Finite Group Databasebeta

← #13175365

Subgroups — #13175365

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×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×#869C211yesmaximal, characteristic, complemented
4#11723532C212yesmaximal, complemented
5#11723526C212yesmaximal, complemented
6#11723413C212yesmaximal, complemented
7#11723404C212yesmaximal, complemented
8#11723392C214yesmaximal, complemented
9#10596205—31nomaximal
10#10593987—34nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C4×C3⋊(C3⋊D3)C2212yes—
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2212yes—
15C2×C32×Dic6C2212yes—
16C2×#6292C2214yes—
17C33×C42C2211yescharacteristic
18#88859C2212yes—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C3×#3752C2212yes—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C32×C4⋊Dic3C2212yes—
23C32×C4⋊Dic3C2212yes—
24#88804C2214yes—
25#88803C2214yes—
26Dic3×C3⋊Dic3C2214yes—
27C32×C4⋊Q8—31no—
28#64156—34no—
29#64104—32no—
30#64104—32no—
31#64101—32no—
32#64101—32no—
33C4×C4⋊(C3⋊D3)—31no—
34C4×C4⋊(C3⋊D3)—34no—
35C4×C4⋊(C3⋊D3)—64no—
36C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
37C3×C4×D12—34no—
38C3×#869D314yescomplemented
39#63916—38no—
40#63909—38no—
41#63907—316no—
42#63849—38no—
43#63841—38no—
44C22×C3⋊(C3⋊D3)C2312yes—
45C22×C3×C3⋊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⋊Dic3C2312yes—
53C3×C4⋊(C3⋊D3)—22no—
54C3×C4×C3⋊D3—24no—
55C2×C32×Dic3C2314yes—
56C32×Dic6—42no—
57#6292—44no—
58#5734—94no—
59#5563—91no—
60C2×C32×Q8—32no—
61C2×C4⋊(C3⋊D3)—31no—
62C2×C4⋊(C3⋊D3)—34no—
63C2×C4⋊(C3⋊D3)—34no—
64C2×C4⋊(C3⋊D3)—64no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C4×C3⋊D3—32no—
67C2×C4×C3⋊D3—38no—
68C2×C4×C3⋊D3—68no—
69C2×C4×C3⋊D3C2×D312yes—
70C2×C3×D12—34no—
71C2×C3×C4×D3—38no—
72C2×C3×Dic6C2×D318yes—
73C2×#626—316no—
74C32×C4⋊C4—32no—
75C32×C4⋊C4—32no—
76C32×C42—24no—
77C32×C42C2×D314yes—
78C32×C42C2×D311yescharacteristic
79#3752—34no—
80#3752—32no—
81#3752—38no—
82#3752—34no—
83#3752—68no—
84#3752C2×D312yes—
85C4⋊(C3⋊Dic3)—31no—
86C4⋊(C3⋊Dic3)—34no—
87C4⋊(C3⋊Dic3)—64no—
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic
89C4×C3⋊Dic3—34no—
90C3×(C2×D3)⋊C4—38no—
91C3×C4⋊Dic3—34no—
92C3×C4⋊Dic3C2×D318yes—
93C3×C4⋊Dic3C2×D318yes—
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×C3×C3⋊D3—22no—
102C2×C3×C3⋊D3—22no—
103C2×C3×C3⋊D3—22no—
104C33×C4—21no—
105C33×C4#4112yes—
106C33×C4#4112yes—
107C3⋊(C3⋊Dic3)—22no—
108C3×C3⋊Dic3—22no—
109C32×Dic3—24no—
110C3×C4⋊Q8—34no—
111C3×C4×D4—91no—
112#947—916no—
113#938—92no—
114#930—92no—
115#926—92no—
116#895—98no—
117#895—98no—
118#892—98no—
119#892—98no—
120#892—92no—
121#885—94no—
122C4×D12—34no—
123C4×D12—94no—
124C4×D12—91no—
125C4×D12—184no—
126#869C3⋊D311yescharacteristic, complemented
