#10211yescharacteristic, complemented
331C23—182no—
332C23—184no—
333C23—362no—
334C23—362no—
335C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
336D4—94no—
337D4—182no—
338D4—184no—
339C2×C4—182no—
340C2×C4—362no—
341C2×C4C4×(C3⋊D3)⋊C312yes—
342C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
343C2×C4#638⋊C312yes—
344C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
345C8—22no—
346C2×C3—31no—
347C2×C3—31no—
348C2×C3—31no—
349C2×C3—31no—
350C2×C3—31no—
351C2×C3—31no—
352C2×C3—34no—
353C2×C3—34no—
354C2×C3—61no—
355C2×C3—61no—
356C2×C3—61no—
357C2×C3—64no—
358C2×C3—364no—
359C2×C3C3×#82114yes—
360C2×C3C2×C3×#20411yescharacteristic
361C2×C3C2×C3×D2411yescharacteristic
362C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
363D3—124no—
364D3—364no—
365C22—184no—
366C22—362no—
367C22—362no—
368C22—368no—
369C22#3744⋊C312yes—
370C22#3745⋊C312yes—
371C22#3752⋊C312yes—
372C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
373C4—362no—
374C4#3752⋊C312yes—
375C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
376C4C2×#638⋊C311yescharacteristic
377C3—31no—
378C3—31no—
379C3—61no—
380C3C2×C3×#82111yescharacteristic
381C2—364no—
382C2#63648⋊C314yesdirect product factor
383C2C2×#3744⋊C311yescharacteristic
384C2C2×#3745⋊C311yescharacteristic
385C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC23C4⋊((C3⋊D3)⋊C3)
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC22×C8×H(3,𝔽3)C2
socleC23×C3C3×D12
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
cosocleC2×C32×C4C23×C3
hypercenterC22×C8(C3⋊D3)⋊C3
hypocenterC32C2×C3×#102
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
2. C22×C8×H(3,𝔽3)C2semidirect
3. C2×C4⋊((C3⋊Dic3)⋊C3)C2semidirect
4. #63648⋊C3C2direct (N × H)
5. C2×#63648C3semidirect
12. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
15. C2×C8×H(3,𝔽3)C22semidirect
17. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
21. C22×C32×C8D3semidirect
22. C22×C32×C8C2×C3semidirect
23. C2×C4⋊(C3⋊Dic3)C2×C3semidirect
25. #63648C2×C3semidirect
40. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
53. C2×C4⋊(C3⋊D3)C3×C4semidirect
64. C2×C32×C8C2×D3semidirect
65. C2×C32×C8C22×C3semidirect
67. C4⋊(C3⋊Dic3)C22×C3semidirect
122. C4⋊(C3⋊D3)C2×C3×C4semidirect
132. C2×H(3,𝔽3)#102semidirect
205. C22×C8(C3⋊D3)⋊C3semidirect
208. H(3,𝔽3)C2×#102semidirect
264. C2×C32#821semidirect
265. C2×C32C3×#102semidirect
282. C2×C8C2×(C3⋊D3)⋊C3semidirect
329. C32C2×#821semidirect
330. C32C2×C3×#102semidirect
382. C2#63648⋊C3direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊((C3⋊D3)⋊C3)2
C22×C8×H(3,𝔽3)2
C2×C4⋊((C3⋊Dic3)⋊C3)2
#63648⋊C32
C2×#636483
C2×C3×#8213