Finite Group Databasebeta

← #13176474

Subgroups — #13176474

Order 1728.

1952 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
1C2×C3⋊D3×Dic6C212yesmaximal, complemented
2#11723929C211yesmaximal, characteristic, complemented
3C3×#64102C211yesmaximal, characteristic
4C32×#888C211yesmaximal, characteristic, complemented
5#11723535C211yesmaximal, characteristic, complemented
6C4×#6293C211yesmaximal, characteristic, complemented
7#11723501C211yesmaximal, characteristic
8#11723497C212yesmaximal, complemented
9Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
10#11723476C212yesmaximal, complemented
11#11723418C212yesmaximal, complemented
12#10596198—31nomaximal
13#10594376—34nomaximal
14C2×#6308C2212yes—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C32×Dic6C2212yes—
18C2×#6293C2212yes—
19C2×#6293C2211yescharacteristic
20C2×Dic3×C3⋊D3C2212yes—
21C2×Dic3×C3⋊D3C2212yes—
22C3⋊D3×Dic6—28no—
23C33×C4⋊C4C2211yescharacteristic
24C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
25#88857C2212yes—
26C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
27C3×#3752C2212yes—
28C3×C4×C3⋊Dic3C2211yescharacteristic
29C32×C4⋊Dic3C2211yescharacteristic
30C32×Dic3⋊C4C2212yes—
31C32×C4×Dic3C2211yescharacteristic
32#88807C2211yescharacteristic
33#88807C2212yes—
34#88806C2211yescharacteristic
35Dic3⋊(C3⋊Dic3)C2211yescharacteristic
36#88803C2212yes—
37#88803C2212yes—
38C2×Q8×C3⋊D3—32no—
39C2×D3×Dic6—38no—
40C32×C4⋊Q8—31no—
41#64155—32no—
42#64105—32no—
43#64105—32no—
44#64102—31no—
45#64102D311yescharacteristic
46#64097—31no—
47#64097—31no—
48#64097—34no—
49#64097—64no—
50C4×C4⋊(C3⋊D3)—31no—
51C4×#634—31no—
52C3×#893—34no—
53C3×#888D314yescomplemented
54#63918—34no—
55C4×#627—34no—
56#63898—34no—
57#63896—38no—
58Dic3×D12—34no—
59#63887—38no—
60#63851—38no—
61C22×C3×C3⋊D3C2312yes—
62C2×C33×C4C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67#6308—24no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C3×C4⋊(C3⋊D3)Q814yescomplemented
71C3×C4×C3⋊D3—24no—
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C32×Dic6—24no—
76#6293—24no—
77#6293D414yescomplemented
78Dic3×C3⋊D3—24no—
79Dic3×C3⋊D3—24no—
80#5727—94no—
81D4×Dic6—91no—
82C2×C32×Q8—32no—
83Q8×C3⋊D3—68no—
84C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
85C2×C4×C3⋊D3—32no—
86C2×C4×C3⋊D3—32no—
87C2×C4×C3⋊D3C2×D312yes—
88C2×#634—31no—
89C2×#634—32no—
90C2×#634—38no—
91C2×#634—32no—
92C2×#634—68no—
93C2×C3×D12—34no—
94C2×C3×C4×D3—38no—
95C2×C3×Dic6C2×D318yes—
96C2×#627—34no—
97C2×#627—38no—
98C2×D3×Dic3—38no—
99C2×D3×Dic3—38no—
100D3×Dic6—632no—
101C32×C4⋊C4—24no—
102C32×C4⋊C4—31no—
103C32×C4⋊C4—32no—
104C32×C4⋊C4C2×D314yes—
105C32×C4⋊C4C2×D311yescharacteristic
106C32×C42—31no—
107#3752—32no—
108#3752—32no—
109#3752C2×D312yes—
110C4⋊(C3⋊Dic3)—31no—
