Finite Group Databasebeta

← #13175360

Subgroups — #13175360

Order 1728.

1792 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
2C3×#64078C211yesmaximal, characteristic, complemented
3C32×C4×D12C211yesmaximal, characteristic, complemented
4#11723531C212yesmaximal, complemented
5#11723521C214yesmaximal, complemented
6#11723516C212yesmaximal, complemented
7#11723424C212yesmaximal, complemented
8#11723413C212yesmaximal, complemented
9#10596034—31nomaximal
10#10593987—34nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2212yes—
13C2×C3×#634C2212yes—
14C2×C32×D12C2211yescharacteristic
15C2×C32×C4×D3C2212yes—
16C2×#6291C2214yes—
17C33×C42C2211yescharacteristic
18#88859C2212yes—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2212yes—
21C3×C4⋊(C3⋊Dic3)C2212yes—
22C32×(C2×D3)⋊C4C2212yes—
23C32×C4⋊Dic3C2211yescharacteristic
24#88804C2214yes—
25(C2×D3)⋊(C3⋊Dic3)C2214yes—
26Dic3×C3⋊Dic3C2214yes—
27C32×C4×D4—31no—
28#64147—32no—
29#64139—32no—
30#64135—32no—
31#64101—32no—
32#64094—34no—
33C4×C4⋊(C3⋊D3)—31no—
34C4×C4⋊(C3⋊D3)—34no—
35C4×C4⋊(C3⋊D3)—64no—
36#64078D311yescharacteristic, complemented
37C3×C4×D12D314yescomplemented
38C3×#869—34no—
39#63916—38no—
40#63909—38no—
41#63907—316no—
42#63849—38no—
43#63841—38no—
44C22×C3⋊(C3⋊D3)C2312yes—
45C22×C32×D3C2312yes—
46C2×C33×C4C2311yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)C2312yes—
50C4⋊(C3⋊(C3⋊D3))—22no—
51C4×C3⋊(C3⋊D3)—24no—
52C2×C3×C3⋊Dic3C2314yes—
53C3×#634—42no—
54C2×C32×Dic3C2312yes—
55C32×D12—22no—
56C32×C4×D3—24no—
57#6291—44no—
58#5734—91no—
59#5563—94no—
60C2×C32×D4—31no—
61C22×C32×C4—32no—
62C2×#638—34no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—34no—
65C2×C4⋊(C3⋊D3)—64no—
66C2×C4×C3⋊D3—32no—
67C2×C4×C3⋊D3—38no—
68C2×C4×C3⋊D3—68no—
69C2×#634C2×D312yes—
70C2×C3×D12C2×D314yes—
71C2×C3×C4×D3C2×D318yes—
72C2×C3×Dic6—38no—
73C2×#626—316no—
74C32×C4⋊C4—31no—
75C32×C22⋊C4—32no—
76C32×C42—24no—
77C32×C42C2×D314yes—
78C32×C42C2×D311yescharacteristic
79#3757—34no—
80#3752—32no—
81#3752—38no—
82#3752—34no—
83#3752—68no—
84C4⋊(C3⋊Dic3)—31no—
85C4⋊(C3⋊Dic3)—34no—
86C4⋊(C3⋊Dic3)—64no—
87C4⋊(C3⋊Dic3)C2×D312yes—
88C4⋊(C3⋊Dic3)C2×D312yes—
89C4×C3⋊Dic3—34no—
90C3×(C2×D3)⋊C4C2×D318yes—
91C3×C4⋊Dic3—38no—
92C3×C4⋊Dic3—38no—
93C3×C4⋊Dic3C2×D314yes—
94#3722—316no—
95(C2×D3)⋊Dic3—316no—
96Dic32—316no—
97C22×C33C2411yescharacteristic
98C2×C3⋊(C3⋊D3)—22no—
99C2×C3⋊(C3⋊D3)—22no—
100C2×C3⋊(C3⋊D3)—22no—
101C2×C32×D3—22no—
102C2×C32×D3—22no—
103C2×C32×D3—22no—
104C33×C4—21no—
105C33×C4#4112yes—
106C33×C4#4112yes—
107C3⋊(C3⋊Dic3)—22no—
108C3×C3⋊Dic3—24no—
109C32×Dic3—22no—
110C3×C4⋊Q8—91no—
111C3×C4×D4—34no—
112#947—94no—
113#938—98no—
114#930—98no—
115#926—98no—
116#895—92no—
117#895—92no—
118#892—98no—
119#892—92no—
120#892—92no—
121#885—916no—
122C4×D12—94no—
123C4×D12—91no—
124C4×D12—184no—
125C4×D12C3⋊D311yescharacteristic, complemented
126#869—34no—
127C23×C32—32no—
128C22×C3⋊D3—32no—
129C22×C3⋊D3—38no—
130C22×C3⋊D3—68no—
