Finite Group Databasebeta

← #13178707

Subgroups — #13178707

Order 1728.

1770 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1770 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#3830C211yesmaximal, characteristic, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic, complemented
3C3×#64149C212yesmaximal, complemented
4#11723729C214yesmaximal, complemented
5#11723717C214yesmaximal, complemented
6#11723677C212yesmaximal, complemented
7#11723673C211yesmaximal, characteristic, complemented
8#10595309—31nomaximal
9#10595303—32nomaximal
10C22×#1113C2212yes—
11C2×C33×D4C2211yescharacteristic, complemented
12C2×C3×#638C2212yes—
13C2×C3×#638C2218yescomplemented
14C22×C3×C3⋊Dic3C2212yescomplemented
15C3×#3830—24no—
16C2×C3×C4⋊(C3⋊D3)C2212yes—
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic, complemented
19C2×#6295C2212yes—
20C4⋊#1113—24no—
21C3×C4×C3⋊Dic3C2212yescomplemented
22#88815C2212yescomplemented
23#88814C2212yes—
24#88814C2218yescomplemented
25C2×#3830D311yescharacteristic, complemented
26C2×C3×#237—32no—
27C2×C3×#237—61no—
28C2×#3802—31no—
29C2×D3×D12—32no—
30#64149D312yescomplemented
31C3×#940—32no—
32C3×#940—32no—
33C3×#940—62no—
34#63983—34no—
35#63983—38no—
36#63980—34no—
37#63978—38no—
38#63874—31no—
39#63864—32no—
40#63863—32no—
41#63860—32no—
42#63855—32no—
43C23×C33C2312yes—
44C22×C3×C3⋊D3C2314yes—
45C22×C3×C3⋊D3C2311yescharacteristic
46C2×#1113—24no—
47C2×#1113—44no—
48C33×D4—22no—
49C2×C33×C4C2311yescharacteristic
50C3×#638—24no—
51C3×#638—48no—
52C2×C3×C3⋊Dic3—24no—
53C2×C3×C3⋊Dic3C2314yes—
54C2×C3×C3⋊Dic3C2312yes—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C3×C4⋊(C3⋊D3)—24no—
57C3×C4×C3⋊D3D414yescomplemented
58C3×#634—22no—
59#6295—24no—
60#5830—91no—
61#5715—92no—
62C22×D32—34no—
63C22×D32—32no—
64C2×C32×D4—21no—
65C2×C32×D4—22no—
66C2×C32×D4—41no—
67C2×C32×D4C2×D311yescharacteristic, complemented
68C2×C32×D4C2×D312yescomplemented
69C2×#638C2×D312yes—
70C2×#638C2×D318yescomplemented
71C22×C3⋊Dic3C2×D312yescomplemented
72#3830—24no—
73C2×C4⋊(C3⋊D3)C2×D312yes—
74C2×C4×C3⋊D3C2×D311yescharacteristic
75C2×#634C2×D311yescharacteristic, complemented
76C2×C3×#67—34no—
77C2×C3×#67—38no—
78C2×C3×#67—38no—
79C2×C3×#67—62no—
80C2×C3×#67—68no—
81C22×C3×Dic3—34no—
82C22×C3×Dic3—62no—
83C3×#237—68no—
84C3×#237—68no—
85C2×C3×D12—32no—
86C2×C3×D12—32no—
87C2×C3×D12—62no—
88C2×C3×C4×D3—32no—
89C2×C3×C4×D3—61no—
90C2×C3×Dic6—32no—
91C2×C3×Dic6—61no—
92C2×#626—34no—
93C2×#625—32no—
94#3802—64no—
95D3×D12—68no—
96C4×C3⋊Dic3C2×D312yescomplemented
97C3×C4×Dic3—32no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—62no—
100#3724—32no—
101Dic3⋊Dic3—34no—
102#3722—38no—
103#3722—32no—
104(C2×D3)⋊Dic3—34no—
105(C2×D3)⋊Dic3—38no—
106(C2×D3)⋊Dic3—38no—
107C22×C33—22no—
