Finite Group Databasebeta

← #13177273

Subgroups — #13177273

Order 1728.

1576 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#11723967C211yesmaximal, characteristic, complemented
2C3×C4×#638C211yesmaximal, characteristic
3C32×#924C211yesmaximal, characteristic, complemented
4#11723596C212yesmaximal, complemented
5#11723591C212yesmaximal
6#11723535C211yesmaximal, characteristic, complemented
7#11723435C211yesmaximal, characteristic, complemented
8#11723433C212yesmaximal, complemented
9#11723404C211yesmaximal, characteristic, complemented
10Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
11#11723391C212yesmaximal, complemented
12#10596112—31nomaximal
13#10594841—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×C3×#638C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C32×Dic6C2211yescharacteristic
18C2×#6293C2212yes—
19#88864C2212yes—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2212yes—
23C3×#3757C2211yescharacteristic
24C3×#3752C2211yescharacteristic
25C3×#3750C2211yescharacteristic
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C32×#217C2212yes—
28C32×C4⋊Dic3C2211yescharacteristic
29C32×Dic3⋊C4C2212yes—
30#88807C2212yes—
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88803C2212yes—
34#88803C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C22⋊Q8—31no—
38#64155—31no—
39Q8×C3⋊Dic3—31no—
40C4×#638D311yescharacteristic
41#64135—31no—
42#64135—34no—
43#64135—64no—
44#64107—32no—
45#64101—31no—
46#64097—31no—
47#64096—32no—
48#64094—32no—
49#64088—32no—
50C3×C4×#67—34no—
51C3×#924D314yescomplemented
52#63959—38no—
53#63956—38no—
54#63918—34no—
55#63862—38no—
56#63856—34no—
57Dic3×Dic6—34no—
58#63841—34no—
59#63838—38no—
60C23×C33C2311yescharacteristic
61C22×C3×C3⋊D3C2311yescharacteristic
62C2×C33×C4—22no—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C3×#638—22no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C3×C4×C3⋊D3—22no—
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2312yes—
74C32×Dic6—22no—
75#6293—24no—
76#5641—94no—
77#5542—91no—
78C2×C32×Q8—31no—
79C22×C32×C4—24no—
80C22×C32×C4C2×D314yes—
81C22×C32×C4C2×D311yescharacteristic
82C2×#638C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×#634—32no—
85C2×C3×#67—34no—
86C2×C3×C4×D3—34no—
87C2×C3×Dic6C2×D314yes—
88C2×#627—38no—
89C32×C4⋊C4—32no—
90C32×C4⋊C4—31no—
91C32×C22⋊C4—32no—
92#3757—32no—
93#3757—38no—
94#3757—68no—
95#3757C2×D311yescharacteristic
96#3752—32no—
97#3752—32no—
98#3752C2×D311yescharacteristic
99C4⋊(C3⋊Dic3)—31no—
100C4⋊(C3⋊Dic3)—31no—
101C4⋊(C3⋊Dic3)—34no—
102C4⋊(C3⋊Dic3)—34no—
103C4⋊(C3⋊Dic3)—32no—
104C4⋊(C3⋊Dic3)—64no—
105C4⋊(C3⋊Dic3)—64no—
106#3750—32no—
107#3750—32no—
108#3750C2×D311yescharacteristic
109C4×C3⋊Dic3—32no—
110C4×C3⋊Dic3—38no—
111C4×C3⋊Dic3—32no—
112C4×C3⋊Dic3—32no—
113C4×C3⋊Dic3—68no—
114C4×C3⋊Dic3C2×D311yescharacteristic
115C3×#217—34no—
116C3×#217C2×D318yes—
117C3×(C2×D3)⋊C4—34no—
118C3×C4⋊Dic3C2×D314yes—
119C3×Dic3⋊C4—34no—
120C3×Dic3⋊C4C2×D318yes—
121C3×C4×Dic3—34no—
122#3724—38no—
123Dic3⋊Dic3—38no—
124Dic3⋊Dic3—38no—
125(C2×D3)⋊Dic3—38no—
126(C2×D3)⋊Dic3—38no—
127Dic32—38no—
128Dic32—38no—
129C22×C33—21no—
130C22×C33—21no—
131C22×C33—21no—
132C22×C33C2411yescharacteristic
133C2×C3×C3⋊D3—21no—
134C2×C3×C3⋊D3—21no—
135C2×C3×C3⋊D3—21no—
136C33×C4—21no—
137C33×C4#4112yes—
138C3⋊(C3⋊Dic3)—22no—
139C3⋊(C3⋊Dic3)—22no—
140C3×C3⋊Dic3—21no—
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3#4112yes—
143C32×Dic3—22no—
144C32×Dic3—22no—
145C3×C22⋊Q8—34no—
146C3×C4×D4—91no—
147#946—94no—
