Finite Group Databasebeta

← #13175168

Subgroups — #13175168

Order 1728.

1892 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
2C2×#89014C211yesmaximal, characteristic
3#11723910C211yesmaximal, characteristic, complemented
4C3×#64081C211yesmaximal, characteristic, complemented
5C32×#875C211yesmaximal, characteristic, complemented
6#11723523C214yesmaximal, complemented
7#11723491C214yesmaximal, complemented
8#11723455C211yesmaximal, characteristic, complemented
9#11723429C211yesmaximal, characteristic, complemented
10#10596146—31nomaximal
11#10593894—34nomaximal
12C2×#6308C2211yescharacteristic
13C2×#6308C2211yescharacteristic
14C2×C3×C4×C3⋊D3C2211yescharacteristic
15C2×C32×D12C2211yescharacteristic
16C2×C32×Dic6C2211yescharacteristic
17C2×#6293C2212yes—
18C2×#6290C2212yes—
19C2×Dic3×C3⋊D3C2212yes—
20C2×D3×C3⋊Dic3C2212yes—
21C3⋊D3×Dic6—24no—
22#89014—24no—
23C33×C42C2211yescharacteristic
24C4⋊(C3⋊(C3⋊Dic3))C2214yes—
25C3×#3752C2212yes—
26C3×#3750C2212yes—
27C3×C4×C3⋊Dic3C2211yescharacteristic
28C32×(C2×D3)⋊C4C2214yes—
29#88807C2214yes—
30(C2×D3)⋊(C3⋊Dic3)C2214yes—
31C2×Q8×C3⋊D3—31no—
32C2×#3830—31no—
33C2×#3795—34no—
34C2×D3×Dic6—34no—
35C32×#124—31no—
36#64153—31no—
37#64145—31no—
38#64092—34no—
39#64087—34no—
40#64081D311yescharacteristic, complemented
41#64078—31no—
42#64078—34no—
43#64078—64no—
44C3×#875D314yescomplemented
45C3×#872—34no—
46#63893—316no—
47#63891—316no—
48#63866—34no—
49#63860—34no—
50C22×C3×C3⋊D3C2311yescharacteristic
51C22×C32×D3C2312yes—
52C2×C33×C4C2312yes—
53C2×C33×C4C2311yescharacteristic
54C2×C3⋊(C3⋊Dic3)C2312yes—
55C2×C3⋊(C3⋊Dic3)C2312yes—
56#6308—22no—
57#6308—22no—
58C2×C3×C3⋊Dic3C2312yes—
59C2×C3×C3⋊Dic3C2311yescharacteristic
60C3×C4×C3⋊D3D414yes—
61C2×C32×Dic3C2312yes—
62C32×D12—22no—
63C32×Dic6—22no—
64#6293—24no—
65#6290—24no—
66Dic3×C3⋊D3—24no—
67D3×C3⋊Dic3—24no—
68#5675—94no—
69#5526—91no—
70C2×C32×Q8—31no—
71C2×C32×D4—31no—
72C2×#638—32no—
73C22×C3⋊Dic3—32no—
74Q8×C3⋊D3—64no—
75#3830—64no—
76C2×C4×C3⋊D3—32no—
77C2×C4×C3⋊D3C2×D311yescharacteristic
78C2×#634—31no—
79C2×#634—31no—
80C2×#634—34no—
81C2×#634—34no—
82C2×#634—32no—
83C2×#634—64no—
84C2×#634—64no—
85C2×C3×D12C2×D314yes—
86C2×C3×C4×D3—34no—
87C2×C3×Dic6C2×D314yes—
88C2×#627—38no—
89C2×#625—38no—
90C2×D3×Dic3—38no—
91C2×D3×Dic3—38no—
92#3795—616no—
