Finite Group Databasebeta

← C2×#63636⋊C3 #13142604

Subgroups — C2×#63636⋊C3 #13142604

Order 1728.

799 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic
2C2×C4⋊C4×H(3,𝔽3)C211yesmaximal, characteristic, complemented
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#63636⋊C3C214yesmaximal, direct product factor
5C2×#63636C311yesmaximal, characteristic, complemented
6C2×C3×#809—31nomaximal
7C23×(C3⋊D3)⋊C3—21no—
8C22×C4×H(3,𝔽3)—21no—
9C22×C4×H(3,𝔽3)C2211yescharacteristic
10C2×C4⋊((C3⋊D3)⋊C3)—21no—
11C2×C4⋊((C3⋊D3)⋊C3)—22no—
12C2×C4⋊((C3⋊D3)⋊C3)C414yescomplemented
13C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
14C4⋊C4×H(3,𝔽3)—21no—
15C4⋊C4×H(3,𝔽3)C2212yescomplemented
16C2×(C3⋊(C3⋊C8))⋊C3—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yescomplemented
18C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
19C22×C3×D12—31no—
20C2×C32×C4⋊C4—21no—
21C2×C32×C4⋊C4D311yescharacteristic, complemented
22C2×C32×C4⋊C4C2×C311yescharacteristic, complemented
23C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
24C22×C3×C3⋊C8—31no—
25#63636C2×C314yescomplemented
26C3×#809—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)—22no—
33C2×C4×H(3,𝔽3)—42no—
34C2×C4×H(3,𝔽3)C2×C412yes—
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)D412yes—
37C2×C4×H(3,𝔽3)C2311yescharacteristic
38C4⋊((C3⋊D3)⋊C3)—24no—
39C4⋊((C3⋊D3)⋊C3)—22no—
40C4⋊((C3⋊D3)⋊C3)C2×C414yescomplemented
41(C3⋊(C3⋊C8))⋊C3—22no—
42C2×C3×#102—91no—
43C2×#809—31no—
44C2×#809—91no—
45C23×C3⋊D3—21no—
46C23×C3×D3—61no—
47C22×C32×C4—21no—
48C22×C32×C4—22no—
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4C2×D311yescharacteristic
52C22×C32×C4C22×C311yescharacteristic
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)—22no—
55C2×C4⋊(C3⋊D3)C3×C414yescomplemented
56C2×C4⋊(C3⋊D3)C22×C312yes—
57C2×C3×D12—34no—
58C2×C3×D12—32no—
59C2×C3×D12—62no—
60C2×C3×D12—61no—
61C32×C4⋊C4—22no—
62C32×C4⋊C4—22no—
63C32×C4⋊C4—21no—
64C32×C4⋊C4—21no—
65C32×C4⋊C4C2×D312yescomplemented
66C32×C4⋊C4C22×C312yescomplemented
67C2×C3⋊(C3⋊C8)—22no—
68C2×C3⋊(C3⋊C8)C22×C312yescomplemented
69C2×C3×C3⋊C8—32no—
70C2×C3×C3⋊C8—62no—
71C22×H(3,𝔽3)C22⋊C412yes—
72C22×H(3,𝔽3)D812yes—
73C22×H(3,𝔽3)SD1612yes—
74C22×H(3,𝔽3)C2×D411yescharacteristic
75C2×(C3⋊D3)⋊C3—24no—
76C2×(C3⋊D3)⋊C3—48no—
77C2×(C3⋊D3)⋊C3—42no—
78C2×(C3⋊D3)⋊C3—42no—
79C4×H(3,𝔽3)—42no—
80C4×H(3,𝔽3)C22⋊C412yes—
81C4×H(3,𝔽3)C22×C411yescharacteristic
82C4×H(3,𝔽3)C2×D411yescharacteristic
83C22×C3×D4—91no—
84C22×D12—31no—
85C22×D12—91no—
86C22×C3×C8—91no—
87C2×C3×C4⋊C4—31no—
88C2×C3×C4⋊C4—31no—
89C2×C3×C4⋊C4—61no—
90C2×C3×C4⋊C4C3×D311yescharacteristic
