Finite Group Databasebeta

← #13175170

Subgroups — #13175170

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×#89017C212yesmaximal, complemented
2C4×C4⋊(C3⋊(C3⋊D3))C211yesmaximal, characteristic, complemented
3C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
4C32×#869C211yesmaximal, characteristic, complemented
5#11723526C212yesmaximal, complemented
6#11723493C214yesmaximal, complemented
7C3⋊D3×C4⋊Dic3C212yesmaximal, complemented
8#11723456C212yesmaximal, complemented
9#10596197—31nomaximal
10#10593896—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—
17C2×Dic3×C3⋊D3C2214yes—
18#89017—28no—
19C33×C42C2211yescharacteristic
20#88859C2212yes—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C3×#3752C2212yes—
23C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
24C32×C4⋊Dic3C2212yes—
25C32×C4⋊Dic3C2212yes—
26#88807C2214yes—
27#88804C2214yes—
28C2×#3832—32no—
29C2×#3796—38no—
30C32×C4⋊Q8—31no—
31#64155—32no—
32#64104—32no—
33#64103—34no—
34C4⋊C4×C3⋊D3—32no—
35C4×C4⋊(C3⋊D3)—31no—
36C4×C4⋊(C3⋊D3)—34no—
37C4×C4⋊(C3⋊D3)—64no—
38C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
39C3×C4×D12—34no—
40C3×#869D314yescomplemented
41#63909—38no—
42#63890—316no—
43D3×C4⋊Dic3—38no—
44#63871—38no—
45C22×C3⋊(C3⋊D3)C2312yes—
46C22×C3×C3⋊D3C2312yes—
47C2×C33×C4C2311yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)C2312yes—
51C4⋊(C3⋊(C3⋊D3))—22no—
52C4×C3⋊(C3⋊D3)—24no—
53C2×C3×C3⋊Dic3C2312yes—
54C3×C4⋊(C3⋊D3)D414yescomplemented
55C3×C4×C3⋊D3—24no—
56C2×C32×Dic3C2314yes—
57C32×Dic6—24no—
58#6292—28no—
59Dic3×C3⋊D3—28no—
60#5726—94no—
61#5555—91no—
62C2×C32×Q8—32no—
63#3832—68no—
64C2×C4⋊(C3⋊D3)—34no—
65C2×C4⋊(C3⋊D3)—31no—
66C2×C4⋊(C3⋊D3)—34no—
67C2×C4⋊(C3⋊D3)—64no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3—34no—
70C2×C4×C3⋊D3—32no—
71C2×C4×C3⋊D3—38no—
72C2×C4×C3⋊D3—68no—
73C2×C4×C3⋊D3C2×D312yes—
74C2×C3×D12—34no—
75C2×C3×C4×D3—38no—
76C2×C3×Dic6C2×D318yes—
77C2×#626—316no—
78C2×D3×Dic3—316no—
79#3796—632no—
80C32×C4⋊C4—32no—
81C32×C4⋊C4—32no—
82C32×C42—24no—
83C32×C42C2×D314yes—
84C32×C42C2×D311yescharacteristic
85#3752—32no—
86#3752—38no—
87#3752—34no—
88#3752—68no—
89#3752C2×D312yes—
90C4⋊(C3⋊Dic3)—31no—
91C4⋊(C3⋊Dic3)—34no—
92C4⋊(C3⋊Dic3)—64no—
93C4⋊(C3⋊Dic3)C2×D311yescharacteristic
94#3750—34no—
95C3×(C2×D3)⋊C4—38no—
96C3×C4⋊Dic3—34no—
97C3×C4⋊Dic3C2×D318yes—
98C3×C4⋊Dic3C2×D318yes—
99Dic3⋊Dic3—316no—
100#3722—316no—
101C22×C33C2411yescharacteristic
102C2×C3⋊(C3⋊D3)—22no—
103C2×C3⋊(C3⋊D3)—22no—
104C2×C3⋊(C3⋊D3)—22no—
105C2×C3×C3⋊D3—22no—
106C2×C3×C3⋊D3—22no—
107C2×C3×C3⋊D3C2×D414yes—
108C33×C4—21no—
109C33×C4C2×D412yes—
110C33×C4#4112yescomplemented
111C3⋊(C3⋊Dic3)—22no—
112C3×C3⋊Dic3—22no—
113C32×Dic3—24no—
114C2×#239—98no—
115C2×#237—92no—
116C3×C4⋊Q8—34no—
117C3×C4×D4—91no—
118#946—98no—
119#930—92no—
120C2×C4⋊Dic3—92no—
121#897—92no—
122#895—98no—
123#894—916no—
124D3×C4⋊C4—98no—
125#886—94no—
126C4×D12—34no—
127C4×D12—94no—
128C4×D12—91no—
129C4×D12—184no—
130#869C3⋊D311yescharacteristic, complemented
131C22×C3⋊D3—32no—
132C22×C3⋊D3—38no—