111C4⋊(C3⋊Dic3)—34no—
112C4⋊(C3⋊Dic3)—31no—
113C4⋊(C3⋊Dic3)—64no—
114#3750—32no—
115#3750—38no—
116#3750—31no—
117#3750—31no—
118#3750—32no—
119#3750—68no—
120C4×C3⋊Dic3—31no—
121C4×C3⋊Dic3—34no—
122C4×C3⋊Dic3—64no—
123C4×C3⋊Dic3C2×D311yescharacteristic
124C3×(C2×D3)⋊C4—38no—
125C3×C4⋊Dic3C2×D314yes—
126C3×Dic3⋊C4C2×D318yes—
127C3×C4×Dic3—34no—
128C3×C4×Dic3C2×D314yes—
129#3724—34no—
130Dic3⋊Dic3—34no—
131Dic3⋊Dic3—34no—
132Dic3⋊Dic3—38no—
133(C2×D3)⋊Dic3—38no—
134(C2×D3)⋊Dic3—38no—
135C22×C33C2411yescharacteristic
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3C2×Q814yescomplemented
139C33×C4—22no—
140C33×C4C2×Q812yescomplemented
141C3⋊(C3⋊Dic3)—22no—
142C3⋊(C3⋊Dic3)—21no—
143C3⋊(C3⋊Dic3)C2×D412yes—
144C3×C3⋊Dic3—21no—
145C3×C3⋊Dic3C2×D412yes—
146C32×Dic3—22no—
147C32×Dic3—21no—
148C32×Dic3C2×D412yes—
149C2×D3×Q8—98no—
150C22×Dic6—92no—
151C3×C4⋊Q8—34no—
152C3×C4×D4—91no—
153#946—98no—
154D4×Dic3—91no—
155D4×Dic3—91no—
156#924—92no—
157#896—98no—
158#896—98no—
159#893—34no—
160#893—94no—
161#888—94no—
162#888—94no—
163#888—91no—
164#888—184no—
165#888C3⋊D311yescharacteristic, complemented
166#878—92no—
167#878—92no—
168C4×D12—94no—
169#869—91no—
170C4×Dic6—94no—
171C4×Dic6—91no—
172C22×C3⋊D3C22×D312yes—
173C22×C3×D3—38no—
174C32×Q8—64no—
175C2×C32×C4—24no—
176C2×C32×C4—28no—
177C2×C32×C4—31no—
178C2×C32×C4—31no—
179C2×C32×C4—32no—
180C2×C32×C4C22×D318yes—
181C2×C32×C4C22×D314yes—
182C2×C32×C4C22×D312yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—31no—
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—34no—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—38no—
190C2×C3⋊Dic3—64no—
191C2×C3⋊Dic3—64no—
192C2×C3⋊Dic3—68no—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C4⋊(C3⋊D3)Dic614yescomplemented
196C4×C3⋊D3—24no—
197C4×C3⋊D3—64no—
198C4×C3⋊D3—64no—
199#634—34no—
200#634—64no—
201#634—64no—
202#634—616no—
203#634—1216no—
204C2×C3×Dic3—34no—
205C2×C3×Dic3—34no—
206C2×C3×Dic3C22×D314yes—
207C2×C3×Dic3C22×D318yes—
208C2×C3×Dic3C22×D314yes—
209C3×D12—316no—
210C3×C4×D3—616no—
211C3×Dic6—216no—
212#627—316no—
213#627—616no—
214D3×Dic3—616no—
215D3×Dic3—616no—
216D4×Q8—271noSylow
217C2×C33C22×D411yescharacteristic
218C2×C33C22×Q811yescharacteristic
219C2×C332−1+411yescharacteristic
220C3×C3⋊D3—24no—
221C2×C3×Q8—38no—
222C2×C3×D4—91no—
223C22×C3×C4—92no—
224C22×Dic3—92no—
225C22×Dic3—92no—
226D3×Q8—1832no—
227C2×D12—34no—
228C2×C4×D3—38no—
229C2×C4×D3—98no—
230C2×C4×D3—98no—
231C2×Dic6—94no—
232C2×Dic6—91no—
233C2×Dic6—98no—
234C2×Dic6—98no—
235C2×Dic6—92no—
236C2×Dic6—92no—
237C2×Dic6—188no—
238C2×Dic6—188no—
239C2×Dic6C2×C3⋊D312yes—
240C3×C4⋊C4—24no—
241C3×C4⋊C4—34no—
242C3×C4⋊C4—38no—
243C3×C4⋊C4D3214yes—
244C3×C4⋊C4C2×C3⋊D311yescharacteristic
245C3×C22⋊C4—92no—
246C3×C42—34no—
247C3×C42—91no—
248#217—92no—
