Finite Group Databasebeta

← #13177277

Subgroups — #13177277

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×#64133C211yesmaximal, characteristic, complemented
3C32×C4×#67C211yesmaximal, characteristic
4#11723600C212yesmaximal
5#11723597C212yesmaximal, complemented
6#11723534C211yesmaximal, characteristic, complemented
7#11723432C212yesmaximal, complemented
8#11723425C211yesmaximal, characteristic, complemented
9#11723424C211yesmaximal, characteristic, complemented
10Dic3×#634C211yesmaximal, characteristic, complemented
11#11723391C212yesmaximal, complemented
12#10596013—31nomaximal
13#10594841—34nomaximal
14C22×C33×C4C2211yescharacteristic
15C2×C3×#634C2211yescharacteristic
16C2×C32×#67C2211yescharacteristic
17C2×C32×C4×D3C2211yescharacteristic
18C2×#6293C2212yes—
19#88864C2212yes—
20C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2212yes—
23C3×#3757C2212yes—
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
25C3×#3750C2212yes—
26C32×#217C2211yescharacteristic
27C32×(C2×D3)⋊C4C2211yescharacteristic
28C32×Dic3⋊C4C2211yescharacteristic
29C32×C4×Dic3C2211yescharacteristic
30#88807C2212yes—
31#88806C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33(C2×D3)⋊(C3⋊Dic3)C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×D4—31no—
38#64145—32no—
39#64135—31no—
40#64135—31no—
41#64135—34no—
42#64135—64no—
43#64133—31no—
44#64133D311yescharacteristic, complemented
45#64096—32no—
46#64088—32no—
47#64088—32no—
48#64078—31no—
49C4×#634—31no—
50C3×C4×#67D314yes—
51C3×#924—34no—
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×C32×D3C2311yescharacteristic
62C2×C33×C4—22no—
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C3×#634—22no—
70C32×#67—22no—
71C2×C32×Dic3C2311yescharacteristic
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C32×C4×D3—22no—
75#6293—24no—
76#5641—91no—
77#5542—94no—
78C2×C32×D4—31no—
79C22×C32×C4—24no—
80C22×C32×C4—31no—
81C22×C32×C4C2×D314yes—
82C22×C32×C4C2×D311yescharacteristic
83C2×#634—32no—
84C2×#634C2×D311yescharacteristic
85C2×C3×#67C2×D314yes—
86C2×C3×C4×D3C2×D314yes—
87C2×C3×Dic6—34no—
88C2×#627—38no—
89C32×C4⋊C4—31no—
90C32×C22⋊C4—31no—
91C32×C22⋊C4—31no—
92C32×C42—31no—
93#3757—32no—
94#3757—38no—
95#3757—32no—
96#3757—32no—
97#3757—68no—
98#3757C2×D312yes—
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—
106C4⋊(C3⋊Dic3)C2×D311yescharacteristic
107#3750—32no—
108#3750—32no—
109#3750C2×D312yes—
110C4×C3⋊Dic3—32no—
111C4×C3⋊Dic3—38no—
112C4×C3⋊Dic3—32no—
113C4×C3⋊Dic3—32no—
114C4×C3⋊Dic3—68no—
115C3×#217—38no—
116C3×#217C2×D314yes—
117C3×(C2×D3)⋊C4C2×D314yes—
118C3×C4⋊Dic3—34no—
119C3×Dic3⋊C4—38no—
120C3×Dic3⋊C4C2×D314yes—
121C3×C4×Dic3C2×D314yes—
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×C32×D3—21no—
134C2×C32×D3—21no—
135C2×C32×D3—21no—
136C33×C4—21no—
137C33×C4#4112yes—
138C3⋊(C3⋊Dic3)—22no—
139C3⋊(C3⋊Dic3)—22no—
140C3×C3⋊Dic3—22no—
141C3×C3⋊Dic3—22no—
142C32×Dic3—21no—
143C32×Dic3—21no—
144C32×Dic3#4112yes—
145C3×C22⋊Q8—91no—
146C3×C4×D4—34no—
147#946—91no—
148Q8×Dic3—91no—
149#936—98no—
150C4×#67C3⋊D311yescharacteristic
151#926—94no—
152#926—94no—
153#926—91no—
154#926—184no—
155#924—34no—
156#924—94no—
157#898—92no—
158#892—91no—
159#888—91no—
160#887—98no—
161#887—92no—
162#885—92no—
163#879—98no—
164#879—98no—
165#879—92no—