148Q8×Dic3—94no—
149#936—92no—
150C4×#67—34no—
151#926—94no—
152#926—91no—
153#926—91no—
154#926—184no—
155#924—91no—
156#924C3⋊D311yescharacteristic, complemented
157#898—98no—
158#892—94no—
159#888—94no—
160#887—98no—
161#887—92no—
162#885—98no—
163#879—98no—
164#879—92no—
165#879—92no—
166#869—91no—
167C4×Dic6—91no—
168C23×C32—24no—
169C23×C32C22×D314yes—
170C23×C32C22×D311yescharacteristic
171C22×C3⋊D3C22×D311yescharacteristic
172C22×C3×D3—34no—
173C32×Q8—62no—
174C2×C32×C4—216no—
175C2×C32×C4—28no—
176C2×C32×C4—22no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—32no—
180C2×C32×C4—32no—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185#638—22no—
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⋊Dic3C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C4×C3⋊D3—22no—
196#634—64no—
197C3×#67—68no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—34no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3C22×D318yes—
202C2×C3×Dic3C22×D318yes—
203C3×C4×D3—68no—
204C3×Dic6—28no—
205#627—616no—
206#556—271noSylow
207C2×C33—41no—
208C2×C33C2×#4111yescharacteristic
209C2×C33C2×#4111yescharacteristic
210C2×C332−1+411yescharacteristic
211C3×C3⋊D3—41no—
212C2×C3×Q8—34no—
213C2×C3×D4—91no—
214C22×C3×C4—24no—
215C22×C3×C4—91no—
216C22×C3×C4D3214yes—
217C22×C3×C4C2×C3⋊D311yescharacteristic
218C2×#67—34no—
219C2×C4×D3—34no—
220C2×Dic6—98no—
221C2×Dic6—92no—
222C2×Dic6C2×C3⋊D311yescharacteristic
223C3×C4⋊C4—38no—
224C3×C4⋊C4—34no—
225C3×C4⋊C4—91no—
226C3×C22⋊C4—38no—
227C3×C22⋊C4—91no—
228C3×C22⋊C4—91no—
229C3×C42—91no—
230#217—34no—
231#217—98no—
232#217—92no—
233#217—92no—
234#217—92no—
235#217—188no—
236#217C2×C3⋊D312yes—
237(C2×D3)⋊C4—34no—
238(C2×D3)⋊C4—98no—
239(C2×D3)⋊C4—98no—
240C4⋊Dic3—98no—
241C4⋊Dic3—94no—
242C4⋊Dic3—94no—
243C4⋊Dic3—91no—
244C4⋊Dic3—91no—
245C4⋊Dic3—92no—
246C4⋊Dic3—184no—
247C4⋊Dic3—184no—
248C4⋊Dic3C2×C3⋊D311yescharacteristic
249Dic3⋊C4—34no—
250Dic3⋊C4—98no—
251Dic3⋊C4—98no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4C2×C3⋊D312yes—
255C4×Dic3—34no—
256C4×Dic3—98no—
257C4×Dic3—98no—
258C4×Dic3—98no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—92no—
262C4×Dic3—188no—
263C22×C32—28no—
264C22×C32—21no—
265C22×C32—24no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—24no—
269C22×C32—24no—
270C22×C32—24no—
271C22×C32—44no—
272C22×C32—44no—
273C22×C32C23×D314yes—
274C22×C32C23×D311yescharacteristic
275C2×C3⋊D3—21no—
276C2×C3⋊D3—21no—
277C2×C3⋊D3—21no—
278C2×C3×D3—64no—
279C2×C3×D3—64no—
280C2×C3×D3—64no—
281C32×C4—28no—
282C32×C4—24no—
283C32×C4—21no—
284C32×C4—28no—
285C32×C4—62no—
286C32×C4—62no—
287C32×C4#23718yes—
288C32×C4#23712yes—
289C3⋊Dic3—21no—
290C3⋊Dic3—21no—
291C3⋊Dic3—62no—
292C3⋊Dic3—62no—
293C3⋊Dic3—68no—
294C3⋊Dic3—68no—
295C3⋊Dic3—128no—
296C3⋊Dic3—128no—
297C3⋊Dic3#23512yes—
298C3×Dic3—28no—
299C3×Dic3—28no—
300C3×Dic3—38no—
301C3×Dic3—64no—
302C3×Dic3—64no—
303C4⋊Q8—271no—
304#126—272no—
305#126—272no—
306#124—272no—
307C22⋊Q8—31no—
308C22⋊Q8—271no—
309C4×Q8—271no—
310C4×Q8—272no—
311C4×D4—91no—
312#117—271no—
313#117—271no—
314C33#55611yescharacteristic, complemented, Sylow
315C23×C3—24no—
316C23×C3—91no—
317C23×C3C2×D3214yes—
318C23×C3C22×C3⋊D311yescharacteristic
319C22×D3—34no—
320C3×Q8—68no—
321C3×D4—182no—
322C2×C3×C4—216no—
323C2×C3×C4—28no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—38no—
328C2×C3×C4—38no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—91no—
332C2×C3×C4—182no—
333C2×C3×C4C2×D3214yes—
334C2×C3×C4C2×D3214yes—
335C2×C3×C4C22×C3⋊D311yescharacteristic
