Finite Group Databasebeta

← #13175368

Subgroups — #13175368

Order 1728.

1568 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723910C211yesmaximal, characteristic, complemented
2C3×#64081C211yesmaximal, characteristic, complemented
3C32×#875C211yesmaximal, characteristic, complemented
4#11723528C212yesmaximal, complemented
5#11723522C211yesmaximal, characteristic
6#11723496C212yesmaximal, complemented
7D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#11723480C212yesmaximal, complemented
9#11723435C211yesmaximal, characteristic, complemented
10#11723430C212yesmaximal, complemented
11Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
12#10596153—31nomaximal
13#10593961—34nomaximal
14C2×#6308C2212yes—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C32×D12C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C2×#6290C2212yes—
19C2×D3×C3⋊Dic3C2212yes—
20C33×C42C2211yescharacteristic
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C4⋊(C3⋊(C3⋊Dic3))C2212yes—
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24C3×#3752C2211yescharacteristic
25C3×#3752C2211yescharacteristic
26C3×#3750C2211yescharacteristic
27C3×#3750C2211yescharacteristic
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C32×(C2×D3)⋊C4C2212yes—
30C32×(C2×D3)⋊C4C2212yes—
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88803C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#124—31no—
38#64155—31no—
39Q8×C3⋊Dic3—31no—
40#64147—31no—
41D4×C3⋊Dic3—31no—
42#64095—32no—
43#64094—32no—
44#64088—32no—
45#64087—32no—
46#64081D311yescharacteristic, complemented
47#64078—31no—
48#64078—34no—
49#64078—64no—
50C3×#875D314yescomplemented
51C3×#872—34no—
52#63913—38no—
53#63908—34no—
54#63896—38no—
55Dic3×D12—34no—
56#63891—38no—
57#63861—38no—
58#63856—34no—
59Dic3×Dic6—34no—
60C22×C3×C3⋊D3C2311yescharacteristic
61C22×C32×D3C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3⋊(C3⋊Dic3)C2312yes—
67#6308—42no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C3×C4×C3⋊D3—22no—
72C2×C32×Dic3C2312yes—
73C32×D12—22no—
74C32×Dic6—22no—
75#6290—42no—
76D3×C3⋊Dic3—42no—
77#5682—94no—
78#5534—91no—
79C2×C32×Q8—31no—
80C2×C32×D4—31no—
81C2×#638—32no—
82C22×C3⋊Dic3—32no—
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×#634—32no—
85C2×#634—38no—
86C2×#634—68no—
87C2×C3×D12C2×D314yes—
88C2×C3×C4×D3—34no—
89C2×C3×Dic6C2×D314yes—
90C2×#625—38no—
91C2×D3×Dic3—38no—
92C32×C22⋊C4—32no—
93C32×C22⋊C4—32no—
94C32×C42—24no—
95C32×C42C2×D314yes—
96C32×C42C2×D311yescharacteristic
97#3757—32no—
98#3757—32no—
99#3752—32no—
100#3752C2×D311yescharacteristic
101#3752C2×D311yescharacteristic
102C4⋊(C3⋊Dic3)—32no—
103C4⋊(C3⋊Dic3)—31no—
104C4⋊(C3⋊Dic3)—31no—
105C4⋊(C3⋊Dic3)—34no—
