Finite Group Databasebeta

← #13168120

Subgroups — #13168120

Order 1728.

1666 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 1666 conjugacy classes of subgroups — too many to draw. Only its 212 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⋊D3×Dic6C211yesmaximal, characteristic, complemented
2C2×C3×#3802C211yesmaximal, characteristic
3C4×#6294C211yesmaximal, characteristic, complemented
4#11723429C212yesmaximal, complemented
5C3×#63863C212yesmaximal, complemented
6#11723004C214yesmaximal, complemented
7#11722902C214yesmaximal, complemented
8#10595432—31nomaximal
9D3×#936—32nomaximal
10C22×C3×D32C2212yes—
11C2×#6308C2211yescharacteristic, complemented
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C32×D12C2212yes—
14C2×C32×Dic6C2211yescharacteristic, complemented
15C2×#6294C2212yes—
16C2×#6293C2212yescomplemented
17C2×Dic3×C3⋊D3C2212yes—
18C3⋊D3×Dic6—24no—
19C2×C3×#625C2212yes—
20C3×#3802—24no—
21C3×C4×C3⋊Dic3C2212yescomplemented
22#88813C2212yescomplemented
23(C2×D3)⋊(C3⋊Dic3)C2218yescomplemented
24C3×(C2×D3)⋊Dic3C2218yescomplemented
25C2×Q8×C3⋊D3—31no—
26C2×C3×D3×D4—32no—
27C2×#3802D311yescharacteristic
28C2×D3×Dic6—32no—
29C2×D3×Dic6—61no—
30#64145—32no—
31C3×#947—32no—
32C3×#936—32no—
33D3×#217—38no—
34#63959—38no—
35#63921—34no—
36#63913—34no—
37C4×#624—31no—
38C4×D3×Dic3—32no—
39#63863D312yescomplemented
40#63860—32no—
41#63860—32no—
42#63860—62no—
43C22×C3×C3⋊D3C2311yescharacteristic
44C22×C32×D3C2314yes—
45C2×C3×D32—24no—
46C2×C3×D32—28no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)C2312yes—
49#6308—22no—
50C2×C3×C3⋊Dic3C2314yes—
51C2×C3×C3⋊Dic3C2311yescharacteristic
52C3×C4×C3⋊D3D414yes—
53C2×C32×Dic3C2312yes—
54C32×D12—24no—
55C32×Dic6—22no—
56#6294—24no—
57#6293—24no—
58Dic3×C3⋊D3—24no—
59C3×#625—24no—
60C2×#936—91no—
61D3×#124—92no—
62C23×C3×D3—34no—
63C22×D32C2×D312yes—
64C2×C32×Q8—31no—
65C2×C32×D4—32no—
66Q8×C3⋊D3—64no—
67C2×C4×C3⋊D3—32no—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×#634—32no—
70C2×#634—31no—
71C2×#634—32no—
72C2×#634—61no—
73C2×#634—62no—
74C2×#634—121no—
75C2×C3×#67—34no—
76C3×D3×D4—68no—
77C2×C3×D12—22no—
78C2×C3×D12C2×D312yes—
79C2×C3×D12C2×D312yes—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—61no—
82C2×C3×Dic6—21no—
83C2×C3×Dic6C2×D312yescomplemented
84C2×#627—34no—
85C2×#627—62no—
86C2×#625C2×D312yes—
87C2×#624—32no—
88C2×D3×Dic3—34no—
89C2×D3×Dic3—34no—
90C2×D3×Dic3—62no—
91#3802—24no—
92D3×Dic6—68no—
93D3×Dic6—68no—
94#3757—38no—
95C4×C3⋊Dic3C2×D312yescomplemented
96C3×#217—38no—
97C3×(C2×D3)⋊C4—38no—
98C3×C4×Dic3—32no—
99C3×C4×Dic3—32no—
100C3×C4×Dic3—62no—