166#869—94no—
167C4×Dic6—94no—
168C23×C32—24no—
169C23×C32—31no—
170C23×C32C22×D314yes—
171C23×C32C22×D311yescharacteristic
172C22×C3×D3C22×D314yes—
173C32×D4—62no—
174C2×C32×C4—216no—
175C2×C32×C4—28no—
176C2×C32×C4—22no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—31no—
180C2×C32×C4—31no—
181C2×C32×C4—31no—
182C2×C32×C4—62no—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—38no—
191C2×C3⋊Dic3—68no—
192C2×C3⋊Dic3—68no—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D312yes—
195#634—22no—
196#634—64no—
197C3×#67—28no—
198C2×C3×Dic3—38no—
199C2×C3×Dic3—38no—
200C2×C3×Dic3C22×D314yes—
201C2×C3×Dic3C22×D314yes—
202C2×C3×Dic3C22×D314yes—
203C3×C4×D3—28no—
204C3×Dic6—68no—
205#627—616no—
206#556—271noSylow
207C2×C33—41no—
208C2×C33C2×#4111yescharacteristic
209C2×C33C2×#4111yescharacteristic
210C2×C332−1+411yescharacteristic
211C32×D3—41no—
212C2×C3×Q8—91no—
213C2×C3×D4—34no—
214C22×C3×C4—24no—
215C22×C3×C4—34no—
216C22×C3×C4D3214yes—
217C22×C3×C4C2×C3⋊D311yescharacteristic
218C2×#67C2×C3⋊D311yescharacteristic
219C2×C4×D3C2×C3⋊D311yescharacteristic
220C2×Dic6—34no—
221C2×Dic6—98no—
222C2×Dic6—92no—
223C3×C4⋊C4—34no—
224C3×C4⋊C4—92no—
225C3×C4⋊C4—91no—
226C3×C22⋊C4—34no—
227C3×C22⋊C4—34no—
228C3×C22⋊C4—92no—
229C3×C42—34no—
230#217—38no—
231#217—98no—
232#217—98no—
233#217—98no—
234#217—92no—
235#217—188no—
236#217C2×C3⋊D311yescharacteristic
237(C2×D3)⋊C4—92no—
238(C2×D3)⋊C4—92no—
239(C2×D3)⋊C4C2×C3⋊D311yescharacteristic
240C4⋊Dic3—34no—
241C4⋊Dic3—98no—
242C4⋊Dic3—94no—
243C4⋊Dic3—94no—
244C4⋊Dic3—91no—
245C4⋊Dic3—91no—
246C4⋊Dic3—92no—
247C4⋊Dic3—184no—
248C4⋊Dic3—184no—
249Dic3⋊C4—38no—
250Dic3⋊C4—98no—
251Dic3⋊C4—98no—
252Dic3⋊C4—92no—
253Dic3⋊C4—92no—
254Dic3⋊C4C2×C3⋊D311yescharacteristic
255C4×Dic3—98no—
256C4×Dic3—98no—
257C4×Dic3—98no—
258C4×Dic3—92no—
259C4×Dic3—92no—
260C4×Dic3—92no—
261C4×Dic3—188no—
262C4×Dic3C2×C3⋊D311yescharacteristic
263C22×C32—28no—
264C22×C32—24no—
265C22×C32—21no—
266C22×C32—21no—
267C22×C32—21no—
268C22×C32—24no—
269C22×C32—24no—
270C22×C32—24no—
271C22×C32—44no—
272C22×C32—44no—
273C22×C32—61no—
274C22×C32—61no—
275C22×C32—61no—
276C22×C32C23×D314yes—
277C22×C32C23×D311yescharacteristic
278C2×C3×D3—24no—
279C2×C3×D3—24no—
280C2×C3×D3—24no—
281C32×C4—28no—
282C32×C4—24no—
283C32×C4—21no—
284C32×C4—28no—
285C32×C4—32no—
286C32×C4—61no—
287C32×C4—61no—
288C32×C4#23718yes—
289C32×C4#23712yes—
290C3⋊Dic3—22no—
291C3⋊Dic3—22no—
292C3⋊Dic3—62no—
293C3⋊Dic3—62no—
294C3⋊Dic3—68no—
295C3⋊Dic3—68no—
296C3⋊Dic3—128no—
297C3⋊Dic3—128no—
298C3×Dic3—24no—
299C3×Dic3—24no—
300C3×Dic3—68no—
301C3×Dic3—68no—
302C3×Dic3#23518yes—
303C4⋊Q8—271no—
304#126—272no—
305#126—272no—
306#124—272no—
307C22⋊Q8—91no—
308C22⋊Q8—271no—
309C4×Q8—271no—
310C4×Q8—272no—
311C4×D4—31no—
312#117—271no—
313#117—271no—
314C33#55611yescharacteristic, complemented, Sylow
315C23×C3—24no—
316C23×C3—34no—
317C23×C3C2×D3214yes—
318C23×C3C22×C3⋊D311yescharacteristic
319C22×D3C22×C3⋊D311yescharacteristic
320C3×Q8—182no—
321C3×D4—68no—
322C2×C3×C4—216no—
323C2×C3×C4—28no—
324C2×C3×C4—22no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—34no—
328C2×C3×C4—34no—
329C2×C3×C4—34no—
330C2×C3×C4—68no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4C2×D3214yes—
334C2×C3×C4C2×D3214yes—
335C2×C3×C4C22×C3⋊D311yescharacteristic