108C22×C33—24no—
109C22×C33C2411yescharacteristic
110C2×C3×C3⋊D3—22no—
111C2×C3×C3⋊D3—24no—
112C2×C3×C3⋊D3—28no—
113C2×C3×C3⋊D3C2×D412yes—
114#1113—48no—
115C33×C4C2×D412yes—
116C3×C3⋊Dic3—22no—
117C3×C3⋊Dic3—24no—
118C3×C3⋊Dic3C2×D412yes—
119C2×C3×#41—91no—
120C2×#237—32no—
121C2×#237—61no—
122C2×D3×D4—92no—
123C22×D12—91no—
124C3×#127—92no—
125#947—92no—
126#940—32no—
127#940—32no—
128#940—62no—
129#939—98no—
130#937—98no—
131#937—94no—
132#936—92no—
133#930—94no—
134#926—91no—
135#875—92no—
136#872—92no—
137C23×C32—22no—
138C23×C32—24no—
139C23×C32—42no—
140C23×C32C22×D314yes—
141C23×C32C22×D312yes—
142C22×C3⋊D3C22×D314yes—
143C22×C3⋊D3C22×D311yescharacteristic
144C22×C3×D3—32no—
145C22×C3×D3—34no—
146C22×C3×D3—34no—
147C22×C3×D3—61no—
148C22×C3×D3—64no—
149C2×D32—68no—
150C2×D32—64no—
151C2×D32—128no—
152C2×D32—124no—
153C32×D4—22no—
154C32×D4—24no—
155C32×D4—24no—
156C32×D4—44no—
157C32×D4—44no—
158C2×C32×C4—22no—
159C2×C32×C4—21no—
160C2×C32×C4—41no—
161C2×C32×C4C22×D312yes—
162C2×C32×C4C22×D311yescharacteristic
163#638—24no—
164#638—48no—
165C2×C3⋊Dic3—24no—
166C2×C3⋊Dic3C22×D312yes—
167C2×C3⋊Dic3C22×D314yes—
168C2×C3⋊Dic3C22×D311yescharacteristic
169C4⋊(C3⋊D3)—24no—
170C4×C3⋊D3#6714yescomplemented
171#634—22no—
172C3×#67—68no—
173C3×#67—68no—
174C3×#67—128no—
175C3×#67—128no—
176C3×#67—1216no—
177C2×C3×Dic3—32no—
178C2×C3×Dic3—34no—
179C2×C3×Dic3—34no—
180C2×C3×Dic3—34no—
181C2×C3×Dic3—64no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—61no—
184C2×C3×Dic3—68no—
185C2×C3×Dic3—68no—
186C3×D12—64no—
187C3×D12—64no—
188C3×D12—124no—
189C3×C4×D3—38no—
190C3×C4×D3—64no—
191C3×Dic6—64no—
192C3×Dic6—64no—
193#626—68no—
194#625—64no—
195#535—271noSylow
196C2×C33—42no—
197C2×C33C22×D411yescharacteristic
198C2×C33H(5,𝔽2)12yes—
199C3×C3⋊D3—22no—
200C3×C3⋊D3—44no—
201C23×D3—92no—
202C23×D3—94no—
203C3×#41—184no—
204C2×C3×Q8—91no—
205C2×C3×D4—21no—
206C2×C3×D4—22no—
207C2×C3×D4—41no—
208C2×C3×D4—92no—
209C2×C3×D4—98no—
210C2×C3×D4—92no—
211C2×C3×D4D3211yescharacteristic, complemented
212C2×C3×D4D3212yescomplemented
213C22×C3×C4—91no—
214C22×C3×C4—92no—
215C2×#67—34no—
216C2×#67—38no—
217C2×#67—38no—
218C2×#67—62no—
219C2×#67—68no—
220C2×#67—94no—
221C22×Dic3—34no—
222C22×Dic3—62no—
223#237—68no—
224#237—68no—
225D3×D4—188no—
226C2×D12—32no—
227C2×D12—32no—
228C2×D12—62no—
229C2×D12—92no—
230C2×D12—184no—
231C2×C4×D3—32no—
232C2×C4×D3—61no—
233C2×Dic6—32no—
234C2×Dic6—61no—
235C3×C42—92no—
236#217—92no—
237#217—98no—
238(C2×D3)⋊C4—94no—
239(C2×D3)⋊C4—98no—
240(C2×D3)⋊C4—98no—
241C4⋊Dic3—92no—
242Dic3⋊C4—94no—
243C4×Dic3—32no—
244C4×Dic3—32no—
245C4×Dic3—62no—
246C22×C32—22no—
247C22×C32—24no—
248C22×C32—24no—
249C22×C32—28no—
250C22×C32—24no—
251C22×C32—21no—
252C22×C32—22no—
253C22×C32—41no—