101(C2×D3)⋊Dic3—38no—
102(C2×D3)⋊Dic3—38no—
103(C2×D3)⋊Dic3—68no—
104(C2×D3)⋊Dic3C2×D318yescomplemented
105Dic32—34no—
106Dic32—32no—
107C22×C33C2411yescharacteristic
108C2×C3×C3⋊D3C2×D412yes—
109C2×C3×C3⋊D3#4114yes—
110C2×C32×D3—24no—
111C2×C32×D3—28no—
112C3×D32—48no—
113C33×C4C2×D412yes—
114C3⋊(C3⋊Dic3)—22no—
115C3×C3⋊Dic3—24no—
116C3×C3⋊Dic3C2×D412yes—
117C32×Dic3—22no—
118C22×C3×D4—91no—
119C2×D3×Q8—92no—
120C2×D3×Q8—181no—
121C2×D3×D4—32no—
122C22×Dic6—91no—
123C3×#124—92no—
124C2×#217—94no—
125#947—32no—
126#936—32no—
127#936—92no—
128#936—92no—
129#936—94no—
130#936—182no—
131C2×C4×Dic3—91no—
132#885—98no—
133D3×C22⋊C4—98no—
134#875—92no—
135C42×D3—92no—
136C23×C32—34no—
137C22×C3⋊D3C22×D311yescharacteristic
138C22×C3×D3—24no—
139C22×C3×D3—32no—
140C22×C3×D3—61no—
141C22×C3×D3—616no—
142C22×C3×D3—68no—
143C22×C3×D3C22×D314yes—
144C22×C3×D3C22×D314yes—
145C2×D32—28no—
146C2×D32—24no—
147C32×Q8—62no—
148C32×D4—64no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—32no—
152C2×C32×C4—41no—
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3—32no—
156C2×C3⋊Dic3—34no—
157C2×C3⋊Dic3—62no—
158C2×C3⋊Dic3—64no—
159C2×C3⋊Dic3—122no—
160C2×C3⋊Dic3C22×D314yes—
161C2×C3⋊Dic3C22×D311yescharacteristic
162C4×C3⋊D3—64no—
163C4×C3⋊D3#6714yes—
164#634—62no—
165#634—64no—
166#634—64no—
167#634—64no—
168#634—124no—
169#634—124no—
170C3×#67—68no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—34no—
174C2×C3×Dic3—34no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—64no—
177C2×C3×Dic3C22×D314yes—
178C3×D12—24no—
179C3×D12—24no—
180C3×D12—44no—
181C3×C4×D3—38no—
182C3×C4×D3—64no—
183C3×Dic6—24no—
184C3×Dic6—24no—
185#627—68no—
186#627—68no—
187#625—24no—
188#624—64no—
189D3×Dic3—68no—
190D3×Dic3—68no—
191D3×Dic3—68no—
192C2×#124—271noSylow
193C2×C33C22×D411yescharacteristic
194C2×C33C2×#4112yes—
195C3×C3⋊D3#12414yescomplemented
196C32×D3—44no—
197C24×C3—92no—
198C23×D3—34no—
199C2×C3×Q8—32no—
200C2×C3×Q8—61no—
201C2×C3×D4—32no—
202C2×C3×D4—32no—
203C2×C3×D4—62no—
204C2×C3×D4—92no—
205C2×C3×D4—184no—
206C22×C3×C4—91no—
207C2×#67—34no—
208C22×Dic3—92no—
209C22×Dic3—92no—
210D3×Q8—188no—
211D3×Q8—188no—
212D3×D4—68no—
213C2×D12D3212yes—
214C2×C4×D3—32no—
215C2×C4×D3—61no—
216C2×C4×D3—94no—
217C2×C4×D3—94no—
218C2×C4×D3—182no—
219C2×Dic6—94no—
220C2×Dic6—92no—
221C2×Dic6—91no—
222C2×Dic6—92no—
223C2×Dic6—182no—
224C2×Dic6—181no—
225C2×Dic6—182no—
226C2×Dic6—184no—
227C2×Dic6—361no—
228C2×Dic6D3211yescharacteristic, complemented
229C3×C22⋊C4—98no—
230C3×C42—92no—
231#217—38no—
232#217—98no—
233#217—98no—
234#217—188no—
235(C2×D3)⋊C4—38no—