91C22×C3⋊C8—31no—
92C22×C3⋊C8—91no—
93C3×#102—94no—
94#809—34no—
95#809—94no—
96C23×C32—21no—
97C23×C32#6711yescharacteristic
98C23×C32C3×D411yescharacteristic
99C22×C3⋊D3—24no—
100C22×C3⋊D3—22no—
101C22×C3⋊D3—42no—
102C22×C3⋊D3—42no—
103C22×C3×D3—64no—
104C22×C3×D3—62no—
105C22×C3×D3—122no—
106C22×C3×D3—122no—
107C2×C32×C4—24no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4—22no—
111C2×C32×C4—22no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—44no—
115C2×C32×C4—42no—
116C2×C32×C4—42no—
117C2×C32×C4C4×D312yes—
118C2×C32×C4D1212yes—
119C2×C32×C4#6711yescharacteristic
120C2×C32×C4C2×C3×C412yes—
121C2×C32×C4C3×D412yes—
122C2×C32×C4C3×D411yescharacteristic
123C2×C32×C4C22×D311yescharacteristic
124C2×C32×C4C23×C311yescharacteristic
125C4⋊(C3⋊D3)—24no—
126C4⋊(C3⋊D3)—22no—
127C4⋊(C3⋊D3)C2×C3×C414yescomplemented
128C3×D12—34no—
129C3×D12—64no—
130C3×D12—62no—
131C3⋊(C3⋊C8)—22no—
132C3×C3⋊C8—62no—
133C2×#102—91noSylow
134C2×H(3,𝔽3)#10214yescomplemented
135C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
136C2×H(3,𝔽3)C2×D811yescharacteristic
137C2×H(3,𝔽3)C2×SD1611yescharacteristic
138(C3⋊D3)⋊C3—44no—
139C24×C3—181no—
140C23×D3—61no—
141C23×D3—181no—
142C2×C3×D4—92no—
143C2×C3×D4—94no—
144C2×C3×D4—182no—
145C2×C3×D4—181no—
146C22×C3×C4—21no—
147C22×C3×C4—31no—
148C22×C3×C4—31no—
149C22×C3×C4—61no—
150C22×C3×C4—61no—
151C22×C3×C4—62no—
152C22×C3×C4—61no—
153C22×C3×C4C2×C3×D311yescharacteristic
154C2×D12—34no—
155C2×D12—32no—
156C2×D12—62no—
157C2×D12—61no—
158C2×D12—92no—
159C2×D12—94no—
160C2×D12—182no—
161C2×D12—181no—
162C2×C3×C8—92no—
163C2×C3×C8—182no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—32no—
166C3×C4⋊C4—32no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4—61no—
169C3×C4⋊C4—62no—
170C3×C4⋊C4—62no—
171C3×C4⋊C4C2×C3×D312yes—
172C2×C3⋊C8—32no—
173C2×C3⋊C8—62no—
174C2×C3⋊C8—92no—
175C2×C3⋊C8—182no—
176C22×C32—22no—
177C22×C32—22no—
178C22×C32—22no—
179C22×C32—21no—
180C22×C32(C2×D3)⋊C412yes—
181C22×C32#21312yes—
182C22×C32#21512yes—
183C22×C32C3×C22⋊C412yes—
184C22×C32C3×D812yes—
185C22×C32C3×SD1612yes—
186C22×C32C2×#6711yescharacteristic
187C22×C32C2×C3×D411yescharacteristic
188C2×C3⋊D3—24no—
189C2×C3⋊D3—48no—
190C2×C3⋊D3—42no—
191C2×C3⋊D3—42no—
192C2×C3×D3—64no—
193C2×C3×D3—128no—
194C2×C3×D3—122no—
195C2×C3×D3—122no—
196C32×C4—22no—
197C32×C4—21no—
198C32×C4—21no—
199C32×C4—44no—
200C32×C4—42no—
201C32×C4—42no—
202C32×C4(C2×D3)⋊C412yes—
203C32×C4C3×C22⋊C412yes—
204C32×C4C2×C4×D311yescharacteristic
205C32×C4C2×D1211yescharacteristic
206C32×C4C22×C3×C411yescharacteristic
207C32×C4C2×C3×D411yescharacteristic