131C22×C3×D3C22×D318yes—
132C32×D4—62no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—32no—
137C2×C32×C4—64no—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D314yes—
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C32×C4C22×D311yescharacteristic
144#638—124no—
145C2×C3⋊Dic3—32no—
146C2×C3⋊Dic3—38no—
147C2×C3⋊Dic3—68no—
148C2×C3⋊Dic3C22×D314yes—
149C4⋊(C3⋊D3)—616no—
150C4⋊(C3⋊D3)—68no—
151C4⋊(C3⋊D3)—62no—
152C4×C3⋊D3—64no—
153C4×C3⋊D3—616no—
154C4×C3⋊D3—1216no—
155#634—42no—
156C2×C3×Dic3—316no—
157C2×C3×Dic3C22×D318yes—
158C3×D12—28no—
159C3×C4×D3—216no—
160C3×Dic6—128no—
161#626—1216no—
162#555—271noSylow
163C2×C33C2×#4111yescharacteristic
164C2×C33C2×#4111yescharacteristic
165C2×C33H(5,𝔽2)11yescharacteristic
166C3⋊(C3⋊D3)—42no—
167C32×D3—42no—
168C2×C3×Q8—92no—
169C2×C3×D4—34no—
170C22×C3×C4—38no—
171C2×#67—916no—
172C2×D12—94no—
173C2×D12—91no—
174C2×D12—94no—
175C2×D12—184no—
176C2×D12C2×C3⋊D311yescharacteristic
177C2×C4×D3—98no—
178C2×C4×D3—92no—
179C2×C4×D3—188no—
180C2×C4×D3C2×C3⋊D312yes—
181C2×Dic6—38no—
182C3×C4⋊C4—34no—
183C3×C4⋊C4—92no—
184C3×C4⋊C4—92no—
185C3×C22⋊C4—38no—
186C3×C42—24no—
187C3×C42D3214yes—
188C3×C42C2×C3⋊D311yescharacteristic
189#217—916no—
190(C2×D3)⋊C4—916no—
191(C2×D3)⋊C4—94no—
192(C2×D3)⋊C4—98no—
193(C2×D3)⋊C4—92no—
194(C2×D3)⋊C4—94no—
195(C2×D3)⋊C4—188no—
196(C2×D3)⋊C4C2×C3⋊D312yes—
197C4⋊Dic3—38no—
198C4⋊Dic3—38no—
199C4⋊Dic3—94no—
200C4⋊Dic3—91no—
201C4⋊Dic3—184no—
202C4⋊Dic3C2×C3⋊D311yescharacteristic
203C4×Dic3—916no—
204C4×Dic3—94no—
205C22×C32—24no—
206C22×C32—62no—
207C22×C32—62no—
208C22×C32—62no—
209C22×C32C23×D314yes—
210C22×C32C23×D311yescharacteristic
211C2×C3⋊D3—616no—
212C2×C3⋊D3—68no—
213C2×C3⋊D3—62no—
214C2×C3⋊D3—62no—
215C2×C3⋊D3—62no—
216C2×C3⋊D3—68no—
217C2×C3⋊D3—68no—
218C2×C3⋊D3—128no—
219C2×C3⋊D3—128no—
220C2×C3×D3—28no—
221C2×C3×D3—28no—
222C2×C3×D3—28no—
223C32×C4—21no—
224C32×C4—28no—
225C32×C4—28no—
226C32×C4—24no—
227C32×C4—44no—
228C32×C4—62no—
229C32×C4#23518yes—
230C32×C4#23718yes—
231C32×C4#23912yes—
232C32×C4#23912yes—
233C3⋊Dic3—24no—
234C3⋊Dic3—62no—
235C3⋊Dic3—68no—
236C3⋊Dic3—128no—
237C3×Dic3—28no—
238C3×Dic3—616no—
239C4⋊Q8—91no—
240#124—274no—
241#121—272no—
242#121—272no—
243C4×D4—31no—
244C4×D4—271no—
245#117—272no—
246#117—272no—
247C33#55511yescharacteristic, complemented, Sylow
248C23×C3—38no—
249C22×D3—98no—
250C22×D3—92no—
251C22×D3—188no—
252C22×D3C22×C3⋊D312yes—
253C3×Q8—362no—
254C3×D4—68no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—38no—
259C2×C3×C4—616no—
260C2×C3×C4—94no—
261C2×C3×C4C2×D3214yes—
262C2×C3×C4C2×D3214yes—
263C2×C3×C4C2×D3214yes—
264C2×C3×C4C22×C3⋊D311yescharacteristic
265C2×C3×C4C22×C3⋊D311yescharacteristic
266C2×C3×C4C22×C3⋊D311yescharacteristic
267#67—3616no—
268C2×Dic3—316no—
269C2×Dic3—98no—
270C2×Dic3—92no—
271C2×Dic3—188no—
272C2×Dic3C22×C3⋊D312yes—
273D12—22no—
274D12—1816no—
275D12—188no—
276D12—182no—
277D12—364no—
278C4×D3—24no—