336C2×C3×C4C22×C3⋊D311yescharacteristic
337#67—68no—
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×Dic3C22×C3⋊D312yes—
348C2×Dic3C22×C3⋊D312yes—
349C4×D3—68no—
350Dic6—22no—
351Dic6—184no—
352Dic6—1816no—
353C2×C32—24no—
354C2×C32—24no—
355C2×C32—24no—
356C2×C32—48no—
357C2×C32—44no—
358C2×C32—41no—
359C2×C32C2×#23511yescharacteristic
360C2×C32C2×#23714yes—
361C2×C32C2×#23711yescharacteristic
362C2×C32#100714yes—
363C2×C32D3×#4114yes—
364C2×C32#101011yescharacteristic
365C3⋊D3—41no—
366C3×D3—124no—
367C2×Q8—31no—
368C2×Q8—272no—
369C2×D4—91no—
370C22×C4—91no—
371C22×C4D3×C3⋊D311yescharacteristic
372C4⋊C4—31no—
373C4⋊C4—32no—
374C4⋊C4—91no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—271no—
379C4⋊C4—271no—
380C22⋊C4—32no—
381C22⋊C4—91no—
382C22⋊C4—91no—
383C22⋊C4—272no—
384C22⋊C4—272no—
385C22⋊C4—272no—
386C42—91no—
387C42—272no—
388C42—272no—
389C42—272no—
390C22×C3—28no—
391C22×C3—24no—
392C22×C3—24no—
393C22×C3—24no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—24no—
398C22×C3—44no—
399C22×C3—44no—
400C22×C3—181no—
401C22×C3—181no—
402C22×C3—181no—
403C22×C3C22×D3214yes—
404C22×C3C23×C3⋊D311yescharacteristic
405C2×D3—64no—
406C2×D3—64no—
407C2×D3—64no—
408C3×C4—28no—
409C3×C4—24no—
410C3×C4—21no—
411C3×C4—28no—
412C3×C4—68no—
413C3×C4—68no—
414C3×C4—92no—
415C3×C4—181no—
416C3×C4—181no—
417C3×C4#380518yes—
418C3×C4#383012yes—
419Dic3—22no—
420Dic3—22no—
421Dic3—38no—
422Dic3—64no—
423Dic3—64no—
424Dic3—188no—
425Dic3—188no—
426Dic3—182no—
427Dic3—182no—
428Dic3—368no—
429Dic3—368no—
430C32—24no—
431C32#554211yescharacteristic, complemented
432C32#564114yescomplemented
433C23—91no—
434C23C2×D3×C3⋊D311yescharacteristic
435Q8—62no—
436Q8—544no—
437D4—182no—
438C2×C4—22no—
439C2×C4—32no—
440C2×C4—32no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—91no—
444C2×C4—182no—
445C2×C4—272no—
446C2×C4—272no—
447C2×C4C2×D3×C3⋊D311yescharacteristic
448C2×C4C2×D3×C3⋊D311yescharacteristic
449C2×C3—24no—
450C2×C3—24no—
451C2×C3—24no—
452C2×C3—48no—
453C2×C3—41no—
454C2×C3—44no—
455C2×C3—361no—
456C2×C3#6429714yes—
457C2×C3D3×#23514yes—
458C2×C3C2×#380514yes—
459C2×C3C2×#383011yescharacteristic
460C2×C3#6436411yescharacteristic
461C2×C3#41×C3⋊D311yescharacteristic
462D3—124no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—181no—
467C22—181no—
468C22—181no—
469C22C22×D3×C3⋊D311yescharacteristic
470C4—21no—
471C4—62no—
472C4—62no—
473C4—92no—
474C4—181no—
475C4—181no—
476C4—542no—
477C4—542no—
478C4#8902912yes—
479C3—24no—
480C3#1059484114yescomplemented
481C3#1059611211yescharacteristic, complemented
482C2—41no—
483C2—361no—
484C2#1172456611yescharacteristic
485C2C3⋊D3×#23511yescharacteristic
486C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
prime radicalC2×C3×#638C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33#556
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11723967C2semidirect
3. C32×#924C2semidirect
4. #11723596C2semidirect
6. #11723535C2semidirect
7. #11723435C2semidirect
8. #11723433C2semidirect
9. #11723404C2semidirect
10. Dic6×C3⋊Dic3C2semidirect
11. #11723391C2semidirect
51. C3×#924D3semidirect
156. #924C3⋊D3semidirect
314. C33#556semidirect
431. C32#5542semidirect
432. C32#5641semidirect
480. C3#10594841semidirect
481. C3#10596112semidirect

Maximal subgroups

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

subgroupindex
#117239672
C3×C4×#6382
C32×#9242
#117235962
#117235912
#117235352
#117234352
#117234332
#117234042
Dic6×C3⋊Dic32
#117233912
#105961123
#105948413