208C22×D4—91no—
209C22×C8—91no—
210C2×C4⋊C4(C3⋊D3)⋊C311yescharacteristic, complemented
211#102—94no—
212H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
213C23×C3—31no—
214C23×C3—31no—
215C23×C3—61no—
216C23×C3—182no—
217C23×C3—184no—
218C23×C3—362no—
219C23×C3—362no—
220C23×C3C3×#6711yescharacteristic
221C22×D3—64no—
222C22×D3—62no—
223C22×D3—122no—
224C22×D3—122no—
225C22×D3—182no—
226C22×D3—184no—
227C22×D3—362no—
228C22×D3—362no—
229C3×D4—94no—
230C3×D4—182no—
231C3×D4—184no—
232C2×C3×C4—22no—
233C2×C3×C4—32no—
234C2×C3×C4—32no—
235C2×C3×C4—32no—
236C2×C3×C4—32no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—31no—
240C2×C3×C4—31no—
241C2×C3×C4—42no—
242C2×C3×C4—62no—
243C2×C3×C4—62no—
244C2×C3×C4—64no—
245C2×C3×C4—62no—
246C2×C3×C4—62no—
247C2×C3×C4—61no—
248C2×C3×C4—61no—
249C2×C3×C4—122no—
250C2×C3×C4—122no—
251C2×C3×C4—124no—
252C2×C3×C4C3×C4×D312yes—
253C2×C3×C4C3×D1212yes—
254C2×C3×C4C3×#6711yescharacteristic
255C2×C3×C4C22×C3×D311yescharacteristic
256D12—34no—
257D12—64no—
258D12—62no—
259D12—94no—
260D12—182no—
261D12—184no—
262C3×C8—182no—
263C3⋊C8—62no—
264C3⋊C8—182no—
265C2×C32—21no—
266C2×C32—21no—
267C2×C32—21no—
268C2×C32—24no—
269C2×C32#80914yescomplemented
270C2×C32C3×#10214yescomplemented
271C2×C32C2×(C2×D3)⋊C411yescharacteristic
272C2×C32C2×#21311yescharacteristic
273C2×C32C2×#21511yescharacteristic
274C2×C32C2×C3×C22⋊C411yescharacteristic
275C2×C32C2×C3×D811yescharacteristic
276C2×C32C2×C3×SD1611yescharacteristic
277C3⋊D3—44no—
278C3×D3—124no—
279C24—181no—
280C2×D4—94no—
281C2×D4—92no—
282C2×D4—182no—
283C2×D4—181no—
284C22×C4—21no—
285C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
286C2×C8—92no—
287C2×C8—182no—
288C4⋊C4—21no—
289C4⋊C4C2×(C3⋊D3)⋊C312yescomplemented
290C22×C3—31no—
291C22×C3—31no—
292C22×C3—32no—
293C22×C3—32no—
294C22×C3—32no—
295C22×C3—32no—
296C22×C3—32no—
297C22×C3—32no—
298C22×C3—61no—
299C22×C3—62no—
300C22×C3—62no—
301C22×C3—62no—
302C22×C3—184no—
303C22×C3—362no—
304C22×C3—362no—
305C22×C3—368no—
306C22×C3C3×(C2×D3)⋊C412yes—
307C22×C3C3×#21312yes—
308C22×C3C3×#21512yes—
309C22×C3C2×C3×#6711yescharacteristic
310C2×D3—64no—
311C2×D3—122no—
312C2×D3—122no—
313C2×D3—128no—
314C2×D3—184no—
315C2×D3—362no—
316C2×D3—362no—
317C2×D3—368no—
318C3×C4—31no—
319C3×C4—31no—
320C3×C4—31no—
321C3×C4—31no—
322C3×C4—32no—
323C3×C4—32no—
324C3×C4—42no—
325C3×C4—61no—
326C3×C4—61no—
327C3×C4—62no—
328C3×C4—122no—
329C3×C4—122no—
330C3×C4—124no—
331C3×C4C3×(C2×D3)⋊C412yes—
332C3×C4C2×C3×C4×D311yescharacteristic
333C3×C4C2×C3×D1211yescharacteristic
334C32—21no—
335C32C2×#80911yescharacteristic, complemented