133C22×C3⋊D3—68no—
134C22×C3⋊D3C22×D312yes—
135C22×C3×D3—38no—
136C32×Q8—64no—
137C2×C32×C4—24no—
138C2×C32×C4—24no—
139C2×C32×C4—24no—
140C2×C32×C4—34no—
141C2×C32×C4C22×D314yes—
142C2×C32×C4C22×D314yes—
143C2×C32×C4C22×D314yes—
144C2×C32×C4C22×D311yescharacteristic
145C2×C32×C4C22×D311yescharacteristic
146C2×C32×C4C22×D311yescharacteristic
147C2×C3⋊Dic3—32no—
148C2×C3⋊Dic3—38no—
149C2×C3⋊Dic3—68no—
150C2×C3⋊Dic3C22×D312yes—
151C4⋊(C3⋊D3)—616no—
152C4⋊(C3⋊D3)—68no—
153C4⋊(C3⋊D3)—62no—
154C4⋊(C3⋊D3)—68no—
155C4⋊(C3⋊D3)D1214yescomplemented
156C4×C3⋊D3—24no—
157C4×C3⋊D3—68no—
158C4×C3⋊D3—64no—
159C4×C3⋊D3—616no—
160C4×C3⋊D3—1216no—
161C2×C3×Dic3—38no—
162C2×C3×Dic3C22×D3116yes—
163C3×D12—316no—
164C3×C4×D3—616no—
165C3×Dic6—216no—
166#626—632no—
167D3×Dic3—632no—
168#547—271noSylow
169C2×C33C22×D411yescharacteristic
170C2×C33C2×#4111yescharacteristic
171C2×C332−1+411yescharacteristic
172C3⋊(C3⋊D3)—42no—
173C3×C3⋊D3—24no—
174C2×C3×Q8—38no—
175C2×C3×D4—91no—
176C22×C3×C4—92no—
177C2×#67—94no—
178C22×Dic3—94no—
179#239—1832no—
180#237—188no—
181C2×D12—34no—
182C2×D12—916no—
183C2×D12—94no—
184C2×D12—91no—
185C2×D12—184no—
186C2×C4×D3—38no—
187C2×C4×D3—916no—
188C2×C4×D3—98no—
189C2×C4×D3—92no—
190C2×C4×D3—188no—
191C2×Dic6C2×C3⋊D312yes—
192C3×C4⋊C4—38no—
193C3×C4⋊C4—38no—
194C3×C4⋊C4—91no—
195C3×C22⋊C4—92no—
196C3×C42—24no—
197C3×C42D3214yes—
198C3×C42C2×C3⋊D311yescharacteristic
199(C2×D3)⋊C4—38no—
200(C2×D3)⋊C4—916no—
201(C2×D3)⋊C4—98no—
202(C2×D3)⋊C4—92no—
203(C2×D3)⋊C4—94no—
204(C2×D3)⋊C4—188no—
205C4⋊Dic3—34no—
206C4⋊Dic3—94no—
207C4⋊Dic3—91no—
208C4⋊Dic3—94no—
209C4⋊Dic3—184no—
210C4⋊Dic3C2×C3⋊D312yes—
211C4⋊Dic3C2×C3⋊D312yes—
212Dic3⋊C4—916no—
213C22×C32—24no—
214C22×C32C23×D314yes—
215C22×C32C23×D311yescharacteristic
216C2×C3⋊D3—22no—
217C2×C3⋊D3—22no—
218C2×C3⋊D3—616no—
219C2×C3⋊D3—62no—
220C2×C3⋊D3—62no—
221C2×C3⋊D3—68no—
222C2×C3⋊D3—62no—
223C2×C3⋊D3—68no—
224C2×C3⋊D3—68no—
225C2×C3⋊D3—128no—
226C2×C3⋊D3—128no—
227C2×C3⋊D3C2×D1214yes—
228C2×C3×D3—316no—
229C2×C3×D3—68no—
230C2×C3×D3—68no—
231C32×C4—21no—
232C32×C4—28no—
233C32×C4—28no—
234C32×C4—24no—
235C32×C4—44no—
236C32×C4—64no—
237C32×C4C2×D1212yes—
238C32×C4D3×D418yes—
239C32×C4#23712yescomplemented
240C32×C4#23918yescomplemented
241C3⋊Dic3—22no—
242C3⋊Dic3—62no—
243C3⋊Dic3—68no—
244C3⋊Dic3—128no—
245C3×Dic3—216no—
246C3×Dic3—68no—
247C2×#41—272no—
248C4⋊Q8—31no—
249#123—274no—
250C22⋊Q8—272no—
251#121—272no—
252C4×D4—91no—
253C4×D4—271no—
254C2×C4⋊C4—272no—
255C33#54711yescharacteristic, complemented, Sylow
256C23×C3—92no—
257C22×D3—38no—
258C22×D3—98no—
259C22×D3—92no—
260C22×D3—188no—
261C3×Q8—616no—
262C3×D4—94no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—316no—
267C2×C3×C4—92no—
268C2×C3×C4—184no—
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—188no—
276C2×Dic3—38no—
277C2×Dic3—98no—
278C2×Dic3—92no—
279C2×Dic3—188no—
280C2×Dic3—188no—
281C2×Dic3C22×C3⋊D314yes—
282D12—316no—
283D12—1816no—
284D12—1832no—
285D12—188no—
286D12—182no—
287C4×D3—616no—
288C4×D3—1832no—
