Finite Group Databasebeta

← #13175367

Subgroups — #13175367

Order 1728.

1688 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723910C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×C4×D12C211yesmaximal, characteristic, complemented
4#11723528C214yesmaximal, complemented
5#11723522C212yesmaximal, complemented
6#11723515C212yesmaximal, complemented
7#11723424C212yesmaximal, complemented
8#11723404C212yesmaximal, complemented
9#10596034—31nomaximal
10#10593988—34nomaximal
11C2×#6308C2212yes—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C32×D12C2211yescharacteristic
15C2×C32×C4×D3C2212yes—
16C2×#6290C2214yes—
17C33×C42C2211yescharacteristic
18C4⋊(C3⋊(C3⋊Dic3))C2212yes—
19C4⋊(C3⋊(C3⋊Dic3))C2212yes—
20C3×#3752C2212yes—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C32×(C2×D3)⋊C4C2212yes—
23C32×C4⋊Dic3C2211yescharacteristic
24#88803C2214yes—
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)D311yescharacteristic, complemented
34#64078—31no—
35#64078—34no—
36#64078—64no—
37C3×C4×D12—34no—
38C3×C4×D12D314yescomplemented
39#63913—316no—
40#63908—38no—
41#63908—38no—
42#63841—38no—
43#63841—38no—
44C22×C3×C3⋊D3C2312yes—
45C22×C32×D3C2312yes—
46C2×C33×C4C2311yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)C2314yes—
50#6308—42no—
51C2×C3×C3⋊Dic3C2312yes—
52C3×C4⋊(C3⋊D3)—22no—
53C3×C4×C3⋊D3—24no—
54C2×C32×Dic3C2312yes—
55C32×D12—22no—
56C32×C4×D3—24no—
57#6290—44no—
58#5563—94no—
59#5563—91no—
60C2×C32×D4—31no—
61C22×C32×C4—32no—
62C2×#638—34no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
64C2×C4×C3⋊D3C2×D312yes—
65C2×#634—32no—
66C2×#634—38no—
67C2×#634—68no—
68C2×C3×D12—34no—
69C2×C3×D12C2×D314yes—
70C2×C3×C4×D3—38no—
71C2×C3×C4×D3C2×D318yes—
72C2×#625—316no—
73C32×C4⋊C4—31no—
74C32×C22⋊C4—32no—
75C32×C42—24no—
76C32×C42C2×D314yes—
77C32×C42C2×D311yescharacteristic
78#3757—34no—
79#3752—34no—
80#3752C2×D312yes—
81C4⋊(C3⋊Dic3)—32no—
82C4⋊(C3⋊Dic3)—32no—
83C4⋊(C3⋊Dic3)—38no—
84C4⋊(C3⋊Dic3)—38no—
85C4⋊(C3⋊Dic3)—68no—
86C4⋊(C3⋊Dic3)—68no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88C4×C3⋊Dic3—34no—
89C3×(C2×D3)⋊C4—38no—
90C3×(C2×D3)⋊C4C2×D318yes—
91C3×C4⋊Dic3—34no—
92C3×C4⋊Dic3C2×D314yes—
93(C2×D3)⋊Dic3—316no—
94(C2×D3)⋊Dic3—316no—
95Dic32—316no—
96C22×C33C2411yescharacteristic
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3—22no—
99C2×C3×C3⋊D3—22no—
100C2×C32×D3—22no—
101C2×C32×D3—22no—
102C2×C32×D3—22no—
103C33×C4—21no—
104C33×C4#4112yes—
105C33×C4#4112yes—
106C3⋊(C3⋊Dic3)—24no—
107C3×C3⋊Dic3—22no—
108C32×Dic3—22no—
109C3×C4×D4—34no—
110C3×C4×D4—91no—
111#938—98no—
112#938—92no—
113#930—98no—
114#930—92no—
115#926—98no—
116#926—92no—
117#892—98no—
118#892—92no—
119#885—916no—
120#885—94no—
121C4×D12—34no—
122C4×D12C3⋊D311yescharacteristic, complemented
123#869—94no—
124#869—91no—
125#869—184no—
126C23×C32—32no—
127C22×C3⋊D3C22×D312yes—
128C22×C3×D3—38no—
