Finite Group Databasebeta

← #13178761

Subgroups — #13178761

Order 1728.

1690 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1690 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×#3832C211yesmaximal, characteristic, complemented
2C2×C4⋊#1113C211yesmaximal, characteristic, complemented
3C3×#64156C212yesmaximal, complemented
4#11723701C214yesmaximal, complemented
5#11723697C212yesmaximal, complemented
6#11723695C214yesmaximal, complemented
7#11723673C211yesmaximal, characteristic, complemented
8#10595433—31nomaximal
9#10595426—32nomaximal
10C22×#1113C2212yes—
11C2×C33×Q8C2211yescharacteristic, complemented
12C3×#3832—24no—
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C2×C3×C4⋊(C3⋊D3)C2212yes—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2212yescomplemented
18C2×#6295C2212yes—
19C2×#6295C2218yescomplemented
20C4⋊#1113—24no—
21C3×#3752C2218yescomplemented
22C3×C4×C3⋊Dic3C2212yescomplemented
23#88815C2212yescomplemented
24#88814C2212yes—
25C2×#3832D311yescharacteristic, complemented
26C2×C3×#239—32no—
27C2×C3×#239—61no—
28C2×#3802—31no—
29C2×D3×D12—32no—
30#64156D312yescomplemented
31C3×#947—32no—
32C3×#947—32no—
33C3×#947—62no—
34#63923—34no—
35#63922—38no—
36#63914—32no—
37#63912—34no—
38#63911—32no—
39#63911—32no—
40#63906—38no—
41#63874—31no—
42#63855—32no—
43C22×C3×C3⋊D3C2314yes—
44C22×C3×C3⋊D3C2312yes—
45C22×C3×C3⋊D3C2311yescharacteristic
46C2×#1113—24no—
47C2×#1113—44no—
48C33×Q8—22no—
49C2×C33×C4C2312yes—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3C2314yes—
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C3×C4⋊(C3⋊D3)—24no—
54C3×C4⋊(C3⋊D3)—22no—
55C3×C4⋊(C3⋊D3)—24no—
56C3×C4×C3⋊D3—24no—
57C3×C4×C3⋊D3D414yescomplemented
58#6295—24no—
59#6295—48no—
60#5830—91no—
61#5674—92no—
62C22×D32—34no—
63C22×D32—32no—
64C2×C32×Q8—21no—
65C2×C32×Q8—22no—
66C2×C32×Q8—41no—
67C2×C32×Q8C2×D311yescharacteristic, complemented
68C2×C32×Q8C2×D312yescomplemented
69#3832—24no—
70C2×C4⋊(C3⋊D3)C2×D312yes—
71C2×C4⋊(C3⋊D3)C2×D312yes—
72C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
73C2×C4×C3⋊D3C2×D311yescharacteristic
74C2×C4×C3⋊D3C2×D312yescomplemented
75C3×#239—68no—
76C3×#239—68no—
77C2×C3×D12—32no—
78C2×C3×D12—32no—
79C2×C3×D12—32no—
80C2×C3×D12—34no—
81C2×C3×D12—62no—
82C2×C3×D12—61no—
83C2×C3×D12—62no—
84C2×C3×C4×D3—32no—
85C2×C3×C4×D3—34no—
86C2×C3×C4×D3—61no—
87C2×C3×C4×D3—62no—
88C2×#626—34no—
89C2×#626—38no—
90C2×#626—38no—
91C2×#625—32no—
92C2×#625—38no—
93#3802—64no—
94D3×D12—68no—
95#3752C2×D318yescomplemented
96C4×C3⋊Dic3C2×D312yescomplemented
97C3×(C2×D3)⋊C4—38no—
98C3×(C2×D3)⋊C4—38no—
99C3×(C2×D3)⋊C4—68no—
100C3×C4×Dic3—32no—
101C3×C4×Dic3—32no—
102C3×C4×Dic3—62no—
103#3724—32no—
104Dic3⋊Dic3—34no—
105#3722—32no—