236(C2×D3)⋊C4—98no—
237C4×Dic3—32no—
238C4×Dic3—32no—
239C4×Dic3—62no—
240C4×Dic3—94no—
241C4×Dic3—92no—
242C22×C32—21no—
243C22×C32—22no—
244C22×C32—41no—
245C22×C32—64no—
246C22×C32—68no—
247C22×C32C23×D312yes—
248C22×C32C23×D311yescharacteristic
249C2×C3⋊D3#23714yes—
250C2×C3⋊D3C2×#6712yes—
251C2×C3×D3—24no—
252C2×C3×D3—28no—
253C2×C3×D3—28no—
254C2×C3×D3—24no—
255C2×C3×D3—34no—
256C2×C3×D3—38no—
257C2×C3×D3—44no—
258C2×C3×D3—48no—
259C2×C3×D3—62no—
260C2×C3×D3—64no—
261C2×C3×D3—1216no—
262D32—48no—
263C32×C4—24no—
264C32×C4—22no—
265C32×C4—42no—
266C32×C4—62no—
267C32×C4D3×D414yes—
268C32×C4C2×#6712yes—
269C3⋊Dic3—24no—
270C3⋊Dic3—62no—
271C3⋊Dic3—64no—
272C3⋊Dic3—64no—
273C3⋊Dic3—124no—
274C3⋊Dic3—124no—
275C3⋊Dic3C2×#6712yes—
276C3×Dic3—24no—
277C3×Dic3—24no—
278C3×Dic3—34no—
279C3×Dic3—62no—
280C3×Dic3—64no—
281C3×Dic3—64no—
282C3×Dic3—124no—
283C22×Q8—271no—
284C22×D4—91no—
285#124—92no—
286#124—272no—
287#124—274no—
288C2×C22⋊C4—274no—
289C2×C42—271no—
290C33C2×#12411yescharacteristic, complemented, Sylow
291C23×C3—34no—
292C23×C3—34no—
293C23×C3—64no—
294C23×C3—91no—
295C23×C3—184no—
296C23×C3—188no—
297C22×D3—32no—
298C22×D3—61no—
299C22×D3—68no—
300C22×D3—616no—
301C22×D3C2×D3214yes—
302C3×Q8—64no—
303C3×Q8—64no—
304C3×D4—64no—
305C3×D4—64no—
306C3×D4—124no—
307C3×D4—184no—
308C2×C3×C4—22no—
309C2×C3×C4—21no—
310C2×C3×C4—34no—
311C2×C3×C4—41no—
312C2×C3×C4—62no—
313C2×C3×C4—91no—
314C2×C3×C4—94no—
315C2×C3×C4—94no—
316C2×C3×C4C2×D3212yes—
317C2×C3×C4C2×D3211yescharacteristic
318#67—68no—
319C2×Dic3—32no—
320C2×Dic3—34no—
321C2×Dic3—34no—
322C2×Dic3—61no—
323C2×Dic3—64no—
324C2×Dic3—94no—
325C2×Dic3—92no—
326C2×Dic3—182no—
327C2×Dic3—184no—
328C2×Dic3—184no—
329C2×Dic3—184no—
330C2×Dic3—362no—
331C2×Dic3C2×D3212yes—
332D12—24no—
333C4×D3—38no—
334C4×D3—64no—
335C4×D3—188no—
336C4×D3—188no—
337C4×D3—188no—
338Dic6—22no—
339Dic6—184no—
340Dic6—188no—
341Dic6—188no—
342Dic6—184no—
343Dic6—184no—
344Dic6—182no—
345Dic6—364no—
346Dic6—364no—
347C2×C32—24no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—21no—
351C2×C32—41no—
352C2×C32—42no—
353C2×C32—124no—
354C2×C32C2×D3×D412yes—
355C2×C32C2×#23712yes—
356C2×C32D3×#4114yes—
357C2×C32C22×#6711yescharacteristic
358C3⋊D3#93614yescomplemented
359C3×D3—38no—
360C3×D3—44no—
361C3×D3—44no—
362C3×D3—64no—
363C3×D3—84no—
364C24—92no—
365C2×Q8—31no—
366C2×Q8—272no—
367C2×Q8—271no—
368C2×Q8—544no—
369C2×D4—32no—
370C2×D4—92no—
371C2×D4—184no—
372C22×C4—91no—
373C22×C4—272no—
374C22×C4—272no—
375C22⋊C4—98no—
376C22⋊C4—278no—
377C42—92no—
378C42—272no—
379C22×C3—22no—
380C22×C3—21no—
381C22×C3—41no—