93D3×Dic6—616no—
94C32×C22⋊C4—34no—
95C32×C42—24no—
96C32×C42C2×D314yes—
97C32×C42C2×D311yescharacteristic
98#3757—34no—
99#3752C2×D312yes—
100C4⋊(C3⋊Dic3)—34no—
101C4⋊(C3⋊Dic3)—316no—
102C4⋊(C3⋊Dic3)—616no—
103#3750—34no—
104#3750C2×D312yes—
105C4×C3⋊Dic3C2×D311yescharacteristic
106C3×(C2×D3)⋊C4—38no—
107C3×(C2×D3)⋊C4C2×D3116yes—
108C3×Dic3⋊C4—38no—
109C3×C4×Dic3—34no—
110Dic3⋊Dic3—316no—
111(C2×D3)⋊Dic3—316no—
112C22×C33C2411yescharacteristic
113C2×C3×C3⋊D3—22no—
114C2×C3×C3⋊D3C2×D412yes—
115C2×C32×D3—22no—
116C2×C32×D3—24no—
117C33×C4—22no—
118C33×C4C2×D412yes—
119C3⋊(C3⋊Dic3)—22no—
120C3⋊(C3⋊Dic3)—22no—
121C3×C3⋊Dic3—22no—
122C3×C3⋊Dic3C2×D412yes—
123C32×Dic3—22no—
124C2×D3×Q8—94no—
125C2×#237—94no—
126C2×#235—91no—
127C22×Dic6—91no—
128C3×#124—34no—
129C3×#117—91no—
130#944—94no—
131#936—94no—
132#897—94no—
133#886—94no—
134#883—916no—
135#878—916no—
136#875—91no—
137#875C3⋊D311yescharacteristic, complemented
138#872—34no—
139#869—91no—
140#869—94no—
141#869—91no—
142#869—184no—
143C23×C32—32no—
144C22×C3⋊D3C22×D311yescharacteristic
145C22×C3×D3—34no—
146C22×C3×D3C22×D318yes—
147C32×Q8—62no—
148C32×D4—62no—
149C2×C32×C4—28no—
150C2×C32×C4—24no—
151C2×C32×C4—32no—
152C2×C32×C4C22×D318yes—
153C2×C32×C4C22×D314yes—
154C2×C32×C4C22×D312yes—
155C2×C32×C4C22×D311yescharacteristic
156#638—64no—
157C2×C3⋊Dic3—32no—
158C2×C3⋊Dic3—32no—
159C2×C3⋊Dic3—38no—
160C2×C3⋊Dic3—38no—
161C2×C3⋊Dic3—68no—
162C2×C3⋊Dic3—68no—
163C2×C3⋊Dic3—64no—
164C2×C3⋊Dic3C22×D312yes—
165C2×C3⋊Dic3C22×D311yescharacteristic
166C4×C3⋊D3—64no—
167C4×C3⋊D3D1214yes—
168#634—62no—
169#634—62no—
170#634—64no—
171#634—68no—
172#634—68no—
173#634—128no—
174#634—128no—
175C2×C3×Dic3—34no—
176C2×C3×Dic3—38no—
177C2×C3×Dic3C22×D318yes—
178C3×D12—28no—
179C3×C4×D3—316no—
180C3×Dic6—28no—
181#627—616no—
182#625—616no—
183D3×Dic3—616no—
184D3×Dic3—616no—
185#536—271noSylow
186C2×C33C22×D411yescharacteristic
187C2×C332−1+412yes—
188C3×C3⋊D3—22no—
189C32×D3—42no—
190C2×C3×Q8—34no—
191C2×C3×D4—34no—
192C22×C3×C4—91no—
193C2×#67—98no—
194C2×#67—92no—
195C22×Dic3—98no—
196C22×Dic3—92no—
197D3×Q8—1816no—
198#237—1816no—
199#235—184no—
200C2×D12C2×C3⋊D311yescharacteristic
201C2×C4×D3—34no—
202C2×C4×D3—98no—
203C2×C4×D3—92no—
204C2×Dic6—98no—
205C2×Dic6—94no—
206C2×Dic6—94no—
207C2×Dic6—91no—