129C22×C3×D3C22×D318yes—
130C32×D4—62no—
131C2×C32×C4—24no—
132C2×C32×C4—24no—
133C2×C32×C4—24no—
134C2×C32×C4—32no—
135C2×C32×C4—64no—
136C2×C32×C4C22×D314yes—
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D314yes—
139C2×C32×C4C22×D311yescharacteristic
140C2×C32×C4C22×D311yescharacteristic
141C2×C32×C4C22×D311yescharacteristic
142#638—124no—
143C2×C3⋊Dic3—34no—
144C2×C3⋊Dic3—316no—
145C2×C3⋊Dic3—616no—
146C2×C3⋊Dic3C22×D312yes—
147C4⋊(C3⋊D3)—22no—
148C4×C3⋊D3—24no—
149#634—122no—
150#634—128no—
151#634—1216no—
152C2×C3×Dic3—38no—
153C2×C3×Dic3C22×D318yes—
154C3×D12—28no—
155C3×D12—68no—
156C3×C4×D3—216no—
157C3×C4×D3—616no—
158#625—1216no—
159#555—271noSylow
160C2×C33C2×#4111yescharacteristic
161C2×C33C2×#4111yescharacteristic
162C2×C33H(5,𝔽2)11yescharacteristic
163C3×C3⋊D3—42no—
164C32×D3—42no—
165C2×C3×D4—34no—
166C2×C3×D4—91no—
167C22×C3×C4—38no—
168C22×C3×C4—92no—
169C2×#67—916no—
170C2×#67—94no—
171C2×D12—34no—
172C2×D12C2×C3⋊D311yescharacteristic
173C2×C4×D3—38no—
174C2×C4×D3C2×C3⋊D312yes—
175C2×Dic6—98no—
176C2×Dic6—92no—
177C2×Dic6—188no—
178C3×C4⋊C4—34no—
179C3×C4⋊C4—91no—
180C3×C22⋊C4—38no—
181C3×C22⋊C4—92no—
182C3×C42—24no—
183C3×C42D3214yes—
184C3×C42C2×C3⋊D311yescharacteristic
185#217—916no—
186#217—94no—
187(C2×D3)⋊C4—38no—
188(C2×D3)⋊C4—916no—
189(C2×D3)⋊C4—94no—
190(C2×D3)⋊C4C2×C3⋊D312yes—
191C4⋊Dic3—34no—
192C4⋊Dic3—98no—
193C4⋊Dic3—98no—
194C4⋊Dic3—92no—
195C4⋊Dic3—92no—
196C4⋊Dic3—188no—
197C4⋊Dic3—188no—
198C4⋊Dic3C2×C3⋊D311yescharacteristic
199C4×Dic3—916no—
200C4×Dic3—94no—
201C22×C32—24no—
202C22×C32—62no—
203C22×C32—62no—
204C22×C32—62no—
205C22×C32C23×D314yes—
206C22×C32C23×D311yescharacteristic
207C2×C3⋊D3—22no—
208C2×C3⋊D3—22no—
209C2×C3⋊D3—22no—
210C2×C3×D3—28no—
211C2×C3×D3—28no—
212C2×C3×D3—28no—
213C2×C3×D3—68no—
214C2×C3×D3—68no—
215C2×C3×D3—68no—
216C32×C4—28no—
217C32×C4—24no—
218C32×C4—21no—
219C32×C4—28no—
220C32×C4—28no—
221C32×C4—62no—
222C32×C4#23518yes—
223C32×C4#23512yes—
224C32×C4#23718yes—
225C32×C4#23712yes—
226C3⋊Dic3—22no—
227C3⋊Dic3—64no—
228C3⋊Dic3—616no—
229C3⋊Dic3—1216no—
230C3×Dic3—28no—
231C3×Dic3—68no—
232C4⋊Q8—271no—
233#124—274no—
234#121—272no—
235#121—272no—
236C4×D4—31no—
237C4×D4—91no—
238#117—272no—
239#117—272no—
240C33#55511yescharacteristic, complemented, Sylow
241C23×C3—38no—
242C23×C3—92no—
243C22×D3—38no—
244C22×D3C22×C3⋊D312yes—
245C3×D4—68no—
246C3×D4—182no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—24no—
250C2×C3×C4—38no—
251C2×C3×C4—616no—
252C2×C3×C4—92no—
253C2×C3×C4—184no—
254C2×C3×C4C2×D3214yes—
255C2×C3×C4C2×D3214yes—
256C2×C3×C4C2×D3214yes—
257C2×C3×C4C22×C3⋊D311yescharacteristic
258C2×C3×C4C22×C3⋊D311yescharacteristic
259C2×C3×C4C22×C3⋊D311yescharacteristic
260#67—364no—
261#67—3616no—
262C2×Dic3—38no—
263C2×Dic3—916no—
264C2×Dic3—94no—
265C2×Dic3—1816no—
266C2×Dic3C22×C3⋊D312yes—
267D12—22no—
268D12—68no—
269C4×D3—24no—
270C4×D3—616no—