279C4×D3—1816no—
280C4×D3—184no—
281C4×D3—3616no—
282Dic6—128no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—24no—
286C2×C32—122no—
287C2×C32C2×#23514yes—
288C2×C32C2×#23714yes—
289C2×C32#100411yescharacteristic
290C2×C32C2×#23911yescharacteristic
291C2×C32C2×#23911yescharacteristic
292C2×C32#100914yes—
293C3⋊D3—1216no—
294C3⋊D3—122no—
295C3⋊D3—128no—
296C3×D3—48no—
297C2×Q8—92no—
298C2×D4—31no—
299C2×D4—274no—
300C2×D4—271no—
301C22×C4—32no—
302C22×C4—272no—
303C4⋊C4—31no—
304C4⋊C4—92no—
305C4⋊C4—92no—
306C4⋊C4—271no—
307C22⋊C4—32no—
308C22⋊C4—274no—
309C22⋊C4—274no—
310C22⋊C4—272no—
311C42—274no—
312C42D3×C3⋊D311yescharacteristic
313C22×C3—24no—
314C22×C3—68no—
315C22×C3—68no—
316C22×C3—68no—
317C22×C3C22×D3214yes—
318C22×C3C23×C3⋊D311yescharacteristic
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—22no—
322C2×D3—1816no—
323C2×D3—188no—
324C2×D3—188no—
325C2×D3—188no—
326C2×D3—182no—
327C2×D3—182no—
328C2×D3—182no—
329C2×D3—368no—
330C2×D3—368no—
331C3×C4—24no—
332C3×C4—21no—
333C3×C4—28no—
334C3×C4—28no—
335C3×C4—44no—
336C3×C4—68no—
337C3×C4—184no—
338C3×C4#379618yes—
339C3×C4#379918yes—
340C3×C4#382812yes—
341C3×C4#383012yes—
342Dic3—22no—
343Dic3—616no—
344Dic3—188no—
345Dic3—182no—
346Dic3—368no—
347C32—24no—
348C32#556314yescomplemented
349C32#573411yescharacteristic, complemented
350C23—32no—
351C23—272no—
352Q8—362no—
353D4—62no—
354D4—542no—
355D4—1084no—
356C2×C4—32no—
357C2×C4—64no—
358C2×C4—94no—
359C2×C4—272no—
360C2×C4—544no—
361C2×C4C2×D3×C3⋊D311yescharacteristic
362C2×C4C2×D3×C3⋊D311yescharacteristic
363C2×C4C2×D3×C3⋊D311yescharacteristic
364C2×C3—24no—
365C2×C3—24no—
366C2×C3—24no—
367C2×C3—128no—
368C2×C3C2×#379614yes—
369C2×C3C2×#379914yes—
370C2×C3#6430714yes—
371C2×C3C2×#382811yescharacteristic
372C2×C3C2×#383011yescharacteristic
373C2×C3#6436611yescharacteristic
374D3—42no—
375D3—3616no—
376D3—362no—
377D3—368no—
378C22—62no—
379C22—62no—
380C22—62no—
381C22—542no—
382C22—542no—
383C22—542no—
384C22C22×D3×C3⋊D311yescharacteristic
385C4—21no—
386C4—62no—
387C4—184no—
388C4—542no—
389C4#8901512yes—
390C4#8902212yes—
391C3—24no—
392C3#1059398714yescomplemented
393C3#1059603411yescharacteristic, complemented
394C2—122no—
395C2—1082no—
396C2C2×#8901511yescharacteristic
397C2C2×#8902211yescharacteristic
398C2#1172458011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#555
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
2. C3×#64078C2semidirect
3. C32×C4×D12C2semidirect
4. #11723531C2semidirect
5. #11723521C2semidirect
6. #11723516C2semidirect
7. #11723424C2semidirect
8. #11723413C2semidirect
36. #64078D3semidirect
37. C3×C4×D12D3semidirect
125. C4×D12C3⋊D3semidirect
247. C33#555semidirect
348. C32#5563semidirect
349. C32#5734semidirect
392. C3#10593987semidirect
393. C3#10596034semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊(C3⋊D3))2
C3×#640782
C32×C4×D122
#117235312
#117235212
#117235162
#117234242
#117234132
#105960343
#105939873