208C2×Dic6—91no—
209C2×Dic6—92no—
210C2×Dic6—184no—
211C2×Dic6—184no—
212C2×Dic6—184no—
213C2×Dic6C2×C3⋊D311yescharacteristic
214C3×C4⋊C4—92no—
215C3×C22⋊C4—316no—
216C3×C22⋊C4—92no—
217C3×C42—24no—
218C3×C42—91no—
219C3×C42D3214yes—
220C3×C42C2×C3⋊D311yescharacteristic
221#217—916no—
222(C2×D3)⋊C4—38no—
223(C2×D3)⋊C4—94no—
224(C2×D3)⋊C4C2×C3⋊D314yes—
225C4⋊Dic3—916no—
226C4⋊Dic3—94no—
227C4⋊Dic3—94no—
228C4⋊Dic3—1816no—
229Dic3⋊C4—38no—
230Dic3⋊C4—916no—
231C4×Dic3—34no—
232C22×C32—24no—
233C22×C32—62no—
234C22×C32—64no—
235C22×C32C23×D314yes—
236C22×C32C23×D311yescharacteristic
237C2×C3⋊D3—22no—
238C2×C3⋊D3C2×D1212yes—
239C2×C3×D3—28no—
240C2×C3×D3—216no—
241C2×C3×D3—38no—
242C2×C3×D3—68no—
243C32×C4—216no—
244C32×C4—28no—
245C32×C4—22no—
246C32×C4—28no—
247C32×C4—62no—
248C32×C4C2×D1212yes—
249C32×C4D3×D418yes—
250C3⋊Dic3—22no—
251C3⋊Dic3—62no—
252C3⋊Dic3—62no—
253C3⋊Dic3—68no—
254C3⋊Dic3—68no—
255C3⋊Dic3—128no—
256C3⋊Dic3—128no—
257C3⋊Dic3C2×D1212yes—
258C3×Dic3—28no—
259C3×Dic3—38no—
260C3×Dic3—68no—
261C2×#41—271no—
262C22×Q8—271no—
263C4⋊Q8—271no—
264C4⋊Q8—271no—
265#124—31no—
266#124—271no—
267#123—274no—
268C22⋊Q8—274no—
269#117—91no—
270C33#53611yescharacteristic, complemented, Sylow
271C23×C3—38no—
272C23×C3—91no—
273C22×D3—34no—
274C22×D3C22×C3⋊D312yes—
275C3×Q8—68no—
276C3×D4—68no—
277C2×C3×C4—28no—
278C2×C3×C4—24no—
279C2×C3×C4—38no—
280C2×C3×C4—91no—
281C2×C3×C4—94no—
282C2×C3×C4—92no—
283C2×C3×C4C2×D3218yes—
284C2×C3×C4C2×D3214yes—
285C2×C3×C4C22×C3⋊D312yes—
286C2×C3×C4C22×C3⋊D311yescharacteristic
287#67—184no—
288#67—1816no—
289C2×Dic3—34no—
290C2×Dic3—38no—
291C2×Dic3—98no—
292C2×Dic3—98no—
293C2×Dic3—92no—
294C2×Dic3—92no—
295C2×Dic3—188no—
296C2×Dic3—188no—
297C2×Dic3—184no—
298C2×Dic3—1816no—
299C2×Dic3C22×C3⋊D312yes—
300D12—22no—
301C4×D3—316no—
302C4×D3—1816no—
303C4×D3—184no—
304Dic6—22no—
305Dic6—184no—
306Dic6—1816no—
307Dic6—188no—
308Dic6—188no—
309Dic6—182no—
310Dic6—182no—
311Dic6—368no—
312Dic6—368no—
313C2×C32—28no—
314C2×C32—24no—
315C2×C32—122no—
316C2×C32C22×D1211yescharacteristic
317C2×C32C2×D3×D414yes—
318C2×C32#101018yes—
319C2×C32#101012yes—
320C3⋊D3—22no—
321C3×D3—48no—
322C3×D3—68no—
323#41—544no—
324C2×Q8—31no—
325C2×Q8—272no—
326C2×Q8—271no—
327C2×Q8—271no—
328C2×Q8—544no—
329C2×D4—31no—
330C2×D4—272no—
331C22×C4—91no—
