Finite Group Databasebeta

← #13175199

Subgroups — #13175199

Order 1728.

2128 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89022C212yesmaximal, complemented
2C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
3C3×#64078C211yesmaximal, characteristic, complemented
4C32×C4×D12C211yesmaximal, characteristic, complemented
5#11723531C212yesmaximal, complemented
6#11723503C212yesmaximal, complemented
7#11723425C212yesmaximal, complemented
8#11723422C214yesmaximal, complemented
9#10596022—31nomaximal
10#10593869—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—
17C2×#6289C2214yes—
18#89022—28no—
19C33×C42C2211yescharacteristic
20#88859C2212yes—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C3×C4⋊(C3⋊Dic3)C2212yes—
23C3×C4⋊(C3⋊Dic3)C2212yes—
24C32×(C2×D3)⋊C4C2212yes—
25C32×C4⋊Dic3C2211yescharacteristic
26#88806C2214yes—
27(C2×D3)⋊(C3⋊Dic3)C2214yes—
28C2×#3828—32no—
29C2×#3799—38no—
30C32×C4×D4—31no—
31#64147—32no—
32#64133—32no—
33C4⋊C4×C3⋊D3—32no—
34#64092—34no—
35C4×C4⋊(C3⋊D3)—31no—
36C4×C4⋊(C3⋊D3)—34no—
37C4×C4⋊(C3⋊D3)—64no—
38#64078D311yescharacteristic, complemented
39C3×C4×D12D314yescomplemented
40C3×#869—34no—
41#63916—38no—
42#63900—38no—
43#63856—38no—
44#63854—316no—
45C22×C3⋊(C3⋊D3)C2312yes—
46C22×C32×D3C2312yes—
47C2×C33×C4C2311yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)C2312yes—
51C4⋊(C3⋊(C3⋊D3))D414yescomplemented
52C4×C3⋊(C3⋊D3)—24no—
53C2×C3×C3⋊Dic3C2314yes—
54C3×#634—24no—
55C2×C32×Dic3C2312yes—
56C32×D12—22no—
57C32×C4×D3—24no—
58#6291—28no—
59#6289—28no—
60#5726—91no—
61#5551—94no—
62C2×C32×D4—31no—
63C22×C32×C4—32no—
64C2×#638—34no—
65#3828—68no—
66C2×C4⋊(C3⋊D3)—31no—
67C2×C4⋊(C3⋊D3)—34no—
68C2×C4⋊(C3⋊D3)—64no—
69C2×C4×C3⋊D3—32no—
70C2×C4×C3⋊D3—38no—
71C2×C4×C3⋊D3—34no—
72C2×C4×C3⋊D3—68no—
73C2×#634C2×D312yes—
74C2×C3×D12C2×D314yes—
75C2×C3×C4×D3C2×D318yes—
76C2×C3×Dic6—38no—
77C2×#626—316no—
78C2×#624—316no—
79#3799—632no—
80C32×C4⋊C4—31no—
81C32×C22⋊C4—32no—
82C32×C42—24no—
83C32×C42C2×D314yes—
84C32×C42C2×D311yescharacteristic
85#3757—34no—
86#3752—32no—
87#3752—38no—
88#3752—68no—
89C4⋊(C3⋊Dic3)—31no—
90C4⋊(C3⋊Dic3)—34no—
91C4⋊(C3⋊Dic3)—64no—
92C4⋊(C3⋊Dic3)C2×D312yes—
93C4⋊(C3⋊Dic3)C2×D312yes—
94#3750—34no—
95C3×(C2×D3)⋊C4C2×D318yes—
96C3×C4⋊Dic3—38no—
97C3×C4⋊Dic3—38no—
98C3×C4⋊Dic3C2×D314yes—
99#3724—316no—
100(C2×D3)⋊Dic3—316no—
101C22×C33C2411yescharacteristic
102C2×C3⋊(C3⋊D3)—22no—
103C2×C3⋊(C3⋊D3)—22no—
104C2×C3⋊(C3⋊D3)C2×D414yes—
105C2×C32×D3—22no—
106C2×C32×D3—22no—
107C2×C32×D3—22no—
108C33×C4—21no—
109C33×C4C2×D412yes—
110C33×C4#4112yescomplemented
111C3⋊(C3⋊Dic3)—22no—
112C3×C3⋊Dic3—24no—
113C32×Dic3—22no—
114C2×#239—92no—
115C2×#235—98no—
116C3×C4⋊Q8—91no—
117C3×C4×D4—34no—
118#946—92no—
119#938—98no—
120#924—98no—
121#895—92no—
122#894—94no—
123D3×C4⋊C4—98no—
124D3×C4⋊C4—92no—
125#883—916no—
126C4×D12—94no—
127C4×D12—91no—
128C4×D12—184no—
129C4×D12C3⋊D311yescharacteristic, complemented
130#869—34no—
131C23×C32—32no—
132C22×C3⋊D3—32no—
133C22×C3⋊D3—38no—
134C22×C3⋊D3—68no—
135C22×C3×D3C22×D318yes—
136C32×D4—62no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—32no—