127C22×C3⋊D3—32no—
128C22×C3⋊D3—38no—
129C22×C3⋊D3—68no—
130C22×C3⋊D3C22×D312yes—
131C22×C3×D3—38no—
132C32×Q8—122no—
133C2×C32×C4—24no—
134C2×C32×C4—24no—
135C2×C32×C4—24no—
136C2×C32×C4—34no—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D314yes—
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142C2×C32×C4C22×D311yescharacteristic
143C2×C3⋊Dic3—32no—
144C2×C3⋊Dic3—38no—
145C2×C3⋊Dic3—68no—
146C2×C3⋊Dic3C22×D312yes—
147C4⋊(C3⋊D3)—22no—
148C4⋊(C3⋊D3)—616no—
149C4⋊(C3⋊D3)—62no—
150C4⋊(C3⋊D3)—68no—
151C4⋊(C3⋊D3)—124no—
152C4×C3⋊D3—24no—
153C4×C3⋊D3—64no—
154C4×C3⋊D3—616no—
155C4×C3⋊D3—1216no—
156C2×C3×Dic3—38no—
157C2×C3×Dic3C22×D3116yes—
158C3×D12—68no—
159C3×C4×D3—616no—
160C3×Dic6—48no—
161#626—1216no—
162#555—271noSylow
163C2×C33C2×#4111yescharacteristic
164C2×C33C2×#4111yescharacteristic
165C2×C33H(5,𝔽2)11yescharacteristic
166C3⋊(C3⋊D3)—42no—
167C3×C3⋊D3—42no—
168C2×C3×Q8—38no—
169C2×C3×D4—91no—
170C22×C3×C4—92no—
171C2×#67—94no—
172C2×D12—34no—
173C2×D12—94no—
174C2×D12—91no—
175C2×D12—916no—
176C2×D12—184no—
177C2×C4×D3—38no—
178C2×C4×D3—98no—
179C2×C4×D3—92no—
180C2×C4×D3—188no—
181C2×Dic6C2×C3⋊D312yes—
182C3×C4⋊C4—38no—
183C3×C4⋊C4—38no—
184C3×C4⋊C4—91no—
185C3×C22⋊C4—92no—
186C3×C42—24no—
187C3×C42D3214yes—
188C3×C42C2×C3⋊D311yescharacteristic
189#217—94no—
190(C2×D3)⋊C4—38no—
191(C2×D3)⋊C4—916no—
192(C2×D3)⋊C4—916no—
193(C2×D3)⋊C4—98no—
194(C2×D3)⋊C4—92no—
195(C2×D3)⋊C4—94no—
196(C2×D3)⋊C4—188no—
197C4⋊Dic3—34no—
198C4⋊Dic3—94no—
199C4⋊Dic3—91no—
200C4⋊Dic3—184no—
201C4⋊Dic3C2×C3⋊D312yes—
202C4⋊Dic3C2×C3⋊D312yes—
203C4×Dic3—916no—
204C4×Dic3—94no—
205C22×C32—24no—
206C22×C32C23×D314yes—
207C22×C32C23×D311yescharacteristic
208C2×C3⋊D3—22no—
209C2×C3⋊D3—22no—
210C2×C3⋊D3—22no—
211C2×C3⋊D3—616no—
212C2×C3⋊D3—62no—
213C2×C3⋊D3—62no—
214C2×C3⋊D3—68no—
215C2×C3⋊D3—62no—
216C2×C3⋊D3—68no—
217C2×C3⋊D3—68no—
218C2×C3⋊D3—128no—
219C2×C3⋊D3—128no—
220C2×C3×D3—68no—
221C2×C3×D3—68no—
222C2×C3×D3—68no—
223C32×C4—21no—
224C32×C4—28no—
225C32×C4—28no—
226C32×C4—24no—
227C32×C4—44no—
228C32×C4—64no—
229C32×C4#23512yes—
230C32×C4#23712yes—
231C32×C4#23918yes—
232C32×C4#23918yes—
233C3⋊Dic3—22no—
234C3⋊Dic3—62no—
235C3⋊Dic3—68no—
236C3⋊Dic3—128no—
237C3×Dic3—216no—
238C3×Dic3—68no—
239C4⋊Q8—31no—
240#124—274no—
241#121—272no—
242#121—272no—
243C4×D4—91no—
244C4×D4—271no—
245#117—272no—
246#117—272no—
247C33#55511yescharacteristic, complemented, Sylow
248C23×C3—92no—
249C22×D3—38no—
250C22×D3—98no—
251C22×D3—92no—
252C22×D3—188no—
253C3×Q8—128no—
254C3×D4—182no—
255C2×C3×C4—24no—
256C2×C3×C4—24no—
257C2×C3×C4—24no—
258C2×C3×C4—316no—
259C2×C3×C4—92no—
260C2×C3×C4—184no—
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—364no—
268C2×Dic3—38no—
269C2×Dic3—98no—