382C22×C3—68no—
383C22×C3—68no—
384C22×C3—64no—
385C22×C3—64no—
386C22×C3—94no—
387C22×C3—92no—
388C22×C3—124no—
389C22×C3—128no—
390C22×C3—368no—
391C22×C3C22×D3212yes—
392C22×C3C22×D3211yescharacteristic
393C2×D3—28no—
394C2×D3—24no—
395C2×D3—38no—
396C2×D3—34no—
397C2×D3—62no—
398C2×D3—64no—
399C2×D3—1216no—
400C3×C4—22no—
401C3×C4—24no—
402C3×C4—42no—
403C3×C4—64no—
404C3×C4—64no—
405C3×C4—92no—
406C3×C4—184no—
407C3×C4D3×#6714yes—
408C3×C4#381112yes—
409Dic3—22no—
410Dic3—34no—
411Dic3—62no—
412Dic3—64no—
413Dic3—64no—
414Dic3—124no—
415Dic3—184no—
416Dic3—184no—
417Dic3—182no—
418Dic3—364no—
419Dic3—364no—
420C32—22no—
421C32—21no—
422C32—41no—
423C32D3×#12412yescomplemented
424C32C2×#93611yescharacteristic, complemented
425C23—34no—
426C23—91no—
427C23—184no—
428C23—188no—
429Q8—62no—
430Q8—544no—
431Q8—542no—
432D4—64no—
433D4—184no—
434C2×C4—32no—
435C2×C4—91no—
436C2×C4—94no—
437C2×C4—94no—
438C2×C4—272no—
439C2×C4—544no—
440C2×C4—544no—
441C2×C4D3311yescharacteristic
442C2×C3—24no—
443C2×C3—22no—
444C2×C3—21no—
445C2×C3—22no—
446C2×C3—42no—
447C2×C3—41no—
448C2×C3—94no—
449C2×C3—124no—
450C2×C3—124no—
451C2×C3—244no—
452C2×C3D3×#23714yes—
453C2×C3#6431912yes—
454C2×C3C2×D3×#6712yes—
455C2×C3C2×#381111yescharacteristic
456D3—38no—
457D3—44no—
458D3—64no—
459C22—68no—
460C22—64no—
461C22—94no—
462C22—92no—
463C22—368no—
464C22C2×D3311yescharacteristic
465C4—62no—
466C4—92no—
467C4—184no—
468C4—542no—
469C4#8895212yes—
470C3—21no—
471C3—22no—
472C3—41no—
473C3D3×#93612yescomplemented
474C3#1059543211yescharacteristic, complemented
475C2—94no—
476C2—124no—
477C2#1172437412yes—
478C2C2×#8895211yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×C3×#3802C2
hypercenterC2×C4D33
hypocenterC33C2×#124
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×Dic6C2semidirect
3. C4×#6294C2semidirect
4. #11723429C2semidirect
5. C3×#63863C2semidirect
6. #11723004C2semidirect
7. #11722902C2semidirect
11. C2×#6308C22semidirect
14. C2×C32×Dic6C22semidirect
16. C2×#6293C22semidirect
21. C3×C4×C3⋊Dic3C22semidirect
22. #88813C22semidirect
23. (C2×D3)⋊(C3⋊Dic3)C22semidirect
24. C3×(C2×D3)⋊Dic3C22semidirect
39. #63863D3semidirect
83. C2×C3×Dic6C2×D3semidirect
95. C4×C3⋊Dic3C2×D3semidirect
104. (C2×D3)⋊Dic3C2×D3semidirect
195. C3×C3⋊D3#124semidirect
228. C2×Dic6D32semidirect
290. C33C2×#124semidirect
358. C3⋊D3#936semidirect
423. C32D3×#124semidirect
424. C32C2×#936semidirect
473. C3D3×#936semidirect
474. C3#10595432semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×C3×#38022
C4×#62942
#117234292
C3×#638632
#117230042
#117229022
#105954323
D3×#9363