Finite Group Databasebeta

← C2×#63652⋊C3 #13143219

Subgroups — C2×#63652⋊C3 #13143219

Order 1728.

663 conjugacy classes of subgroups.

The subgroup lattice has 458 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
3C2×C4×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4#63652⋊C3C214yesmaximal, direct product factor
5C2×#63652C311yesmaximal, characteristic, complemented
6C2×C3×#825—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C2×#638⋊C3—21no—
9C22×(C3⋊Dic3)⋊C3—21no—
10#3828⋊C3—21no—
11#3828⋊C3—21no—
12#3828⋊C3C412yescomplemented
13#3828⋊C3C2212yes—
14C2×C4⋊((C3⋊D3)⋊C3)C411yescharacteristic, complemented
15C2×C4×(C3⋊D3)⋊C3—21no—
16C2×#634⋊C3C411yescharacteristic, complemented
17M16×H(3,𝔽3)—21no—
18M16×H(3,𝔽3)C2212yescomplemented
19C2×C8×H(3,𝔽3)—21no—
20C4×(C3⋊Dic3)⋊C3—21no—
21C4×(C3⋊Dic3)⋊C3C2212yescomplemented
22C2×#3828C2×C311yescharacteristic
23C2×C3×#235—31no—
24C2×C32×M16—21no—
25C2×C32×M16D311yescharacteristic, complemented
26C2×C32×M16C2×C311yescharacteristic, complemented
27C2×C4×C3⋊Dic3C2×C311yescharacteristic, complemented
28C2×C3×C4×Dic3—31no—
29#63652C2×C314yescomplemented
30C3×#825—34no—
31C23×H(3,𝔽3)D411yescharacteristic
32C22×(C3⋊D3)⋊C3—21no—
33C2×C4×H(3,𝔽3)C2×C411yescharacteristic
34C2×C4×H(3,𝔽3)C2×C411yescharacteristic
35C2×C4×H(3,𝔽3)D411yescharacteristic
36C2×C4×H(3,𝔽3)D411yescharacteristic
37C2×C4×H(3,𝔽3)D411yescharacteristic
38C2×C4×H(3,𝔽3)C2311yescharacteristic
39#638⋊C3—22no—
40#638⋊C3—22no—
41C2×(C3⋊Dic3)⋊C3—24no—
42C2×(C3⋊Dic3)⋊C3—21no—
43C2×(C3⋊Dic3)⋊C3—22no—
44C4⋊((C3⋊D3)⋊C3)—21no—
45C4⋊((C3⋊D3)⋊C3)C2×C412yescomplemented
46C4×(C3⋊D3)⋊C3—22no—
47C4×(C3⋊D3)⋊C3—22no—
48#634⋊C3—21no—
49#634⋊C3C2×C412yescomplemented
50C8×H(3,𝔽3)—22no—
51C2×C3×#104—91no—
52C2×#825—31no—
53C2×#825—91no—
54C22×C32×C4—21no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C22×C311yescharacteristic
57C2×#638—21no—
58C22×C3⋊Dic3—21no—
59#3828—21no—
60#3828—21no—
61#3828C3×C412yescomplemented
62#3828C22×C312yes—
63C2×C4⋊(C3⋊D3)C3×C411yescharacteristic, complemented
64C2×C4×C3⋊D3—21no—
65C2×#634C3×C411yescharacteristic, complemented
66C2×C3×#67—61no—
67C22×C3×Dic3—61no—
68C3×#235—32no—
69C3×#235—32no—
70C3×#235—61no—
71C3×#235—61no—
72C2×C3×D12—31no—
73C2×C3×C4×D3—61no—
74C2×C3×Dic6—31no—
75C32×M16—22no—
76C32×M16—22no—
77C32×M16—21no—
78C32×M16—21no—
79C32×M16C2×D312yescomplemented
80C32×M16C22×C312yescomplemented
81C2×C32×C8—22no—
82C2×C32×C8—21no—
83C2×C32×C8—21no—
84C4×C3⋊Dic3—21no—
85C4×C3⋊Dic3C22×C312yescomplemented
86C3×C4×Dic3—32no—
87C3×C4×Dic3—61no—
88C22×H(3,𝔽3)—22no—
89C22×H(3,𝔽3)C22⋊C411yescharacteristic
90C22×H(3,𝔽3)C22⋊C411yescharacteristic
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C2×(C3⋊D3)⋊C3—22no—
93C2×(C3⋊D3)⋊C3—42no—
94C4×H(3,𝔽3)C22⋊C411yescharacteristic
95C4×H(3,𝔽3)C22⋊C411yescharacteristic
96C4×H(3,𝔽3)C22×C411yescharacteristic
97C4×H(3,𝔽3)C2×D411yescharacteristic
98(C3⋊Dic3)⋊C3—24no—
99(C3⋊Dic3)⋊C3—22no—
100C2×C3×#41—91no—
