Finite Group Databasebeta

← #10595996⋊C3 #13152519

Subgroups — #10595996⋊C3 #13152519

Order 1728.

1245 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828⋊C3C211yesmaximal, characteristic, complemented
2C22×C4⋊((C3⋊D3)⋊C3)C211yesmaximal, characteristic, complemented
3H(3,𝔽3)×#117C211yesmaximal, characteristic, complemented
4#64104⋊C3C214yesmaximal, complemented
5#64091⋊C3C214yesmaximal, complemented
6#64084⋊C3C212yesmaximal, complemented
7#11721385C212yesmaximal, complemented
8#10595996C311yesmaximal, characteristic, complemented
9C3×#5525—31nomaximal
10C23×(C3⋊D3)⋊C3C2212yes—
11C22×C4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3C2212yes—
13#3828⋊C3—24no—
14C2×C4⋊((C3⋊D3)⋊C3)—24no—
15C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
16C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
18C2×C4⋊((C3⋊D3)⋊C3)C2218yescomplemented
19C2×C4×(C3⋊D3)⋊C3C2212yescomplemented
20C2×#634⋊C3C2211yescharacteristic, complemented
21C4⋊C4×H(3,𝔽3)C2212yescomplemented
22C22⋊C4×H(3,𝔽3)C2212yes—
23C42×H(3,𝔽3)C2212yescomplemented
24#3752⋊C3C2218yescomplemented
25C2×#3828C2×C311yescharacteristic, complemented
26C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
27C2×C3×#235—31no—
28C22×C3×D12—31no—
29C32×#117—21no—
30C32×#117D311yescharacteristic, complemented
31C32×#117C2×C311yescharacteristic, complemented
32#64104C2×C314yescomplemented
33#64091C2×C314yescomplemented
34#64084C2×C312yescomplemented
35#64083C2×C312yescomplemented
36C3×#895—34no—
37C3×#882—34no—
38C3×#875—32no—
39C3×#874—32no—
40C23×H(3,𝔽3)C2311yescharacteristic
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—44no—
43C22×(C3⋊D3)⋊C3C2314yes—
44C22×(C3⋊D3)⋊C3C2312yes—
45C2×C4×H(3,𝔽3)D414yescomplemented
46C2×C4×H(3,𝔽3)C2314yes—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C2×C4×H(3,𝔽3)C2311yescharacteristic
49#638⋊C3—24no—
50C2×(C3⋊Dic3)⋊C3C2312yes—
51C4⋊((C3⋊D3)⋊C3)—24no—
52C4⋊((C3⋊D3)⋊C3)—24no—
53C4⋊((C3⋊D3)⋊C3)—22no—
54C4⋊((C3⋊D3)⋊C3)—48no—
55C4×(C3⋊D3)⋊C3—24no—
56#634⋊C3—22no—
57C3×#535—91no—
58#5525—31no—
59#5525—91no—
60C23×C3⋊D3C22×C312yes—
61C23×C3×D3—32no—
62C22×C32×C4—21no—
63C22×C32×C4C2×D311yescharacteristic
64C22×C32×C4C22×C311yescharacteristic
65C2×#638C22×C312yes—
66#3828—24no—
67C2×C4⋊(C3⋊D3)—24no—
68C2×C4⋊(C3⋊D3)C22×C312yes—
69C2×C4⋊(C3⋊D3)C22×C312yes—
70C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
71C2×C4⋊(C3⋊D3)C22×C318yescomplemented
72C2×C4×C3⋊D3C22×C312yescomplemented
73C2×#634C22×C311yescharacteristic, complemented
74C2×C3×#67—32no—
75C3×#235—64no—
76C2×C3×D12—32no—
77C2×C3×D12—32no—
78C2×C3×D12—31no—
79C2×C3×D12—38no—
80C2×C3×D12—64no—
81C2×C3×C4×D3—32no—
82C2×C3×Dic6—31no—
83C32×C4⋊C4—22no—
84C32×C4⋊C4C2×D312yescomplemented
85C32×C4⋊C4C22×C312yescomplemented
86C32×C22⋊C4—22no—
87C32×C22⋊C4C2×D312yes—
88C32×C22⋊C4C22×C312yes—
89C32×C42—22no—
90C32×C42C2×D312yescomplemented
91C32×C42C22×C312yescomplemented