254C22×C32—44no—
255C22×C32—44no—
256C22×C32—44no—
257C22×C32—48no—
258C22×C32—84no—
259C22×C32C23×D312yes—
260C22×C32C23×D311yescharacteristic
261C2×C3⋊D3—22no—
262C2×C3⋊D3—24no—
263C2×C3⋊D3—28no—
264C2×C3⋊D3C2×#6712yes—
265C2×C3×D3—34no—
266C2×C3×D3—62no—
267C2×C3×D3—64no—
268C2×C3×D3—64no—
269C2×C3×D3—68no—
270C2×C3×D3—68no—
271C2×C3×D3—64no—
272C2×C3×D3—122no—
273C2×C3×D3—124no—
274C2×C3×D3—128no—
275D32—1216no—
276D32—128no—
277C32×C4—24no—
278C32×C4—22no—
279C32×C4—42no—
280C32×C4D3×D414yes—
281C32×C4C2×#6712yes—
282C3⋊Dic3—24no—
283C3⋊Dic3—22no—
284C3⋊Dic3C2×#6712yes—
285C3×Dic3—34no—
286C3×Dic3—62no—
287C3×Dic3—64no—
288C3×Dic3—68no—
289C3×Dic3—64no—
290C3×Dic3—64no—
291C3×Dic3—64no—
292C2×#41—91no—
293C22×D4—271no—
294#127—92no—
295#124—272no—
296#121—274no—
297#120—274no—
298#117—271no—
299C33#53511yescharacteristic, complemented, Sylow
300C23×C3—22no—
301C23×C3—24no—
302C23×C3—42no—
303C23×C3—91no—
304C23×C3—94no—
305C23×C3C2×D3214yes—
306C23×C3C2×D3212yes—
307C22×D3—32no—
308C22×D3—34no—
309C22×D3—34no—
310C22×D3—61no—
311C22×D3—64no—
312C22×D3—188no—
313C22×D3—184no—
314C22×D3—368no—
315C22×D3—364no—
316C3×Q8—182no—
317C3×D4—24no—
318C3×D4—24no—
319C3×D4—22no—
320C3×D4—44no—
321C3×D4—44no—
322C3×D4—184no—
323C3×D4—184no—
324C3×D4—368no—
325C2×C3×C4—22no—
326C2×C3×C4—21no—
327C2×C3×C4—41no—
328C2×C3×C4—91no—
329C2×C3×C4—94no—
330C2×C3×C4—94no—
331C2×C3×C4—92no—
332C2×C3×C4—184no—
333C2×C3×C4C2×D3212yes—
334C2×C3×C4C2×D3211yescharacteristic
335#67—68no—
336#67—68no—
337#67—128no—
338#67—128no—
339#67—1216no—
340#67—188no—
341C2×Dic3—32no—
342C2×Dic3—34no—
343C2×Dic3—34no—
344C2×Dic3—34no—
345C2×Dic3—64no—
346C2×Dic3—62no—
347C2×Dic3—61no—
348C2×Dic3—68no—
349C2×Dic3—68no—
350D12—64no—
351D12—64no—
352D12—124no—
353D12—184no—
354C4×D3—38no—
355C4×D3—64no—
356Dic6—64no—
357Dic6—64no—
358C2×C32—22no—
359C2×C32—24no—
360C2×C32—22no—
361C2×C32—21no—
362C2×C32—41no—
363C2×C32—42no—
364C2×C32—42no—
365C2×C32—44no—
366C2×C32—44no—
367C2×C32—84no—
368C2×C32—84no—
369C2×C32C2×D3×D412yes—
370C2×C32#100414yes—
371C2×C32#100912yes—
372C2×C32C22×#6711yescharacteristic
373C3⋊D3—22no—
374C3⋊D3—44no—
375C3×D3—64no—
376C3×D3—122no—
377C3×D3—124no—
378C3×D3—124no—
379C3×D3—244no—
380C24—272no—
381#41—184no—
382C2×Q8—91no—
383C2×D4—92no—
384C2×D4—98no—
385C2×D4—92no—
386C2×D4—272no—
387C2×D4—544no—
388C2×D4#111311yescharacteristic, complemented
389C22×C4—91no—
390C22×C4—92no—
391C4⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—278no—
394C42—92no—
395C22×C3—24no—
396C22×C3—24no—
397C22×C3—28no—
398C22×C3—24no—
399C22×C3—22no—
400C22×C3—21no—
401C22×C3—22no—
402C22×C3—44no—
403C22×C3—48no—
404C22×C3—44no—
405C22×C3—44no—
406C22×C3—41no—
407C22×C3—84no—
408C22×C3—92no—
409C22×C3—182no—