101C2×#235—31no—
102C2×#235—91no—
103C2×C3×M16—31no—
104C2×C3×M16—31no—
105C2×C3×M16—61no—
106C2×C3×M16C3×D311yescharacteristic
107C2×C3×C42—91no—
108C2×C4×Dic3—31no—
109C2×C4×Dic3—91no—
110C3×#104—94no—
111#825—34no—
112#825—94no—
113C23×C32—21no—
114C23×C32D1211yescharacteristic
115C23×C32C3×D411yescharacteristic
116C22×C3⋊D3—21no—
117C22×C3×D3—61no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—21no—
122C2×C32×C4—21no—
123C2×C32×C4—21no—
124C2×C32×C4C4×D311yescharacteristic
125C2×C32×C4C4×D311yescharacteristic
126C2×C32×C4D1211yescharacteristic
127C2×C32×C4#6711yescharacteristic
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4C2×C3×C411yescharacteristic
130C2×C32×C4C2×C3×C411yescharacteristic
131C2×C32×C4C3×D411yescharacteristic
132C2×C32×C4C3×D411yescharacteristic
133C2×C32×C4C3×D411yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C23×C311yescharacteristic
136#638—22no—
137#638—22no—
138C2×C3⋊Dic3—24no—
139C2×C3⋊Dic3—21no—
140C2×C3⋊Dic3—22no—
141C4⋊(C3⋊D3)—21no—
142C4⋊(C3⋊D3)C2×C3×C412yescomplemented
143C4×C3⋊D3—22no—
144C4×C3⋊D3—22no—
145#634—21no—
146#634C2×C3×C412yescomplemented
147C3×#67—62no—
148C3×#67—62no—
149C2×C3×Dic3—64no—
150C2×C3×Dic3—61no—
151C2×C3×Dic3—62no—
152C3×D12—32no—
153C3×D12—61no—
154C3×C4×D3—62no—
155C3×C4×D3—62no—
156C3×Dic6—32no—
157C3×Dic6—61no—
158C32×C8—24no—
159C32×C8—22no—
160C32×C8—22no—
161C2×#104—91noSylow
162C2×H(3,𝔽3)—21no—
163C2×H(3,𝔽3)—21no—
164C2×H(3,𝔽3)#10412yescomplemented
165C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
166(C3⋊D3)⋊C3—42no—
167C3×#41—92no—
168C3×#41—92no—
169C3×#41—181no—
170C3×#41—181no—
171C2×C3×Q8—91no—
172C2×C3×D4—91no—
173C2×C3×D4—181no—
174C22×C3×C4—31no—
175C22×C3×C4—31no—
176C22×C3×C4—61no—
177C22×C3×C4—181no—
178C22×C3×C4—181no—
179C22×C3×C4C2×C3×D311yescharacteristic
180C2×#67—61no—
181C2×#67—181no—
182C22×Dic3—61no—
183C22×Dic3—181no—
184#235—32no—
185#235—32no—
186#235—61no—
187#235—61no—
188#235—92no—
189#235—92no—
190#235—181no—
191#235—181no—
192C2×D12—31no—
193C2×D12—91no—
194C2×C4×D3—61no—
195C2×C4×D3—181no—
196C2×Dic6—31no—
197C2×Dic6—91no—
198C3×M16—21no—
199C3×M16—32no—
200C3×M16—32no—
201C3×M16—61no—
202C3×M16—61no—
203C3×M16—62no—
204C3×M16—62no—
205C3×M16C2×C3×D312yes—
206C2×C3×C8—21no—
207C2×C3×C8—61no—
208C2×C3×C8—61no—
209C2×C3×C8—62no—
210C3×C42—92no—
211C3×C42—181no—
212C4×Dic3—32no—
213C4×Dic3—61no—
214C4×Dic3—92no—
215C4×Dic3—181no—
216C22×C32—22no—
217C22×C32—22no—
218C22×C32—21no—
219C22×C32—21no—
220C22×C32—21no—
221C22×C32—42no—
222C22×C32(C2×D3)⋊C411yescharacteristic
223C22×C32(C2×D3)⋊C411yescharacteristic
224C22×C32C3×C22⋊C411yescharacteristic
225C22×C32C3×C22⋊C411yescharacteristic
226C22×C32C2×D1211yescharacteristic
227C22×C32C2×C3×D411yescharacteristic
228C2×C3⋊D3—22no—
229C2×C3⋊D3—42no—
230C2×C3×D3—62no—
231C2×C3×D3—122no—
232C32×C4—21no—
233C32×C4—21no—
234C32×C4—21no—
235C32×C4—21no—
236C32×C4(C2×D3)⋊C411yescharacteristic
237C32×C4(C2×D3)⋊C411yescharacteristic