92#3752C22×C318yescomplemented
93C3×(C2×D3)⋊C4—38no—
94C22×H(3,𝔽3)—22no—
95C22×H(3,𝔽3)C2×D412yes—
96C22×H(3,𝔽3)C2411yescharacteristic
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—24no—
99C2×(C3⋊D3)⋊C3—24no—
100C2×(C3⋊D3)⋊C3—28no—
101C2×(C3⋊D3)⋊C3—48no—
102C4×H(3,𝔽3)—24no—
103C4×H(3,𝔽3)C2×D412yes—
104C4×H(3,𝔽3)C2×D412yes—
105(C3⋊Dic3)⋊C3—22no—
106C2×C3×#41—91no—
107C22×C3×D4—91no—
108C2×#235—31no—
109C2×#235—91no—
110C22×D12—31no—
111C22×D12—91no—
112C3×#127—92no—
113C3×#124—92no—
114C3×#121—94no—
115C3×#120—94no—
116C3×#117—31no—
117C3×#117—31no—
118C3×#117—61no—
119C3×#117C3×D311yescharacteristic
120#895—34no—
121#895—94no—
122#882—34no—
123#882—94no—
124#875—32no—
125#875—92no—
126#874—32no—
127#874—92no—
128C23×C32—21no—
129C23×C32C22×D311yescharacteristic
130C23×C32C23×C311yescharacteristic
131C22×C3⋊D3—24no—
132C22×C3⋊D3—44no—
133C22×C3⋊D3C23×C314yes—
134C22×C3⋊D3C23×C312yes—
135C22×C3×D3—34no—
136C22×C3×D3—32no—
137C22×C3×D3—64no—
138C22×C3×D3—124no—
139C2×C32×C4—24no—
140C2×C32×C4—21no—
141C2×C32×C4—21no—
142C2×C32×C4—24no—
143C2×C32×C4D1214yescomplemented
144C2×C32×C4C3×D414yescomplemented
145C2×C32×C4C22×D314yes—
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148C2×C32×C4C23×C314yes—
149C2×C32×C4C23×C311yescharacteristic
150C2×C32×C4C23×C311yescharacteristic
151#638—24no—
152C2×C3⋊Dic3C23×C312yes—
153C4⋊(C3⋊D3)—24no—
154C4⋊(C3⋊D3)—24no—
155C4⋊(C3⋊D3)—22no—
156C4⋊(C3⋊D3)—48no—
157C4×C3⋊D3—24no—
158#634—22no—
159C3×#67—64no—
160C2×C3×Dic3—32no—
161C3×D12—64no—
162C3×D12—64no—
163C3×D12—62no—
164C3×D12—128no—
165C3×C4×D3—64no—
166C3×Dic6—62no—
167#535—91noSylow
168C2×H(3,𝔽3)—22no—
169C2×H(3,𝔽3)C22×D411yescharacteristic
170C2×H(3,𝔽3)H(5,𝔽2)12yes—
171(C3⋊D3)⋊C3—42no—
172(C3⋊D3)⋊C3—44no—
173C24×C3—92no—
174C23×D3—32no—
175C23×D3—92no—
176C3×#41—184no—
177C2×C3×Q8—91no—
178C2×C3×D4—92no—
179C2×C3×D4—92no—
180C2×C3×D4—91no—
181C2×C3×D4—98no—
182C2×C3×D4—92no—
183C2×C3×D4—184no—
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C22×C3×C4—61no—
187C22×C3×C4—92no—
188C22×C3×C4C2×C3×D311yescharacteristic
189C2×#67—32no—
190C2×#67—92no—
191#235—64no—
192#235—184no—
193C2×D12—31no—
194C2×D12—32no—
195C2×D12—32no—
196C2×D12—38no—
197C2×D12—64no—
198C2×D12—91no—
199C2×D12—92no—
200C2×D12—92no—
201C2×D12—98no—
202C2×D12—184no—
203C2×C4×D3—32no—
204C2×C4×D3—92no—
205C2×Dic6—31no—
206C2×Dic6—91no—
207C3×C4⋊C4—32no—
208C3×C4⋊C4—32no—
209C3×C4⋊C4—62no—
210C3×C4⋊C4C2×C3×D312yes—
211C3×C22⋊C4—32no—
212C3×C22⋊C4—32no—
213C3×C22⋊C4—62no—
214C3×C22⋊C4—98no—
215C3×C22⋊C4C2×C3×D312yes—
216C3×C42—32no—
217C3×C42—32no—
218C3×C42—62no—
219C3×C42C2×C3×D312yes—
220(C2×D3)⋊C4—38no—
221(C2×D3)⋊C4—98no—
222C22×C32—22no—
223C22×C32—22no—
224C22×C32—21no—
225C22×C32—22no—
226C22×C32—42no—
227C22×C32C2×D1212yes—
228C22×C32C2×C3×D412yes—
229C22×C32C23×D311yescharacteristic
230C22×C32C24×C311yescharacteristic