410C22×C3—188no—
411C22×C3—184no—
412C22×C3C22×D3211yescharacteristic
413C22×C3C22×D3212yes—
414C2×D3—34no—
415C2×D3—62no—
416C2×D3—64no—
417C2×D3—68no—
418C2×D3—68no—
419C2×D3—64no—
420C2×D3—64no—
421C2×D3—122no—
422C2×D3—124no—
423C2×D3—128no—
424C2×D3—3616no—
425C2×D3—368no—
426C3×C4—22no—
427C3×C4—24no—
428C3×C4—42no—
429C3×C4—92no—
430C3×C4—184no—
431C3×C4—182no—
432C3×C4D3×#6714yes—
433C3×C4#381112yes—
434Dic3—34no—
435Dic3—62no—
436Dic3—64no—
437Dic3—68no—
438Dic3—64no—
439Dic3—64no—
440Dic3—64no—
441C32—22no—
442C32—21no—
443C32—41no—
444C32#571512yescomplemented
445C32#583011yescharacteristic, complemented
446C23—91no—
447C23—94no—
448C23—544no—
449C23—1084no—
450C23C2×#111312yes—
451Q8—182no—
452D4—22no—
453D4—184no—
454D4—184no—
455D4—368no—
456D4—544no—
457C2×C4—91no—
458C2×C4—94no—
459C2×C4—94no—
460C2×C4—92no—
461C2×C4—184no—
462C2×C4C2×#111311yescharacteristic
463C2×C3—24no—
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—22no—
467C2×C3—44no—
468C2×C3—44no—
469C2×C3—42no—
470C2×C3—42no—
471C2×C3—41no—
472C2×C3—84no—
473C2×C3—84no—
474C2×C3—182no—
475C2×C3—364no—
476C2×C3#6431312yes—
477C2×C3#6431414yes—
478C2×C3C2×D3×#6712yes—
479C2×C3C2×#381111yescharacteristic
480D3—64no—
481D3—122no—
482D3—124no—
483D3—124no—
484D3—244no—
485C22—24no—
486C22—22no—
487C22—92no—
488C22—182no—
489C22—188no—
490C22—184no—
491C22—1088no—
492C22C22×#111311yescharacteristic
493C4—92no—
494C4—184no—
495C4—182no—
496C4#8904912yes—
497C3—21no—
498C3—22no—
499C3—41no—
500C3#1059530312yescomplemented
501C3#1059530911yescharacteristic, complemented
502C2—42no—
503C2—182no—
504C2—364no—
505C2#1172462312yes—
506C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
hypercenterC2×D4#1113
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3830C2semidirect
2. C2×C4⋊#1113C2semidirect
3. C3×#64149C2semidirect
4. #11723729C2semidirect
5. #11723717C2semidirect
6. #11723677C2semidirect
7. #11723673C2semidirect
11. C2×C33×D4C22semidirect
13. C2×C3×#638C22semidirect
14. C22×C3×C3⋊Dic3C22semidirect
18. C2×C3×#634C22semidirect
21. C3×C4×C3⋊Dic3C22semidirect
22. #88815C22semidirect
24. #88814C22semidirect
25. C2×#3830D3semidirect
30. #64149D3semidirect
57. C3×C4×C3⋊D3D4semidirect
67. C2×C32×D4C2×D3semidirect
68. C2×C32×D4C2×D3semidirect
70. C2×#638C2×D3semidirect
71. C22×C3⋊Dic3C2×D3semidirect
75. C2×#634C2×D3semidirect
96. C4×C3⋊Dic3C2×D3semidirect
170. C4×C3⋊D3#67semidirect
211. C2×C3×D4D32semidirect
212. C2×C3×D4D32semidirect
299. C33#535semidirect
388. C2×D4#1113semidirect
444. C32#5715semidirect
445. C32#5830semidirect
500. C3#10595303semidirect
501. C3#10595309semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38302
C2×C4⋊#11132
C3×#641492
#117237292
#117237172
#117236772
#117236732
#105953093
#105953033