238C32×C4C3×C22⋊C411yescharacteristic
239C32×C4C3×C22⋊C411yescharacteristic
240C32×C4C2×C4×D311yescharacteristic
241C32×C4C2×#6711yescharacteristic
242C32×C4C22×C3×C411yescharacteristic
243C32×C4C2×C3×D411yescharacteristic
244C3⋊Dic3—24no—
245C3⋊Dic3—22no—
246C3×Dic3—64no—
247C3×Dic3—62no—
248C2×#41—91no—
249C2×M16(C3⋊D3)⋊C311yescharacteristic, complemented
250C2×C42—91no—
251#104—94no—
252H(3,𝔽3)C2×#10411yescharacteristic, complemented, Sylow
253C23×C3—31no—
254C23×C3—31no—
255C23×C3—61no—
256C23×C3—181no—
257C23×C3C3×D1211yescharacteristic
258C22×D3—61no—
259C22×D3—181no—
260C3×Q8—92no—
261C3×Q8—181no—
262C3×D4—92no—
263C3×D4—181no—
264C3×D4—182no—
265C3×D4—182no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—31no—
272C2×C3×C4—31no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—61no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—182no—
285C2×C3×C4—182no—
286C2×C3×C4—182no—
287C2×C3×C4—181no—
288C2×C3×C4—184no—
289C2×C3×C4C3×C4×D311yescharacteristic
290C2×C3×C4C3×C4×D311yescharacteristic
291C2×C3×C4C3×D1211yescharacteristic
292C2×C3×C4C3×#6711yescharacteristic
293C2×C3×C4C3×#6711yescharacteristic
294C2×C3×C4C22×C3×D311yescharacteristic
295#67—62no—
296#67—62no—
297#67—182no—
298#67—182no—
299C2×Dic3—64no—
300C2×Dic3—61no—
301C2×Dic3—62no—
302C2×Dic3—182no—
303C2×Dic3—181no—
304C2×Dic3—184no—
305D12—32no—
306D12—61no—
307D12—92no—
308D12—181no—
309C4×D3—62no—
310C4×D3—62no—
311C4×D3—182no—
312C4×D3—182no—
313Dic6—32no—
314Dic6—61no—
315Dic6—92no—
316Dic6—181no—
317C3×C8—22no—
318C3×C8—62no—
319C3×C8—62no—
320C3×C8—64no—
321C2×C32—21no—
322C2×C32—21no—
323C2×C32—21no—
324C2×C32—21no—
325C2×C32—21no—
326C2×C32—22no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32#82512yescomplemented
330C2×C32C3×#10412yescomplemented
331C2×C32C2×(C2×D3)⋊C411yescharacteristic
332C2×C32C2×C3×C22⋊C411yescharacteristic
333C3⋊D3—42no—
334C3×D3—122no—
335#41—92no—
336#41—92no—
337#41—181no—
338#41—181no—
339C2×Q8—91no—
340C2×D4—91no—
341C2×D4—181no—
342C22×C4—181no—
343C22×C4—181no—
344C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
345M16—21no—
346M16C2×(C3⋊D3)⋊C312yescomplemented
347C2×C8—21no—
348C42—92no—
349C42—181no—
350C22×C3—22no—
351C22×C3—31no—
352C22×C3—31no—
353C22×C3—31no—
354C22×C3—31no—
355C22×C3—31no—
356C22×C3—31no—
357C22×C3—62no—
358C22×C3—62no—
359C22×C3—61no—
360C22×C3—61no—
361C22×C3—61no—
362C22×C3—122no—
363C22×C3—182no—
364C22×C3—362no—
365C22×C3C3×(C2×D3)⋊C411yescharacteristic
366C22×C3C3×(C2×D3)⋊C411yescharacteristic
367C22×C3C2×C3×D1211yescharacteristic
368C2×D3—62no—
369C2×D3—122no—
370C2×D3—182no—
371C2×D3—362no—
372C3×C4—31no—
373C3×C4—31no—
374C3×C4—31no—
375C3×C4—31no—
376C3×C4—31no—
377C3×C4—31no—
378C3×C4—31no—
379C3×C4—31no—
380C3×C4—61no—
381C3×C4—61no—
382C3×C4—61no—
383C3×C4—61no—
384C3×C4—182no—
385C3×C4—184no—
386C3×C4C3×(C2×D3)⋊C411yescharacteristic
387C3×C4C3×(C2×D3)⋊C411yescharacteristic
388C3×C4C2×C3×C4×D311yescharacteristic