249#217—92no—
250(C2×D3)⋊C4—38no—
251(C2×D3)⋊C4—98no—
252(C2×D3)⋊C4—98no—
253C4⋊Dic3—94no—
254C4⋊Dic3—94no—
255C4⋊Dic3—91no—
256C4⋊Dic3—91no—
257C4⋊Dic3—92no—
258C4⋊Dic3—184no—
259C4⋊Dic3C2×C3⋊D311yescharacteristic
260Dic3⋊C4—94no—
261Dic3⋊C4—94no—
262Dic3⋊C4—98no—
263Dic3⋊C4—98no—
264Dic3⋊C4—92no—
265Dic3⋊C4—91no—
266Dic3⋊C4—91no—
267Dic3⋊C4—188no—
268Dic3⋊C4C2×C3⋊D312yes—
269C4×Dic3—34no—
270C4×Dic3—94no—
271C4×Dic3—91no—
272C4×Dic3—184no—
273C4×Dic3C2×C3⋊D311yescharacteristic
274C22×C32—24no—
275C22×C32C23×D314yes—
276C22×C32C23×D311yescharacteristic
277C2×C3⋊D3—22no—
278C2×C3⋊D3—22no—
279C2×C3⋊D3C2×Dic614yescomplemented
280C2×C3×D3—316no—
281C2×C3×D3—68no—
282C2×C3×D3—68no—
283C32×C4—22no—
284C32×C4—28no—
285C32×C4—28no—
286C32×C4—32no—
287C32×C4—48no—
288C32×C4—62no—
289C32×C4—61no—
290C32×C4C2×Dic612yescomplemented
291C32×C4D3×Q818yescomplemented
292C3⋊Dic3—21no—
293C3⋊Dic3—32no—
294C3⋊Dic3—38no—
295C3⋊Dic3—68no—
296C3⋊Dic3—68no—
297C3⋊Dic3—62no—
298C3⋊Dic3—64no—
299C3⋊Dic3—61no—
300C3⋊Dic3—68no—
301C3⋊Dic3—128no—
302C3⋊Dic3D3×D412yes—
303C3×Dic3—28no—
304C3×Dic3—24no—
305C3×Dic3—38no—
306C3×Dic3—64no—
307C3×Dic3D3×D418yes—
308C22×Q8—272no—
309C4⋊Q8—31no—
310C4⋊Q8—271no—
311C4⋊Q8—271no—
312C22⋊Q8—272no—
313C22⋊Q8—272no—
314C22⋊Q8—272no—
315C4×Q8—271no—
316C4×D4—91no—
317C4×D4—271no—
318C4×D4—271no—
319C33D4×Q811yescharacteristic, complemented, Sylow
320C23×C3—92no—
321C22×D3—38no—
322C3×Q8—616no—
323C3×D4—94no—
324C2×C3×C4—24no—
325C2×C3×C4—28no—
326C2×C3×C4—34no—
327C2×C3×C4—34no—
328C2×C3×C4—38no—
329C2×C3×C4—91no—
330C2×C3×C4—91no—
331C2×C3×C4—184no—
332C2×C3×C4C2×D3218yes—
333C2×C3×C4C2×D3214yes—
334C2×C3×C4C22×C3⋊D312yes—
335C2×C3×C4C22×C3⋊D311yescharacteristic
336C2×Dic3—34no—
337C2×Dic3—34no—
338C2×Dic3—94no—
339C2×Dic3—91no—
340C2×Dic3—94no—
341C2×Dic3—98no—
342C2×Dic3—91no—
343C2×Dic3—92no—
344C2×Dic3—184no—
345C2×Dic3—184no—
346C2×Dic3—188no—
347C2×Dic3—184no—
348C2×Dic3—184no—
349C2×Dic3C22×C3⋊D311yescharacteristic
350C2×Dic3C22×C3⋊D312yes—
351C2×Dic3C22×C3⋊D311yescharacteristic
352D12—316no—
353C4×D3—616no—
354C4×D3—1816no—
355C4×D3—1816no—
356Dic6—24no—
357Dic6—916no—
358Dic6—94no—
359Dic6—184no—
360Dic6—1816no—
361Dic6—184no—
362Dic6—1816no—
363Dic6—3616no—
364C2×C32—24no—
365C2×C32—24no—
366C2×C32—24no—
367C2×C32C22×Dic611yescharacteristic
368C2×C32C2×D3×D414yes—
369C2×C32C2×D3×D411yescharacteristic
370C2×C32C2×D3×Q814yes—
371C2×C32#100714yes—
372C2×C32#101011yescharacteristic
373C3⋊D3—24no—
374C3×D3—616no—
375C2×Q8—32no—
376C2×Q8—271no—
377C2×Q8—272no—
378C2×Q8—272no—
379C2×Q8—548no—
380C2×D4—91no—
381C22×C4—92no—
382C22×C4—272no—
383C22×C4—272no—
384C4⋊C4—31no—
385C4⋊C4—32no—
386C4⋊C4—271no—
387C4⋊C4—271no—
388C4⋊C4—271no—
389C4⋊C4—272no—
390C4⋊C4—271no—
391C4⋊C4—272no—
392C4⋊C4D3×C3⋊D311yescharacteristic