106C4⋊(C3⋊Dic3)—34no—
107C4⋊(C3⋊Dic3)—38no—
108C4⋊(C3⋊Dic3)—32no—
109C4⋊(C3⋊Dic3)—68no—
110C4⋊(C3⋊Dic3)—64no—
111C4⋊(C3⋊Dic3)—64no—
112#3750—32no—
113#3750C2×D311yescharacteristic
114#3750C2×D311yescharacteristic
115C4×C3⋊Dic3—32no—
116C4×C3⋊Dic3C2×D311yescharacteristic
117C3×(C2×D3)⋊C4—34no—
118C3×(C2×D3)⋊C4—34no—
119C3×(C2×D3)⋊C4C2×D318yes—
120C3×(C2×D3)⋊C4C2×D318yes—
121C3×Dic3⋊C4—34no—
122C3×Dic3⋊C4—34no—
123C3×C4×Dic3—34no—
124#3724—38no—
125Dic3⋊Dic3—38no—
126(C2×D3)⋊Dic3—38no—
127(C2×D3)⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129Dic32—38no—
130C22×C33C2411yescharacteristic
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3—21no—
134C2×C32×D3—22no—
135C2×C32×D3—22no—
136C2×C32×D3—22no—
137C33×C4—21no—
138C33×C4—21no—
139C33×C4#4112yes—
140C3⋊(C3⋊Dic3)—22no—
141C3⋊(C3⋊Dic3)—22no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3—21no—
145C32×Dic3—22no—
146C3×#124—34no—
147C3×#117—91no—
148#946—94no—
149Q8×Dic3—94no—
150#938—94no—
151D4×Dic3—94no—
152#930—91no—
153#924—91no—
154#898—92no—
155#896—92no—
156#886—98no—
157#885—98no—
158#885—92no—
159#883—92no—
160#879—98no—
161#878—98no—
162#875C3⋊D311yescharacteristic, complemented
163C4×D12—91no—
164#872—34no—
165#869—94no—
166#869—91no—
167#869—184no—
168C4×Dic6—91no—
169C23×C32—32no—
170C22×C3⋊D3C22×D311yescharacteristic
171C22×C3×D3—34no—
172C22×C3×D3C22×D318yes—
173C32×Q8—62no—
174C32×D4—62no—
175C2×C32×C4—24no—
176C2×C32×C4—24no—
177C2×C32×C4—24no—
178C2×C32×C4—32no—
179C2×C32×C4C22×D314yes—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185#638—122no—
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—38no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—68no—
191C2×C3⋊Dic3—68no—
192C2×C3⋊Dic3—122no—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D311yescharacteristic
196C4×C3⋊D3—22no—
197#634—122no—
198#634—128no—
199#634—1216no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3—34no—
203C2×C3×Dic3C22×D318yes—
204C3×D12—28no—
205C3×C4×D3—68no—
206C3×Dic6—28no—
207#625—128no—
208D3×Dic3—128no—
209#542—271noSylow
210C2×C33C2×#4111yescharacteristic
211C2×C33H(5,𝔽2)11yescharacteristic
212C2×C332−1+411yescharacteristic
213C3×C3⋊D3—41no—
214C32×D3—42no—
215C2×C3×Q8—34no—
216C2×C3×D4—34no—
217C22×C3×C4—91no—
218C2×#67—98no—
219C2×#67—92no—
220C22×Dic3—98no—
221C2×D12C2×C3⋊D311yescharacteristic
222C2×C4×D3—34no—
223C2×C4×D3—92no—
224C2×Dic6—98no—
225C2×Dic6—92no—
226C2×Dic6—188no—
227C2×Dic6C2×C3⋊D311yescharacteristic
228C3×C4⋊C4—91no—
229C3×C4⋊C4—91no—
230C3×C22⋊C4—38no—
231C3×C22⋊C4—38no—
232C3×C22⋊C4—91no—
233C3×C22⋊C4—91no—
234C3×C42—24no—
235C3×C42—91no—
236C3×C42D3214yes—
237C3×C42C2×C3⋊D311yescharacteristic
238#217—98no—
239#217—98no—
240#217—92no—