336C32C2×C3×#10211yescharacteristic, complemented
337C23—182no—
338C23—184no—
339C23—362no—
340C23—362no—
341C23#638⋊C311yescharacteristic
342D4—94no—
343D4—182no—
344D4—184no—
345C2×C4—22no—
346C2×C4—42no—
347C2×C4C4×(C3⋊D3)⋊C312yes—
348C2×C4C4⋊((C3⋊D3)⋊C3)12yes—
349C2×C4#638⋊C311yescharacteristic
350C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
351C8—182no—
352C2×C3—31no—
353C2×C3—31no—
354C2×C3—31no—
355C2×C3—31no—
356C2×C3—31no—
357C2×C3—31no—
358C2×C3—34no—
359C2×C3—34no—
360C2×C3—61no—
361C2×C3—61no—
362C2×C3—61no—
363C2×C3—64no—
364C2×C3—364no—
365C2×C3C3×#80914yes—
366C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
367C2×C3C2×C3×#21311yescharacteristic
368C2×C3C2×C3×#21511yescharacteristic
369D3—124no—
370D3—364no—
371C22—184no—
372C22—362no—
373C22—362no—
374C22—368no—
375C22#3752⋊C312yes—
376C22#3753⋊C312yes—
377C22#3755⋊C312yes—
378C22C2×#638⋊C311yescharacteristic
379C4—42no—
380C4#3752⋊C312yes—
381C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
382C4C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
383C3—31no—
384C3—31no—
385C3—61no—
386C3C2×C3×#80911yescharacteristic
387C2—364no—
388C2#63636⋊C314yesdirect product factor
389C2C2×#3752⋊C311yescharacteristic
390C2C2×#3753⋊C311yescharacteristic
391C2C2×#3755⋊C311yescharacteristic
subgroupG/H
centerC23#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×C4⋊C4×H(3,𝔽3)C2
socleC23×C3C3×#67
prime radicalC22×C4⋊((C3⋊D3)⋊C3)C2
cosocleC2×C32×C4C23×C3
hypercenterC2×C4⋊C4(C3⋊D3)⋊C3
hypocenterC32C2×C3×#102
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×H(3,𝔽3)C2semidirect
3. C22×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. #63636⋊C3C2direct (N × H)
5. C2×#63636C3semidirect
12. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
15. C4⋊C4×H(3,𝔽3)C22semidirect
17. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
21. C2×C32×C4⋊C4D3semidirect
22. C2×C32×C4⋊C4C2×C3semidirect
23. C22×C3⋊(C3⋊C8)C2×C3semidirect
25. #63636C2×C3semidirect
40. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
55. C2×C4⋊(C3⋊D3)C3×C4semidirect
65. C32×C4⋊C4C2×D3semidirect
66. C32×C4⋊C4C22×C3semidirect
68. C2×C3⋊(C3⋊C8)C22×C3semidirect
127. C4⋊(C3⋊D3)C2×C3×C4semidirect
134. C2×H(3,𝔽3)#102semidirect
210. C2×C4⋊C4(C3⋊D3)⋊C3semidirect
212. H(3,𝔽3)C2×#102semidirect
269. C2×C32#809semidirect
270. C2×C32C3×#102semidirect
289. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
335. C32C2×#809semidirect
336. C32C2×C3×#102semidirect
388. C2#63636⋊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
C2×C4⋊C4×H(3,𝔽3)2
C22×(C3⋊(C3⋊C8))⋊C32
#63636⋊C32
C2×#636363
C2×C3×#8093