270C2×Dic3—92no—
271C2×Dic3—188no—
272C2×Dic3C22×C3⋊D314yes—
273D12—68no—
274D12—1816no—
275D12—188no—
276D12—182no—
277D12—3616no—
278C4×D3—616no—
279C4×D3—1816no—
280C4×D3—184no—
281C4×D3—3616no—
282Dic6—42no—
283C2×C32—24no—
284C2×C32—24no—
285C2×C32—24no—
286C2×C32C2×#23511yescharacteristic
287C2×C32C2×#23711yescharacteristic
288C2×C32#100414yes—
289C2×C32C2×#23914yes—
290C2×C32C2×#23914yes—
291C2×C32#100911yescharacteristic
292C3⋊D3—42no—
293C3⋊D3—1216no—
294C3⋊D3—128no—
295C3⋊D3—122no—
296C3×D3—128no—
297C2×Q8—32no—
298C2×D4—91no—
299C2×D4—274no—
300C2×D4—271no—
301C22×C4—92no—
302C22×C4—272no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—91no—
306C4⋊C4—271no—
307C22⋊C4—92no—
308C22⋊C4—274no—
309C22⋊C4—274no—
310C22⋊C4—272no—
311C42—274no—
312C42D3×C3⋊D311yescharacteristic
313C22×C3—24no—
314C22×C3—182no—
315C22×C3—182no—
316C22×C3—182no—
317C22×C3C22×D3214yes—
318C22×C3C23×C3⋊D311yescharacteristic
319C2×D3—68no—
320C2×D3—68no—
321C2×D3—68no—
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—616no—
337C3×C4—182no—
338C3×C4#379618yes—
339C3×C4#379918yes—
340C3×C4#383212yes—
341C3×C4#383212yes—
342Dic3—24no—
343Dic3—68no—
344Dic3—188no—
345Dic3—182no—
346Dic3—368no—
347C32—24no—
348C32#556311yescharacteristic, complemented
349C32#573414yescomplemented
350C23—92no—
351C23—272no—
352Q8—122no—
353D4—182no—
354D4—542no—
355D4—1084no—
356C2×C4—34no—
357C2×C4—92no—
358C2×C4—184no—
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—362no—
368C2×C3C2×#379614yes—
369C2×C3C2×#379914yes—
370C2×C3#6430714yes—
371C2×C3#6436111yescharacteristic
372C2×C3C2×#383211yescharacteristic
373C2×C3C2×#383211yescharacteristic
374D3—128no—
375D3—3616no—
376D3—362no—
377D3—368no—
378C22—182no—
379C22—182no—
380C22—182no—
381C22—542no—
382C22—542no—
383C22—542no—
384C22C22×D3×C3⋊D311yescharacteristic
385C4—21no—
386C4—64no—
387C4—182no—
388C4—542no—
389C4#8901712yes—
390C4#8901812yes—
391C3—24no—
392C3#1059398714yescomplemented
393C3#1059620511yescharacteristic, complemented
394C2—362no—
395C2—1082no—
396C2C2×#8901711yescharacteristic
397C2C2×#8901811yescharacteristic
398C2#1172457911yescharacteristic
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×C4×C4⋊(C3⋊D3)C2semidirect
3. C32×#869C2semidirect
4. #11723532C2semidirect
5. #11723526C2semidirect
6. #11723413C2semidirect
7. #11723404C2semidirect
8. #11723392C2semidirect
36. C4×C4⋊(C3⋊D3)D3semidirect
38. C3×#869D3semidirect
126. #869C3⋊D3semidirect
247. C33#555semidirect
348. C32#5563semidirect
349. C32#5734semidirect
392. C3#10593987semidirect
393. C3#10596205semidirect

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×C4×C4⋊(C3⋊D3)2
C32×#8692
#117235322
#117235262
#117234132
#117234042
#117233922
#105962053
#105939873