241(C2×D3)⋊C4—34no—
242(C2×D3)⋊C4—34no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4C2×C3⋊D312yes—
247(C2×D3)⋊C4C2×C3⋊D312yes—
248C4⋊Dic3—98no—
249C4⋊Dic3—98no—
250C4⋊Dic3—94no—
251C4⋊Dic3—94no—
252C4⋊Dic3—91no—
253C4⋊Dic3—91no—
254C4⋊Dic3—92no—
255C4⋊Dic3—184no—
256C4⋊Dic3—184no—
257C4⋊Dic3—188no—
258Dic3⋊C4—34no—
259Dic3⋊C4—34no—
260Dic3⋊C4—98no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263C4×Dic3—34no—
264C4×Dic3—98no—
265C4×Dic3—92no—
266C22×C32—24no—
267C22×C32—62no—
268C22×C32—62no—
269C22×C32—62no—
270C22×C32C23×D314yes—
271C22×C32C23×D311yescharacteristic
272C2×C3⋊D3—21no—
273C2×C3⋊D3—21no—
274C2×C3⋊D3—21no—
275C2×C3×D3—28no—
276C2×C3×D3—28no—
277C2×C3×D3—28no—
278C2×C3×D3—64no—
279C2×C3×D3—64no—
280C2×C3×D3—64no—
281C32×C4—28no—
282C32×C4—28no—
283C32×C4—24no—
284C32×C4—21no—
285C32×C4—24no—
286C32×C4—21no—
287C32×C4—28no—
288C32×C4—62no—
289C32×C4#23512yes—
290C32×C4#23718yes—
291C3⋊Dic3—21no—
292C3⋊Dic3—21no—
293C3⋊Dic3—21no—
294C3⋊Dic3—62no—
295C3⋊Dic3—62no—
296C3⋊Dic3—68no—
297C3⋊Dic3—68no—
298C3⋊Dic3—128no—
299C3⋊Dic3—128no—
300C3×Dic3—28no—
301C3×Dic3—64no—
302C3×Dic3—64no—
303C3×Dic3—64no—
304C4⋊Q8—271no—
305#126—272no—
306#124—31no—
307#124—272no—
308#123—272no—
309C22⋊Q8—272no—
310C22⋊Q8—271no—
311#121—271no—
312C4×Q8—271no—
313C4×D4—271no—
314#117—91no—
315C33#54211yescharacteristic, complemented, Sylow
316C23×C3—38no—
317C23×C3—91no—
318C22×D3—34no—
319C22×D3C22×C3⋊D312yes—
320C3×Q8—68no—
321C3×D4—68no—
322C2×C3×C4—24no—
323C2×C3×C4—24no—
324C2×C3×C4—24no—
325C2×C3×C4—38no—
326C2×C3×C4—91no—
327C2×C3×C4—91no—
328C2×C3×C4—91no—
329C2×C3×C4—182no—
330C2×C3×C4C2×D3214yes—
331C2×C3×C4C2×D3214yes—
332C2×C3×C4C2×D3214yes—
333C2×C3×C4C22×C3⋊D311yescharacteristic
334C2×C3×C4C22×C3⋊D311yescharacteristic
335C2×C3×C4C22×C3⋊D311yescharacteristic
336#67—362no—
337#67—368no—
338C2×Dic3—34no—
339C2×Dic3—34no—
340C2×Dic3—34no—
341C2×Dic3—98no—
342C2×Dic3—98no—
343C2×Dic3—92no—
344C2×Dic3—92no—
345C2×Dic3—188no—
346C2×Dic3—188no—
347C2×Dic3—368no—
348C2×Dic3C22×C3⋊D312yes—
349D12—22no—
350C4×D3—68no—
351C4×D3—362no—
352Dic6—22no—
353Dic6—368no—
354Dic6—362no—
355Dic6—3616no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—24no—
359C2×C32—122no—
360C2×C32C2×#23511yescharacteristic
361C2×C32C2×#23714yes—
362C2×C32#100914yes—
363C2×C32#100911yescharacteristic
364C2×C32#101014yes—
365C2×C32#101011yescharacteristic
366C3⋊D3—41no—
367C3×D3—48no—
368C3×D3—124no—
369C2×Q8—31no—
370C2×Q8—272no—
371C2×D4—31no—
372C2×D4—272no—
373C22×C4—91no—
374C22×C4—272no—
375C4⋊C4—91no—
376C4⋊C4—91no—
377C4⋊C4—272no—
378C4⋊C4—272no—
379C4⋊C4—271no—
380C4⋊C4—271no—
381C4⋊C4—272no—
382C22⋊C4—32no—
383C22⋊C4—32no—
384C22⋊C4—91no—
385C22⋊C4—91no—