332C22×C4—272no—
333C22×C4—272no—
334C4⋊C4—92no—
335C4⋊C4—274no—
336C4⋊C4—274no—
337C22⋊C4—34no—
338C22⋊C4—92no—
339C22⋊C4—274no—
340C42—91no—
341C42D3×C3⋊D311yescharacteristic
342C22×C3—24no—
343C22×C3—616no—
344C22×C3—68no—
345C22×C3—92no—
346C22×C3—182no—
347C22×C3C22×D3214yes—
348C22×C3C23×C3⋊D311yescharacteristic
349C2×D3—22no—
350C2×D3—24no—
351C2×D3—38no—
352C2×D3—68no—
353C3×C4—216no—
354C3×C4—28no—
355C3×C4—22no—
356C3×C4—28no—
357C3×C4—68no—
358C3×C4—92no—
359C3×C4—182no—
360C3×C4D3×D1218yes—
361C3×C4D4×C3⋊D312yes—
362Dic3—22no—
363Dic3—38no—
364Dic3—68no—
365Dic3—188no—
366Dic3—188no—
367Dic3—182no—
368Dic3—182no—
369Dic3—368no—
370Dic3—368no—
371C32—24no—
372C32#552611yescharacteristic, complemented
373C32#567514yescomplemented
374C23—32no—
375C23—91no—
376Q8—62no—
377Q8—544no—
378Q8—542no—
379Q8—542no—
380D4—62no—
381D4—544no—
382C2×C4—32no—
383C2×C4—91no—
384C2×C4—94no—
385C2×C4—92no—
386C2×C4—272no—
387C2×C4—272no—
388C2×C4—544no—
389C2×C4—544no—
390C2×C4C2×D3×C3⋊D312yes—
391C2×C4C2×D3×C3⋊D311yescharacteristic
392C2×C3—28no—
393C2×C3—24no—
394C2×C3—128no—
395C2×C3—182no—
396C2×C3#6429818yes—
397C2×C3C2×D3×D1214yes—
398C2×C3C2×D4×C3⋊D311yescharacteristic
399C2×C3#6436712yes—
400D3—42no—
401D3—68no—
402C22—64no—
403C22—62no—
404C22—92no—
405C22—182no—
406C22C22×D3×C3⋊D311yescharacteristic
407C4—22no—
408C4—62no—
409C4—92no—
410C4—182no—
411C4—542no—
412C4—542no—
413C4C3⋊D3×D1212yes—
414C3—24no—
415C3#1059389414yescomplemented
416C3#1059614611yescharacteristic, complemented
417C2—122no—
418C2—182no—
419C2#1172456312yes—
420C2C2×C3⋊D3×D1211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radicalC2×#89014C2
hypercenterC42D3×C3⋊D3
hypocenterC33#536
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×Dic6C2semidirect
3. #11723910C2semidirect
4. C3×#64081C2semidirect
5. C32×#875C2semidirect
6. #11723523C2semidirect
7. #11723491C2semidirect
8. #11723455C2semidirect
9. #11723429C2semidirect
40. #64081D3semidirect
44. C3×#875D3semidirect
137. #875C3⋊D3semidirect
270. C33#536semidirect
372. C32#5526semidirect
373. C32#5675semidirect
415. C3#10593894semidirect
416. C3#10596146semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×#890142
#117239102
C3×#640812
C32×#8752
#117235232
#117234912
#117234552
#117234292
#105961463
#105938943