336C2×C3×C4C22×C3⋊D311yescharacteristic
337#67—22no—
338C2×Dic3—38no—
339C2×Dic3—38no—
340C2×Dic3—98no—
341C2×Dic3—98no—
342C2×Dic3—92no—
343C2×Dic3—92no—
344C2×Dic3—188no—
345C2×Dic3—188no—
346C2×Dic3C22×C3⋊D311yescharacteristic
347C2×Dic3C22×C3⋊D311yescharacteristic
348C2×Dic3C22×C3⋊D311yescharacteristic
349C4×D3—22no—
350Dic6—68no—
351Dic6—1816no—
352Dic6—184no—
353C2×C32—24no—
354C2×C32—24no—
355C2×C32—24no—
356C2×C32—48no—
357C2×C32—41no—
358C2×C32—44no—
359C2×C32—121no—
360C2×C32C2×#23514yes—
361C2×C32C2×#23714yes—
362C2×C32C2×#23711yescharacteristic
363C2×C32#100711yescharacteristic
364C2×C32D3×#4111yescharacteristic
365C2×C32#101014yes—
366C3×D3—44no—
367C2×Q8—91no—
368C2×Q8—272no—
369C2×D4—31no—
370C22×C4—31no—
371C22×C4D3×C3⋊D311yescharacteristic
372C4⋊C4—31no—
373C4⋊C4—91no—
374C4⋊C4—92no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—272no—
378C4⋊C4—271no—
379C4⋊C4—271no—
380C22⋊C4—31no—
381C22⋊C4—31no—
382C22⋊C4—92no—
383C22⋊C4—272no—
384C22⋊C4—272no—
385C22⋊C4—272no—
386C42—31no—
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—64no—
401C22×C3—64no—
402C22×C3—64no—
403C22×C3C22×D3214yes—
404C22×C3C23×C3⋊D311yescharacteristic
405C2×D3—21no—
406C2×D3—21no—
407C2×D3—21no—
408C3×C4—28no—
409C3×C4—24no—
410C3×C4—21no—
411C3×C4—28no—
412C3×C4—38no—
413C3×C4—64no—
414C3×C4—64no—
415C3×C4—182no—
416C3×C4—182no—
417C3×C4#380518yes—
418C3×C4#383012yes—
419Dic3—21no—
420Dic3—21no—
421Dic3—68no—
422Dic3—68no—
423Dic3—188no—
424Dic3—188no—
425Dic3—182no—
426Dic3—182no—
427Dic3—368no—
428Dic3—368no—
429Dic3#382812yes—
430C32—24no—
431C32#554214yescomplemented
432C32#564111yescharacteristic, complemented
433C23—31no—
434C23C2×D3×C3⋊D311yescharacteristic
435Q8—182no—
436Q8—544no—
437D4—62no—
438C2×C4—22no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—62no—
443C2×C4—92no—
444C2×C4—92no—
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—44no—
454C2×C3—41no—
455C2×C3—124no—
456C2×C3#6429714yes—
457C2×C3D3×#23514yes—
458C2×C3C2×#380514yes—
459C2×C3C2×#382811yescharacteristic
460C2×C3C2×#383011yescharacteristic
461C2×C3#6436711yescharacteristic
462D3—41no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—61no—
467C22—61no—
468C22—61no—
469C22C22×D3×C3⋊D311yescharacteristic
470C4—21no—
471C4—32no—
472C4—61no—
473C4—61no—
474C4—182no—
475C4—182no—
476C4—542no—
477C4—542no—
478C4#8902912yes—
479C3—24no—
480C3#1059484114yescomplemented
481C3#1059601311yescharacteristic, complemented
482C2—41no—
483C2—121no—
484C2#1172456211yescharacteristic
485C2D3×#382811yescharacteristic
486C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC22×C33×C4C22
prime radicalC2×C32×#67C22
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
2. C3×#64133C2semidirect
5. #11723597C2semidirect
6. #11723534C2semidirect
7. #11723432C2semidirect
8. #11723425C2semidirect
9. #11723424C2semidirect
10. Dic3×#634C2semidirect
11. #11723391C2semidirect
44. #64133D3semidirect
314. C33#556semidirect
431. C32#5542semidirect
432. C32#5641semidirect
480. C3#10594841semidirect
481. C3#10596013semidirect

Maximal subgroups

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

subgroupindex
#117239672
C3×#641332
C32×C4×#672
#117236002
#117235972
#117235342
#117234322
#117234252
#117234242
Dic3×#6342
#117233912
#105960133
#105948413