106(C2×D3)⋊Dic3—34no—
107C22×C33C2411yescharacteristic
108C2×C3×C3⋊D3—22no—
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3—24no—
111C2×C3×C3⋊D3—24no—
112C2×C3×C3⋊D3—28no—
113C2×C3×C3⋊D3C2×D412yes—
114#1113—48no—
115C33×C4—22no—
116C33×C4C2×D412yes—
117C3×C3⋊Dic3—24no—
118C3×C3⋊Dic3C2×D412yes—
119C2×C3×#41—91no—
120C2×#239—32no—
121C2×#239—61no—
122C2×D3×D4—92no—
123C22×D12—91no—
124C3×#124—92no—
125#947—32no—
126#947—32no—
127#947—62no—
128#940—92no—
129#940—92no—
130#937—94no—
131#930—94no—
132#926—91no—
133#884—98no—
134#882—98no—
135#874—92no—
136#872—92no—
137C22×C3⋊D3C22×D314yes—
138C22×C3⋊D3C22×D312yes—
139C22×C3⋊D3C22×D311yescharacteristic
140C22×C3×D3—32no—
141C22×C3×D3—34no—
142C22×C3×D3—34no—
143C22×C3×D3—34no—
144C22×C3×D3—61no—
145C22×C3×D3—62no—
146C22×C3×D3—64no—
147C2×D32—68no—
148C2×D32—64no—
149C2×D32—128no—
150C2×D32—124no—
151C32×Q8—22no—
152C32×Q8—24no—
153C32×Q8—24no—
154C32×Q8—44no—
155C32×Q8—44no—
156C2×C32×C4—22no—
157C2×C32×C4—22no—
158C2×C32×C4—21no—
159C2×C32×C4—24no—
160C2×C32×C4—41no—
161C2×C32×C4—42no—
162C2×C32×C4C22×D312yes—
163C2×C32×C4C22×D314yes—
164C2×C32×C4C22×D312yes—
165C2×C32×C4C22×D311yescharacteristic
166C2×C3⋊Dic3C22×D314yes—
167C2×C3⋊Dic3C22×D311yescharacteristic
168C4⋊(C3⋊D3)—24no—
169C4⋊(C3⋊D3)—22no—
170C4⋊(C3⋊D3)—24no—
171C4×C3⋊D3—24no—
172C4×C3⋊D3#6714yescomplemented
173C2×C3×Dic3—32no—
174C2×C3×Dic3—34no—
175C2×C3×Dic3—34no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—64no—
178C3×D12—68no—
179C3×D12—68no—
180C3×D12—64no—
181C3×D12—64no—
182C3×D12—64no—
183C3×D12—64no—
184C3×D12—124no—
185C3×C4×D3—38no—
186C3×C4×D3—64no—
187C3×C4×D3—68no—
188C3×C4×D3—68no—
189#626—68no—
190#626—128no—
191#626—128no—
192#625—64no—
193#625—128no—
194#535—271noSylow
195C2×C33C22×D411yescharacteristic
196C2×C33H(5,𝔽2)12yes—
197C3×C3⋊D3—22no—
198C3×C3⋊D3—42no—
199C3×C3⋊D3—44no—
200C23×D3—94no—
201C23×D3—92no—
202C3×#41—184no—
203C2×C3×Q8—21no—
204C2×C3×Q8—22no—
205C2×C3×Q8—41no—
206C2×C3×Q8D3211yescharacteristic, complemented
207C2×C3×Q8D3212yescomplemented
208C2×C3×D4—92no—
209C2×C3×D4—91no—
210C2×C3×D4—92no—
211C22×C3×C4—91no—
212C22×C3×C4—92no—
213C2×#67—94no—
214C2×#67—98no—
215C2×#67—98no—
216#239—68no—
217#239—68no—
218D3×D4—188no—
219C2×D12—32no—
220C2×D12—32no—
221C2×D12—34no—
222C2×D12—32no—
223C2×D12—62no—
224C2×D12—61no—
225C2×D12—62no—
226C2×D12—92no—
227C2×D12—98no—
228C2×D12—184no—
229C2×C4×D3—32no—
230C2×C4×D3—34no—
231C2×C4×D3—61no—
232C2×C4×D3—62no—
233C3×C22⋊C4—98no—
234C3×C42—92no—
235#217—92no—
236(C2×D3)⋊C4—38no—
237(C2×D3)⋊C4—38no—
238(C2×D3)⋊C4—68no—
239(C2×D3)⋊C4—94no—
240C4⋊Dic3—92no—
241Dic3⋊C4—94no—
242C4×Dic3—32no—
243C4×Dic3—32no—
244C4×Dic3—62no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—41no—