386C22⋊C4—272no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C42—91no—
390C42—272no—
391C42D3×C3⋊D311yescharacteristic
392C22×C3—24no—
393C22×C3—68no—
394C22×C3—68no—
395C22×C3—68no—
396C22×C3—181no—
397C22×C3—181no—
398C22×C3—181no—
399C22×C3C22×D3214yes—
400C22×C3C23×C3⋊D311yescharacteristic
401C2×D3—22no—
402C2×D3—22no—
403C2×D3—22no—
404C2×D3—64no—
405C2×D3—64no—
406C2×D3—64no—
407C3×C4—28no—
408C3×C4—28no—
409C3×C4—24no—
410C3×C4—24no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—28no—
414C3×C4—68no—
415C3×C4—181no—
416C3×C4—181no—
417C3×C4—181no—
418C3×C4#379518yes—
419C3×C4#383012yes—
420Dic3—22no—
421Dic3—64no—
422Dic3—64no—
423Dic3—64no—
424Dic3—188no—
425Dic3—188no—
426Dic3—182no—
427Dic3—182no—
428Dic3—368no—
429Dic3—368no—
430C32—24no—
431C32#553411yescharacteristic, complemented
432C32#568214yescomplemented
433C23—32no—
434C23—91no—
435Q8—62no—
436Q8—1082no—
437D4—62no—
438D4—1082no—
439C2×C4—32no—
440C2×C4—91no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—182no—
444C2×C4—272no—
445C2×C4—272no—
446C2×C4—1082no—
447C2×C4C2×D3×C3⋊D311yescharacteristic
448C2×C4C2×D3×C3⋊D311yescharacteristic
449C2×C4C2×D3×C3⋊D311yescharacteristic
450C2×C3—24no—
451C2×C3—24no—
452C2×C3—24no—
453C2×C3—128no—
454C2×C3—361no—
455C2×C3C2×#379514yes—
456C2×C3#6429814yes—
457C2×C3#6430814yes—
458C2×C3C2×#383011yescharacteristic
459C2×C3#6436611yescharacteristic
460C2×C3#6436711yescharacteristic
461D3—42no—
462D3—124no—
463C22—62no—
464C22—62no—
465C22—62no—
466C22—181no—
467C22—181no—
468C22—181no—
469C22C22×D3×C3⋊D311yescharacteristic
470C4—21no—
471C4—21no—
472C4—62no—
473C4—181no—
474C4—181no—
475C4—181no—
476C4—542no—
477C4—542no—
478C4#8901412yes—
479C3—24no—
480C3#1059396114yescomplemented
481C3#1059615311yescharacteristic, complemented
482C2—122no—
483C2—361no—
484C2C2×#8901411yescharacteristic
485C2#1172456311yescharacteristic
486C2#1172458111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radical#11723522C2
hypercenterC42D3×C3⋊D3
hypocenterC33#542
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×#64081C2semidirect
3. C32×#875C2semidirect
4. #11723528C2semidirect
6. #11723496C2semidirect
7. D12×C3⋊Dic3C2semidirect
8. #11723480C2semidirect
9. #11723435C2semidirect
10. #11723430C2semidirect
11. Dic6×C3⋊Dic3C2semidirect
46. #64081D3semidirect
50. C3×#875D3semidirect
162. #875C3⋊D3semidirect
315. C33#542semidirect
431. C32#5534semidirect
432. C32#5682semidirect
480. C3#10593961semidirect
481. C3#10596153semidirect

Maximal subgroups

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

subgroupindex
#117239102
C3×#640812
C32×#8752
#117235282
#117235222
#117234962
D12×C3⋊Dic32
#117234802
#117234352
#117234302
Dic6×C3⋊Dic32
#105961533
#105939613