389C3×C4C2×C3×#6711yescharacteristic
390Dic3—64no—
391Dic3—62no—
392Dic3—182no—
393Dic3—184no—
394C32—21no—
395C32C2×#82511yescharacteristic, complemented
396C32C2×C3×#10411yescharacteristic, complemented
397C23—181no—
398C23C4⋊((C3⋊D3)⋊C3)11yescharacteristic
399Q8—92no—
400Q8—181no—
401D4—92no—
402D4—182no—
403D4—182no—
404D4—181no—
405C2×C4—184no—
406C2×C4—182no—
407C2×C4—181no—
408C2×C4—182no—
409C2×C4—182no—
410C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
411C2×C4C4×(C3⋊D3)⋊C311yescharacteristic
412C2×C4C4⋊((C3⋊D3)⋊C3)11yescharacteristic
413C2×C4#638⋊C311yescharacteristic
414C2×C4#638⋊C311yescharacteristic
415C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
416C8—22no—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3—32no—
420C2×C3—32no—
421C2×C3—31no—
422C2×C3—31no—
423C2×C3—61no—
424C2×C3—61no—
425C2×C3—61no—
426C2×C3—61no—
427C2×C3—62no—
428C2×C3—61no—
429C2×C3—121no—
430C2×C3—121no—
431C2×C3—362no—
432C2×C3C3×#82512yes—
433C2×C3C2×C3×(C2×D3)⋊C411yescharacteristic
434D3—122no—
435D3—362no—
436C22—22no—
437C22—182no—
438C22—362no—
439C22#3752⋊C311yescharacteristic
440C22#3752⋊C311yescharacteristic
441C22C2×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
442C4—184no—
443C4—182no—
444C4#3752⋊C311yescharacteristic
445C4#3752⋊C311yescharacteristic
446C4C2×C4×(C3⋊D3)⋊C311yescharacteristic
447C4C2×#638⋊C311yescharacteristic
448C3—31no—
449C3—31no—
450C3—61no—
451C3C2×C3×#82511yescharacteristic
452C2—21no—
453C2—21no—
454C2—362no—
455C2#63652⋊C312yesdirect product factor
456C2C2×#3752⋊C311yescharacteristic
subgroupG/H
centerC2×C4#638⋊C3
commutatorC32×C4C22×C3×C4
FrattiniC2×C3×C4C22×C3×D3
FittingC2×M16×H(3,𝔽3)C2
socleC22×C3C3×(C2×D3)⋊C4
prime radicalC2×#3828⋊C3C2
cosocleC2×C32×C4C23×C3
hypercenterC2×M16(C3⋊D3)⋊C3
hypocenterC32C2×C3×#104
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×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. C2×C4×(C3⋊Dic3)⋊C3C2semidirect
4. #63652⋊C3C2direct (N × H)
5. C2×#63652C3semidirect
12. #3828⋊C3C4semidirect
14. C2×C4⋊((C3⋊D3)⋊C3)C4semidirect
16. C2×#634⋊C3C4semidirect
18. M16×H(3,𝔽3)C22semidirect
21. C4×(C3⋊Dic3)⋊C3C22semidirect
25. C2×C32×M16D3semidirect
26. C2×C32×M16C2×C3semidirect
27. C2×C4×C3⋊Dic3C2×C3semidirect
29. #63652C2×C3semidirect
45. C4⋊((C3⋊D3)⋊C3)C2×C4semidirect
49. #634⋊C3C2×C4semidirect
61. #3828C3×C4semidirect
63. C2×C4⋊(C3⋊D3)C3×C4semidirect
65. C2×#634C3×C4semidirect
79. C32×M16C2×D3semidirect
80. C32×M16C22×C3semidirect
85. C4×C3⋊Dic3C22×C3semidirect
142. C4⋊(C3⋊D3)C2×C3×C4semidirect
146. #634C2×C3×C4semidirect
164. C2×H(3,𝔽3)#104semidirect
249. C2×M16(C3⋊D3)⋊C3semidirect
252. H(3,𝔽3)C2×#104semidirect
329. C2×C32#825semidirect
330. C2×C32C3×#104semidirect
346. M16C2×(C3⋊D3)⋊C3semidirect
395. C32C2×#825semidirect
396. C32C2×C3×#104semidirect
455. C2#63652⋊C3direct (N × H)

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
C2×C4×(C3⋊Dic3)⋊C32
#63652⋊C32
C2×#636523
C2×C3×#8253