248C22×C32C23×D312yes—
249C22×C32C23×D311yescharacteristic
250C2×C3⋊D3—22no—
251C2×C3⋊D3—22no—
252C2×C3⋊D3—24no—
253C2×C3⋊D3—24no—
254C2×C3⋊D3—28no—
255C2×C3⋊D3C2×#6712yes—
256C2×C3×D3—34no—
257C2×C3×D3—62no—
258C2×C3×D3—64no—
259C2×C3×D3—64no—
260C2×C3×D3—64no—
261C2×C3×D3—64no—
262C2×C3×D3—68no—
263C2×C3×D3—68no—
264C2×C3×D3—68no—
265C2×C3×D3—64no—
266C2×C3×D3—122no—
267C2×C3×D3—124no—
268C2×C3×D3—124no—
269C2×C3×D3—128no—
270D32—1216no—
271D32—128no—
272C32×C4—22no—
273C32×C4—24no—
274C32×C4—24no—
275C32×C4—24no—
276C32×C4—22no—
277C32×C4—42no—
278C32×C4—44no—
279C32×C4—44no—
280C32×C4D3×D414yes—
281C32×C4C2×#6712yes—
282C3⋊Dic3—24no—
283C3⋊Dic3C2×#6712yes—
284C3×Dic3—34no—
285C3×Dic3—62no—
286C3×Dic3—64no—
287C3×Dic3—64no—
288C3×Dic3—124no—
289C2×#41—91no—
290C22×D4—271no—
291#127—272no—
292#124—92no—
293#121—274no—
294#120—274no—
295#117—271no—
296C33#53511yescharacteristic, complemented, Sylow
297C23×C3—91no—
298C23×C3—94no—
299C23×C3—92no—
300C22×D3—32no—
301C22×D3—34no—
302C22×D3—34no—
303C22×D3—34no—
304C22×D3—61no—
305C22×D3—62no—
306C22×D3—64no—
307C22×D3—188no—
308C22×D3—184no—
309C22×D3—368no—
310C22×D3—364no—
311C3×Q8—24no—
312C3×Q8—24no—
313C3×Q8—22no—
314C3×Q8—44no—
315C3×Q8—44no—
316C3×D4—184no—
317C3×D4—184no—
318C3×D4—182no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—21no—
322C2×C3×C4—24no—
323C2×C3×C4—42no—
324C2×C3×C4—41no—
325C2×C3×C4—91no—
326C2×C3×C4—94no—
327C2×C3×C4—94no—
328C2×C3×C4—184no—
329C2×C3×C4C2×D3212yes—
330C2×C3×C4C2×D3214yes—
331C2×C3×C4C2×D3212yes—
332C2×C3×C4C2×D3211yescharacteristic
333#67—188no—
334#67—368no—
335#67—368no—
336C2×Dic3—32no—
337C2×Dic3—34no—
338C2×Dic3—34no—
339C2×Dic3—61no—
340C2×Dic3—64no—
341D12—68no—
342D12—68no—
343D12—64no—
344D12—64no—
345D12—64no—
346D12—64no—
347D12—124no—
348D12—184no—
349D12—368no—
350C4×D3—38no—
351C4×D3—64no—
352C4×D3—68no—
353C4×D3—68no—
354C2×C32—24no—
355C2×C32—22no—
356C2×C32—22no—
357C2×C32—21no—
358C2×C32—41no—
359C2×C32—42no—
360C2×C32C2×D3×D412yes—
361C2×C32#100914yes—
362C2×C32#100912yes—
363C2×C32C22×#6711yescharacteristic
364C3⋊D3—22no—
365C3⋊D3—42no—
366C3⋊D3—44no—
367C3×D3—64no—
368C3×D3—122no—
369C3×D3—124no—
370C3×D3—124no—
371C3×D3—124no—
372C3×D3—124no—
373C3×D3—244no—
374C24—272no—
375#41—184no—
376C2×Q8#111311yescharacteristic, complemented
377C2×D4—92no—
378C2×D4—92no—
379C2×D4—91no—
380C2×D4—272no—
381C2×D4—278no—
382C2×D4—544no—
383C22×C4—91no—
384C22×C4—92no—
385C4⋊C4—272no—
386C22⋊C4—98no—
387C22⋊C4—272no—
388C42—92no—
389C22×C3—22no—
390C22×C3—21no—
391C22×C3—41no—
392C22×C3—92no—
393C22×C3—182no—
394C22×C3—184no—
395C22×C3—188no—
396C22×C3—184no—
397C22×C3—182no—
398C22×C3C22×D3212yes—
399C22×C3C22×D3211yescharacteristic
400C2×D3—34no—