289C4×D3—1816no—
290C4×D3—184no—
291C4×D3—3616no—
292Dic6—24no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—24no—
296C2×C32C22×D1211yescharacteristic
297C2×C32C2×D3×D414yes—
298C2×C32C2×#23711yescharacteristic
299C2×C32C2×#23914yes—
300C2×C32#100714yes—
301C2×C32#101011yescharacteristic
302C3⋊D3—24no—
303C3⋊D3—1216no—
304C3⋊D3—128no—
305C3⋊D3—122no—
306C3×D3—616no—
307#41—548no—
308C2×Q8—32no—
309C2×D4—91no—
310C2×D4—274no—
311C2×D4—271no—
312C22×C4—92no—
313C22×C4—274no—
314C22×C4—272no—
315C4⋊C4—32no—
316C4⋊C4—32no—
317C4⋊C4—91no—
318C4⋊C4—274no—
319C4⋊C4—271no—
320C22⋊C4—92no—
321C22⋊C4—274no—
322C22⋊C4—272no—
323C42D3×C3⋊D311yescharacteristic
324C22×C3—24no—
325C22×C3—94no—
326C22×C3—182no—
327C22×C3—182no—
328C22×C3C22×D3214yes—
329C22×C3C23×C3⋊D311yescharacteristic
330C2×D3—316no—
331C2×D3—68no—
332C2×D3—68no—
333C2×D3—1816no—
334C2×D3—188no—
335C2×D3—188no—
336C2×D3—188no—
337C2×D3—182no—
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—616no—
348C3×C4—182no—
349C3×C4#379618yescomplemented
350C3×C4D3×D1218yes—
351C3×C4D4×C3⋊D312yes—
352C3×C4#383212yescomplemented
353Dic3—24no—
354Dic3—68no—
355Dic3—188no—
356Dic3—182no—
357Dic3—368no—
358C32—24no—
359C32#555511yescharacteristic, complemented
360C32#572614yescomplemented
361C23—92no—
362C23—272no—
363Q8—64no—
364D4—94no—
365D4—548no—
366D4—542no—
367C2×C4—34no—
368C2×C4—92no—
369C2×C4—184no—
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—184no—
380C2×C3C2×#379614yes—
381C2×C3#6429714yes—
382C2×C3C2×D3×D1214yes—
383C2×C3C2×D4×C3⋊D311yescharacteristic
384C2×C3C2×#383211yescharacteristic
385C2×C3#6436411yescharacteristic
386D3—616no—
387D3—3616no—
388D3—362no—
389D3—368no—
390C22—94no—
391C22—182no—
392C22—182no—
393C22—542no—
394C22—542no—
395C22—542no—
396C22C22×D3×C3⋊D311yescharacteristic
397C4—21no—
398C4—64no—
399C4—182no—
400C4—542no—
401C4#8901712yescomplemented
402C4C3⋊D3×D1212yes—
403C3—24no—
404C3#1059389614yescomplemented
405C3#1059619711yescharacteristic, complemented
406C2—184no—
407C2—1082no—
408C2C2×#8901711yescharacteristic
409C2#1172456611yescharacteristic
410C2C2×C3⋊D3×D1211yescharacteristic
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×#89017C2semidirect
2. C4×C4⋊(C3⋊(C3⋊D3))C2semidirect
3. C3×C4×C4⋊(C3⋊D3)C2semidirect
4. C32×#869C2semidirect
5. #11723526C2semidirect
6. #11723493C2semidirect
7. C3⋊D3×C4⋊Dic3C2semidirect
8. #11723456C2semidirect
38. C4×C4⋊(C3⋊D3)D3semidirect
40. C3×#869D3semidirect
54. C3×C4⋊(C3⋊D3)D4semidirect
110. C33×C4#41semidirect
130. #869C3⋊D3semidirect
155. C4⋊(C3⋊D3)D12semidirect
239. C32×C4#237semidirect
240. C32×C4#239semidirect
255. C33#547semidirect
349. C3×C4#3796semidirect
352. C3×C4#3832semidirect
359. C32#5555semidirect
360. C32#5726semidirect
401. C4#89017semidirect
404. C3#10593896semidirect
405. C3#10596197semidirect

Maximal subgroups

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

subgroupindex
C2×#890172
C4×C4⋊(C3⋊(C3⋊D3))2
C3×C4×C4⋊(C3⋊D3)2
C32×#8692
#117235262
#117234932
C3⋊D3×C4⋊Dic32
#117234562
#105961973
#105938963