141C2×C32×C4—64no—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D314yes—
144C2×C32×C4C22×D314yes—
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C32×C4C22×D311yescharacteristic
148#638—68no—
149C2×C3⋊Dic3—32no—
150C2×C3⋊Dic3—38no—
151C2×C3⋊Dic3—68no—
152C2×C3⋊Dic3C22×D314yes—
153C4⋊(C3⋊D3)—34no—
154C4⋊(C3⋊D3)—316no—
155C4⋊(C3⋊D3)—616no—
156C4×C3⋊D3—68no—
157C4×C3⋊D3—64no—
158C4×C3⋊D3—616no—
159C4×C3⋊D3—1216no—
160#634—24no—
161C2×C3×Dic3—316no—
162C2×C3×Dic3C22×D318yes—
163C3×D12—28no—
164C3×C4×D3—216no—
165C3×Dic6—616no—
166#626—632no—
167#624—632no—
168#547—271noSylow
169C2×C33C22×D411yescharacteristic
170C2×C33C2×#4111yescharacteristic
171C2×C332−1+411yescharacteristic
172C3⋊(C3⋊D3)—24no—
173C32×D3—42no—
174C2×C3×Q8—92no—
175C2×C3×D4—34no—
176C22×C3×C4—38no—
177C2×#67—916no—
178#239—188no—
179#235—1832no—
180C2×D12—94no—
181C2×D12—91no—
182C2×D12—94no—
183C2×D12—184no—
184C2×D12C2×C3⋊D311yescharacteristic
185C2×C4×D3—916no—
186C2×C4×D3—98no—
187C2×C4×D3—92no—
188C2×C4×D3—94no—
189C2×C4×D3—188no—
190C2×C4×D3C2×C3⋊D312yes—
191C2×Dic6—38no—
192C3×C4⋊C4—34no—
193C3×C4⋊C4—92no—
194C3×C4⋊C4—92no—
195C3×C22⋊C4—38no—
196C3×C42—24no—
197C3×C42D3214yes—
198C3×C42C2×C3⋊D311yescharacteristic
199#217—916no—
200(C2×D3)⋊C4—94no—
201(C2×D3)⋊C4—98no—
202(C2×D3)⋊C4—92no—
203(C2×D3)⋊C4—188no—
204(C2×D3)⋊C4C2×C3⋊D312yes—
205C4⋊Dic3—38no—
206C4⋊Dic3—38no—
207C4⋊Dic3—94no—
208C4⋊Dic3—91no—
209C4⋊Dic3—184no—
210C4⋊Dic3C2×C3⋊D311yescharacteristic
211Dic3⋊C4—916no—
212Dic3⋊C4—94no—
213C22×C32—24no—
214C22×C32—62no—
215C22×C32—62no—
216C22×C32—62no—
217C22×C32C23×D314yes—
218C22×C32C23×D311yescharacteristic
219C2×C3⋊D3—34no—
220C2×C3⋊D3—316no—
221C2×C3⋊D3—616no—
222C2×C3⋊D3—62no—
223C2×C3⋊D3—62no—
224C2×C3⋊D3—68no—
225C2×C3⋊D3—68no—
226C2×C3⋊D3—128no—
227C2×C3⋊D3—128no—
228C2×C3×D3—28no—
229C2×C3×D3—28no—
230C2×C3×D3—28no—
231C32×C4—21no—
232C32×C4—28no—
233C32×C4—28no—
234C32×C4—24no—
235C32×C4—44no—
236C32×C4—62no—
237C32×C4#23518yescomplemented
238C32×C4D3×D418yes—
239C32×C4D3×D412yes—
240C32×C4#23912yescomplemented
241C3⋊Dic3—24no—
242C3⋊Dic3—62no—
243C3⋊Dic3—68no—
244C3⋊Dic3—128no—
245C3×Dic3—28no—
246C3×Dic3—616no—
247C2×#41—272no—
248C4⋊Q8—91no—
249#123—274no—
250C22⋊Q8—272no—
251#121—272no—
252C4×D4—31no—
253C4×D4—271no—
254C2×C4⋊C4—272no—
255C33#54711yescharacteristic, complemented, Sylow
256C23×C3—38no—
257C22×D3—98no—
258C22×D3—92no—
259C22×D3—188no—
260C22×D3C22×C3⋊D312yes—
261C3×Q8—184no—
262C3×D4—68no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—38no—
267C2×C3×C4—616no—
268C2×C3×C4—94no—
269C2×C3×C4C2×D3214yes—
270C2×C3×C4C2×D3214yes—
271C2×C3×C4C2×D3214yes—
272C2×C3×C4C22×C3⋊D311yescharacteristic
273C2×C3×C4C22×C3⋊D311yescharacteristic
274C2×C3×C4C22×C3⋊D311yescharacteristic
275#67—1832no—
276C2×Dic3—316no—
277C2×Dic3—98no—
278C2×Dic3—92no—
279C2×Dic3—188no—
280C2×Dic3C22×C3⋊D312yes—
281D12—22no—
282D12—916no—
283D12—94no—
284D12—1816no—
285D12—188no—
286C4×D3—24no—
287C4×D3—188no—
288C4×D3—1832no—
289C4×D3—1816no—
290C4×D3—184no—
291C4×D3—3616no—
292Dic6—616no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32—122no—
297C2×C32C2×#23514yes—
298C2×C32C2×D3×D414yes—