231C2×C3⋊D3—22no—
232C2×C3⋊D3—24no—
233C2×C3⋊D3—24no—
234C2×C3⋊D3—28no—
235C2×C3⋊D3—48no—
236C2×C3×D3—62no—
237C2×C3×D3—64no—
238C2×C3×D3—64no—
239C2×C3×D3—68no—
240C2×C3×D3—128no—
241C32×C4—24no—
242C32×C4—24no—
243C32×C4—22no—
244C32×C4—22no—
245C32×C4—44no—
246C32×C4C2×D1212yes—
247C32×C4C2×D1212yes—
248C32×C4C2×C3×D412yes—
249C32×C4C2×C3×D412yes—
250C3⋊Dic3—22no—
251C3×Dic3—62no—
252C2×#41—91no—
253C22×D4—91no—
254#127—92no—
255#124—92no—
256#121—94no—
257#120—94no—
258#117(C3⋊D3)⋊C311yescharacteristic, complemented
259H(3,𝔽3)#53511yescharacteristic, complemented, Sylow
260C23×C3—31no—
261C23×C3—31no—
262C23×C3—61no—
263C23×C3—94no—
264C23×C3—92no—
265C23×C3—184no—
266C23×C3—364no—
267C23×C3C22×C3×D311yescharacteristic
268C22×D3—34no—
269C22×D3—32no—
270C22×D3—64no—
271C22×D3—94no—
272C22×D3—92no—
273C22×D3—124no—
274C22×D3—184no—
275C22×D3—364no—
276C3×Q8—182no—
277C3×D4—184no—
278C3×D4—184no—
279C3×D4—182no—
280C3×D4—184no—
281C3×D4—368no—
282C2×C3×C4—34no—
283C2×C3×C4—34no—
284C2×C3×C4—31no—
285C2×C3×C4—31no—
286C2×C3×C4—31no—
287C2×C3×C4—31no—
288C2×C3×C4—34no—
289C2×C3×C4—34no—
290C2×C3×C4—64no—
291C2×C3×C4—61no—
292C2×C3×C4—61no—
293C2×C3×C4—64no—
294C2×C3×C4—92no—
295C2×C3×C4—184no—
296C2×C3×C4C3×D1214yes—
297C2×C3×C4C22×C3×D314yes—
298C2×C3×C4C22×C3×D311yescharacteristic
299C2×C3×C4C22×C3×D311yescharacteristic
300#67—64no—
301#67—184no—
302C2×Dic3—32no—
303C2×Dic3—92no—
304D12—64no—
305D12—64no—
306D12—62no—
307D12—128no—
308D12—184no—
309D12—184no—
310D12—182no—
311D12—368no—
312C4×D3—64no—
313C4×D3—184no—
314Dic6—62no—
315Dic6—182no—
316C2×C32—22no—
317C2×C32—22no—
318C2×C32—21no—
319C2×C32—22no—
320C2×C32—42no—
321C2×C32C22×D1211yescharacteristic
322C2×C32#100912yes—
323C2×C32C22×C3×D411yescharacteristic
324C2×C32C3×H(5,𝔽2)12yes—
325C3⋊D3—42no—
326C3⋊D3—44no—
327C3×D3—122no—
328C3×D3—124no—
329C24—92no—
330#41—184no—
331C2×Q8—91no—
332C2×D4—92no—
333C2×D4—92no—
334C2×D4—92no—
335C2×D4—91no—
336C2×D4—98no—
337C2×D4—184no—
338C22×C4—92no—
339C22×C4C2×(C3⋊D3)⋊C311yescharacteristic
340C4⋊C4C2×(C3⋊D3)⋊C312yescomplemented
341C22⋊C4—98no—
342C22⋊C4C2×(C3⋊D3)⋊C312yes—
343C42C2×(C3⋊D3)⋊C312yescomplemented
344C22×C3—22no—
345C22×C3—31no—
346C22×C3—31no—
347C22×C3—32no—
348C22×C3—32no—
349C22×C3—62no—
350C22×C3—62no—
351C22×C3—61no—
352C22×C3—62no—
353C22×C3—122no—
354C22×C3—182no—
355C22×C3—184no—
356C22×C3—184no—
357C22×C3—188no—
358C22×C3—368no—
359C22×C3C2×C3×D1212yes—
360C22×C3C23×C3×D311yescharacteristic
361C2×D3—62no—
362C2×D3—64no—
363C2×D3—64no—
364C2×D3—68no—
365C2×D3—128no—
366C2×D3—182no—
367C2×D3—184no—
368C2×D3—184no—
369C2×D3—188no—
370C2×D3—368no—
371C3×C4—24no—
372C3×C4—32no—
373C3×C4—32no—
374C3×C4—32no—
375C3×C4—32no—
376C3×C4—64no—
377C3×C4—64no—
378C3×C4—62no—
379C3×C4—62no—
380C3×C4—124no—
381C3×C4—182no—
382C3×C4C2×C3×D1212yes—
383C3×C4C2×C3×D1212yes—
384Dic3—62no—
385Dic3—182no—