393C22⋊C4—92no—
394C22⋊C4—272no—
395C22⋊C4—272no—
396C42—31no—
397C42—91no—
398C42—271no—
399C22×C3—24no—
400C22×C3—94no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3C22×D3214yes—
404C22×C3C23×C3⋊D311yescharacteristic
405C2×D3—316no—
406C2×D3—68no—
407C2×D3—68no—
408C3×C4—28no—
409C3×C4—22no—
410C3×C4—28no—
411C3×C4—38no—
412C3×C4—48no—
413C3×C4—68no—
414C3×C4—64no—
415C3×C4—92no—
416C3×C4—181no—
417C3×C4D3×Dic618yescomplemented
418C3×C4Q8×C3⋊D312yescomplemented
419Dic3—22no—
420Dic3—21no—
421Dic3—38no—
422Dic3—64no—
423Dic3—98no—
424Dic3—92no—
425Dic3—188no—
426Dic3—188no—
427Dic3—188no—
428Dic3—184no—
429Dic3—182no—
430Dic3—181no—
431Dic3—368no—
432Dic3D4×C3⋊D312yes—
433C32—24no—
434C32D4×Dic611yescharacteristic, complemented
435C32#572714yescomplemented
436C23—92no—
437Q8—64no—
438Q8—274no—
439Q8—544no—
440Q8—544no—
441D4—94no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—32no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—184no—
448C2×C4—271no—
449C2×C4—271no—
450C2×C4—272no—
451C2×C4—544no—
452C2×C4—544no—
453C2×C4C2×D3×C3⋊D312yes—
454C2×C4C2×D3×C3⋊D311yescharacteristic
455C2×C3—24no—
456C2×C3—24no—
457C2×C3—24no—
458C2×C3—184no—
459C2×C3C2×D3×Dic614yes—
460C2×C3D32×D414yes—
461C2×C3#6431514yes—
462C2×C3C2×D4×C3⋊D311yescharacteristic
463C2×C3C2×Q8×C3⋊D311yescharacteristic
464C2×C3#6436411yescharacteristic
465D3—616no—
466C22—94no—
467C22—182no—
468C22—182no—
469C22C22×D3×C3⋊D311yescharacteristic
470C4—22no—
471C4—32no—
472C4—62no—
473C4—61no—
474C4—92no—
475C4—181no—
476C4—272no—
477C4—542no—
478C4—541no—
479C4C3⋊D3×Dic612yescomplemented
480C3—24no—
481C3#1059437614yescomplemented
482C3#1059619811yescharacteristic, complemented
483C2—184no—
484C2C2×C3⋊D3×Dic611yescharacteristic
485C2D3×D4×C3⋊D311yescharacteristic
486C2#1172459211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33D4×Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×Dic6C2semidirect
2. #11723929C2semidirect
4. C32×#888C2semidirect
5. #11723535C2semidirect
6. C4×#6293C2semidirect
8. #11723497C2semidirect
10. #11723476C2semidirect
11. #11723418C2semidirect
53. C3×#888D3semidirect
70. C3×C4⋊(C3⋊D3)Q8semidirect
77. #6293D4semidirect
138. C2×C3×C3⋊D3C2×Q8semidirect
140. C33×C4C2×Q8semidirect
165. #888C3⋊D3semidirect
195. C4⋊(C3⋊D3)Dic6semidirect
279. C2×C3⋊D3C2×Dic6semidirect
290. C32×C4C2×Dic6semidirect
291. C32×C4D3×Q8semidirect
319. C33D4×Q8semidirect
417. C3×C4D3×Dic6semidirect
418. C3×C4Q8×C3⋊D3semidirect
434. C32D4×Dic6semidirect
435. C32#5727semidirect
479. C4C3⋊D3×Dic6semidirect
481. C3#10594376semidirect
482. C3#10596198semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
#117239292
C3×#641022
C32×#8882
#117235352
C4×#62932
#117235012
#117234972
Dic3×C4⋊(C3⋊D3)2
#117234762
#117234182
#105961983
#105943763