271Dic6—368no—
272Dic6—362no—
273Dic6—3616no—
274C2×C32—24no—
275C2×C32—24no—
276C2×C32—24no—
277C2×C32—122no—
278C2×C32C2×#23514yes—
279C2×C32C2×#23511yescharacteristic
280C2×C32C2×#23714yes—
281C2×C32C2×#23711yescharacteristic
282C2×C32#100914yes—
283C2×C32#100911yescharacteristic
284C3⋊D3—42no—
285C3×D3—48no—
286C3×D3—128no—
287C2×Q8—272no—
288C2×D4—31no—
289C2×D4—91no—
290C2×D4—274no—
291C22×C4—32no—
292C22×C4—92no—
293C4⋊C4—31no—
294C4⋊C4—91no—
295C4⋊C4—272no—
296C4⋊C4—272no—
297C22⋊C4—32no—
298C22⋊C4—92no—
299C22⋊C4—274no—
300C22⋊C4—274no—
301C42—274no—
302C42D3×C3⋊D311yescharacteristic
303C22×C3—24no—
304C22×C3—68no—
305C22×C3—68no—
306C22×C3—68no—
307C22×C3—182no—
308C22×C3—182no—
309C22×C3—182no—
310C22×C3C22×D3214yes—
311C22×C3C23×C3⋊D311yescharacteristic
312C2×D3—22no—
313C2×D3—22no—
314C2×D3—22no—
315C2×D3—68no—
316C2×D3—68no—
317C2×D3—68no—
318C3×C4—28no—
319C3×C4—24no—
320C3×C4—21no—
321C3×C4—28no—
322C3×C4—28no—
323C3×C4—68no—
324C3×C4—182no—
325C3×C4#379518yes—
326C3×C4#379518yes—
327C3×C4#382812yes—
328C3×C4#383012yes—
329Dic3—22no—
330Dic3—68no—
331Dic3—1816no—
332Dic3—184no—
333Dic3—3616no—
334C32—24no—
335C32#556314yescomplemented
336C32#556311yescharacteristic, complemented
337C23—32no—
338C23—92no—
339Q8—1082no—
340D4—62no—
341D4—182no—
342D4—1084no—
343C2×C4—32no—
344C2×C4—64no—
345C2×C4—92no—
346C2×C4—184no—
347C2×C4—274no—
348C2×C4C2×D3×C3⋊D311yescharacteristic
349C2×C4C2×D3×C3⋊D311yescharacteristic
350C2×C4C2×D3×C3⋊D311yescharacteristic
351C2×C3—24no—
352C2×C3—24no—
353C2×C3—24no—
354C2×C3—128no—
355C2×C3—362no—
356C2×C3C2×#379514yes—
357C2×C3C2×#379514yes—
358C2×C3#6430814yes—
359C2×C3C2×#382811yescharacteristic
360C2×C3C2×#383011yescharacteristic
361C2×C3#6436611yescharacteristic
362D3—42no—
363D3—128no—
364C22—62no—
365C22—62no—
366C22—62no—
367C22—182no—
368C22—182no—
369C22—182no—
370C22C22×D3×C3⋊D311yescharacteristic
371C4—21no—
372C4—62no—
373C4—182no—
374C4—544no—
375C4#8901412yes—
376C4#8902112yes—
377C3—24no—
378C3#1059398814yescomplemented
379C3#1059603411yescharacteristic, complemented
380C2—122no—
381C2—362no—
382C2C2×#8901411yescharacteristic
383C2C2×#8902111yescharacteristic
384C2#1172458111yescharacteristic
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. #11723910C2semidirect
2. C3×C4×C4⋊(C3⋊D3)C2semidirect
3. C32×C4×D12C2semidirect
4. #11723528C2semidirect
5. #11723522C2semidirect
6. #11723515C2semidirect
7. #11723424C2semidirect
8. #11723404C2semidirect
33. C4×C4⋊(C3⋊D3)D3semidirect
38. C3×C4×D12D3semidirect
122. C4×D12C3⋊D3semidirect
240. C33#555semidirect
335. C32#5563semidirect
336. C32#5563semidirect
378. C3#10593988semidirect
379. C3#10596034semidirect

Maximal subgroups

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

subgroupindex
#117239102
C3×C4×C4⋊(C3⋊D3)2
C32×C4×D122
#117235282
#117235222
#117235152
#117234242
#117234042
#105960343
#105939883