299C2×C32C2×D3×D411yescharacteristic
300C2×C32C2×#23911yescharacteristic
301C2×C32#100711yescharacteristic
302C2×C32#101014yes—
303C3⋊D3—64no—
304C3⋊D3—616no—
305C3⋊D3—1216no—
306C3×D3—48no—
307#41—548no—
308C2×Q8—92no—
309C2×D4—31no—
310C2×D4—274no—
311C2×D4—271no—
312C22×C4—32no—
313C22×C4—274no—
314C22×C4—272no—
315C4⋊C4—31no—
316C4⋊C4—92no—
317C4⋊C4—92no—
318C4⋊C4—274no—
319C4⋊C4—271no—
320C22⋊C4—32no—
321C22⋊C4—274no—
322C22⋊C4—272no—
323C42D3×C3⋊D311yescharacteristic
324C22×C3—24no—
325C22×C3—68no—
326C22×C3—68no—
327C22×C3—68no—
328C22×C3C22×D3214yes—
329C22×C3C23×C3⋊D311yescharacteristic
330C2×D3—22no—
331C2×D3—22no—
332C2×D3—22no—
333C2×D3—916no—
334C2×D3—94no—
335C2×D3—1816no—
336C2×D3—188no—
337C2×D3—188no—
338C2×D3—182no—
339C2×D3—182no—
340C2×D3—368no—
341C2×D3—368no—
342C3×C4—24no—
343C3×C4—21no—
344C3×C4—28no—
345C3×C4—28no—
346C3×C4—44no—
347C3×C4—68no—
348C3×C4—184no—
349C3×C4#379918yescomplemented
350C3×C4#380218yes—
351C3×C4#382812yescomplemented
352C3×C4D4×C3⋊D312yes—
353Dic3—22no—
354Dic3—616no—
355Dic3—188no—
356Dic3—182no—
357Dic3—368no—
358C32—24no—
359C32#555114yescomplemented
360C32#572611yescharacteristic, complemented
361C23—32no—
362C23—272no—
363Q8—184no—
364D4—62no—
365D4—274no—
366D4—548no—
367C2×C4—32no—
368C2×C4—64no—
369C2×C4—94no—
370C2×C4—272no—
371C2×C4—548no—
372C2×C4—544no—
373C2×C4C2×D3×C3⋊D311yescharacteristic
374C2×C4C2×D3×C3⋊D311yescharacteristic
375C2×C4C2×D3×C3⋊D311yescharacteristic
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—24no—
379C2×C3—128no—
380C2×C3#6429714yes—
381C2×C3C2×#379914yes—
382C2×C3C2×#380214yes—
383C2×C3C2×#382811yescharacteristic
384C2×C3C2×D4×C3⋊D311yescharacteristic
385C2×C3#6436711yescharacteristic
386D3—42no—
387D3—1816no—
388D3—184no—
389D3—3616no—
390C22—62no—
391C22—62no—
392C22—62no—
393C22—274no—
394C22—542no—
395C22—542no—
396C22C22×D3×C3⋊D311yescharacteristic
397C4—21no—
398C4—62no—
399C4—184no—
400C4—542no—
401C4#8902212yescomplemented
402C4#8902612yes—
403C3—24no—
404C3#1059386914yescomplemented
405C3#1059602211yescharacteristic, complemented
406C2—122no—
407C2—544no—
408C2#1172456211yescharacteristic
409C2C2×#8902211yescharacteristic
410C2C2×#8902611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#547
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×#89022C2semidirect
2. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
3. C3×#64078C2semidirect
4. C32×C4×D12C2semidirect
5. #11723531C2semidirect
6. #11723503C2semidirect
7. #11723425C2semidirect
8. #11723422C2semidirect
38. #64078D3semidirect
39. C3×C4×D12D3semidirect
51. C4⋊(C3⋊(C3⋊D3))D4semidirect
110. C33×C4#41semidirect
129. C4×D12C3⋊D3semidirect
237. C32×C4#235semidirect
240. C32×C4#239semidirect
255. C33#547semidirect
349. C3×C4#3799semidirect
351. C3×C4#3828semidirect
359. C32#5551semidirect
360. C32#5726semidirect
401. C4#89022semidirect
404. C3#10593869semidirect
405. C3#10596022semidirect

Maximal subgroups

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

subgroupindex
C2×#890222
C4×C4⋊(C3⋊(C3⋊D3))2
C3×#640782
C32×C4×D122
#117235312
#117235032
#117234252
#117234222
#105960223
#105938693