401C2×D3—62no—
402C2×D3—64no—
403C2×D3—64no—
404C2×D3—68no—
405C2×D3—68no—
406C2×D3—68no—
407C2×D3—64no—
408C2×D3—64no—
409C2×D3—64no—
410C2×D3—122no—
411C2×D3—124no—
412C2×D3—124no—
413C2×D3—128no—
414C2×D3—3616no—
415C2×D3—368no—
416C3×C4—24no—
417C3×C4—24no—
418C3×C4—22no—
419C3×C4—22no—
420C3×C4—24no—
421C3×C4—44no—
422C3×C4—44no—
423C3×C4—42no—
424C3×C4—92no—
425C3×C4—184no—
426C3×C4D3×#6714yes—
427C3×C4#381112yes—
428Dic3—34no—
429Dic3—62no—
430Dic3—64no—
431Dic3—64no—
432Dic3—124no—
433C32—22no—
434C32—21no—
435C32—41no—
436C32#567412yescomplemented
437C32#583011yescharacteristic, complemented
438C23—91no—
439C23—94no—
440C23—92no—
441C23—544no—
442C23—1084no—
443Q8—22no—
444D4—184no—
445D4—184no—
446D4—182no—
447D4—544no—
448D4—1088no—
449C2×C4—91no—
450C2×C4—94no—
451C2×C4—94no—
452C2×C4—184no—
453C2×C4C2×#111312yes—
454C2×C4C2×#111311yescharacteristic
455C2×C3—24no—
456C2×C3—22no—
457C2×C3—21no—
458C2×C3—22no—
459C2×C3—42no—
460C2×C3—41no—
461C2×C3—182no—
462C2×C3—364no—
463C2×C3—362no—
464C2×C3#6432014yes—
465C2×C3#6432012yes—
466C2×C3C2×D3×#6712yes—
467C2×C3C2×#381111yescharacteristic
468D3—64no—
469D3—122no—
470D3—124no—
471D3—124no—
472D3—124no—
473D3—124no—
474D3—244no—
475C22—92no—
476C22—182no—
477C22—184no—
478C22—188no—
479C22—182no—
480C22—184no—
481C22—1088no—
482C22C22×#111311yescharacteristic
483C4—22no—
484C4—92no—
485C4—184no—
486C4#8904912yes—
487C3—21no—
488C3—22no—
489C3—41no—
490C3#1059542612yescomplemented
491C3#1059543311yescharacteristic, complemented
492C2—182no—
493C2—364no—
494C2—362no—
495C2#1172462712yes—
496C2C2×#8904911yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33#535
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×#3832C2semidirect
2. C2×C4⋊#1113C2semidirect
3. C3×#64156C2semidirect
4. #11723701C2semidirect
5. #11723697C2semidirect
6. #11723695C2semidirect
7. #11723673C2semidirect
11. C2×C33×Q8C22semidirect
15. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C2×C3×C4×C3⋊D3C22semidirect
19. C2×#6295C22semidirect
21. C3×#3752C22semidirect
22. C3×C4×C3⋊Dic3C22semidirect
23. #88815C22semidirect
25. C2×#3832D3semidirect
30. #64156D3semidirect
57. C3×C4×C3⋊D3D4semidirect
67. C2×C32×Q8C2×D3semidirect
68. C2×C32×Q8C2×D3semidirect
72. C2×C4⋊(C3⋊D3)C2×D3semidirect
74. C2×C4×C3⋊D3C2×D3semidirect
95. #3752C2×D3semidirect
96. C4×C3⋊Dic3C2×D3semidirect
172. C4×C3⋊D3#67semidirect
206. C2×C3×Q8D32semidirect
207. C2×C3×Q8D32semidirect
296. C33#535semidirect
376. C2×Q8#1113semidirect
436. C32#5674semidirect
437. C32#5830semidirect
490. C3#10595426semidirect
491. C3#10595433semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#38322
C2×C4⋊#11132
C3×#641562
#117237012
#117236972
#117236952
#117236732
#105954333
#105954263