386C32—21no—
387C32#552511yescharacteristic, complemented
388C32C3×#53511yescharacteristic, complemented
389C23—94no—
390C23—92no—
391C23—184no—
392C23—364no—
393C23C22×(C3⋊D3)⋊C311yescharacteristic
394Q8—182no—
395D4—184no—
396D4—184no—
397D4—184no—
398D4—182no—
399D4—368no—
400C2×C4—92no—
401C2×C4—184no—
402C2×C4C4⋊((C3⋊D3)⋊C3)14yescomplemented
403C2×C4C22×(C3⋊D3)⋊C314yes—
404C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
405C2×C4C22×(C3⋊D3)⋊C311yescharacteristic
406C2×C3—22no—
407C2×C3—31no—
408C2×C3—31no—
409C2×C3—32no—
410C2×C3—32no—
411C2×C3—62no—
412C2×C3—62no—
413C2×C3—61no—
414C2×C3—62no—
415C2×C3—122no—
416C2×C3—364no—
417C2×C3—362no—
418C2×C3C22×C3×D1211yescharacteristic
419C2×C3C3×#100912yes—
420D3—124no—
421D3—122no—
422D3—364no—
423D3—362no—
424C22—22no—
425C22—182no—
426C22—184no—
427C22—184no—
428C22—188no—
429C22—368no—
430C22C2×C4⋊((C3⋊D3)⋊C3)12yes—
431C22C23×(C3⋊D3)⋊C311yescharacteristic
432C4—24no—
433C4—182no—
434C4C2×C4⋊((C3⋊D3)⋊C3)12yes—
435C4C2×C4⋊((C3⋊D3)⋊C3)12yes—
436C3—31no—
437C3—31no—
438C3—61no—
439C3C3×#552511yescharacteristic
440C2—22no—
441C2—364no—
442C2—362no—
443C2C22×C4⋊((C3⋊D3)⋊C3)11yescharacteristic
444C2#1172263812yes—
subgroupG/H
centerC22C23×(C3⋊D3)⋊C3
commutator, cosocleC22×C32C24×C3
Frattini, socleC22×C3C23×C3×D3
FittingH(3,𝔽3)×#117C2
hypercenter#117(C3⋊D3)⋊C3
hypocenterC32C3×#535
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C2×#3828⋊C3C2semidirect
2. C22×C4⋊((C3⋊D3)⋊C3)C2semidirect
3. H(3,𝔽3)×#117C2semidirect
4. #64104⋊C3C2semidirect
5. #64091⋊C3C2semidirect
6. #64084⋊C3C2semidirect
7. #11721385C2semidirect
8. #10595996C3semidirect
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
18. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
19. C2×C4×(C3⋊D3)⋊C3C22semidirect
20. C2×#634⋊C3C22semidirect
21. C4⋊C4×H(3,𝔽3)C22semidirect
23. C42×H(3,𝔽3)C22semidirect
24. #3752⋊C3C22semidirect
25. C2×#3828C2×C3semidirect
26. C22×C4⋊(C3⋊D3)C2×C3semidirect
30. C32×#117D3semidirect
31. C32×#117C2×C3semidirect
32. #64104C2×C3semidirect
33. #64091C2×C3semidirect
34. #64084C2×C3semidirect
35. #64083C2×C3semidirect
45. C2×C4×H(3,𝔽3)D4semidirect
70. C2×C4⋊(C3⋊D3)C22×C3semidirect
71. C2×C4⋊(C3⋊D3)C22×C3semidirect
72. C2×C4×C3⋊D3C22×C3semidirect
73. C2×#634C22×C3semidirect
84. C32×C4⋊C4C2×D3semidirect
85. C32×C4⋊C4C22×C3semidirect
90. C32×C42C2×D3semidirect
91. C32×C42C22×C3semidirect
92. #3752C22×C3semidirect
143. C2×C32×C4D12semidirect
144. C2×C32×C4C3×D4semidirect
258. #117(C3⋊D3)⋊C3semidirect
259. H(3,𝔽3)#535semidirect
340. C4⋊C4C2×(C3⋊D3)⋊C3semidirect
343. C42C2×(C3⋊D3)⋊C3semidirect
387. C32#5525semidirect
388. C32C3×#535semidirect
402. C2×C4C4⋊((C3⋊D3)⋊C3)semidirect

Maximal subgroups

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

subgroupindex
C2×#3828⋊C32
C22×C4⋊((C3⋊D3)⋊C3)2
H(3,𝔽3)×#1172
#64104⋊C32
#64091⋊C32
#64084⋊C32
#117213852
#105959963
C3×#55253