Finite Group Databasebeta

← #13178747

Subgroups — #13178747

Order 1728.

1038 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64155C212yesmaximal, complemented
2C3×#64153C211yesmaximal, characteristic, complemented
3#11723694C212yesmaximal
4#11723676C212yesmaximal, complemented
5#11723675C212yesmaximal, complemented
6#11723671C212yesmaximal, complemented
7#11723670C211yesmaximal, characteristic
8#11723669C212yesmaximal, complemented
9#11723666C211yesmaximal, characteristic, complemented
10#10595416—31nomaximal
11#10595345—32nomaximal
12C2×C33×Q8C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C3×#634C2211yescharacteristic
15C2×#6296C2211yescharacteristic
16C2×#6295C2211yescharacteristic
17C3×#3752C2212yes—
18C3×#3752C2212yes—
19C3×C4⋊(C3⋊Dic3)C2212yes—
20C3×#3750C2212yes—
21C3×#3750C2212yes—
22C3×#3750C2212yes—
23C3×#3750C2212yes—
24C3×C4×C3⋊Dic3C2211yescharacteristic
25#88815C2212yes—
26#88815C2212yes—
27#88815C2212yes—
28#88814C2212yes—
29#88814C2212yes—
30#88814C2212yes—
31#88813C2211yescharacteristic
32#88813C2211yescharacteristic
33#64155D312yescomplemented
34#64153D311yescharacteristic, complemented
35C3×#946—32no—
36C3×#946—32no—
37C3×#946—62no—
38C3×#944—32no—
39C3×#944—61no—
40#63913—31no—
41#63907—32no—
42#63905—32no—
43#63868—32no—
44#63865—32no—
45#63862—32no—
46#63861—32no—
47#63859—32no—
48#63858—32no—
49#63853—32no—
50#63851—32no—
51#63851—32no—
52#63850—32no—
53#63848—32no—
54#63847—32no—
55#63846—32no—
56#63845—32no—
57#63840—31no—
58#63839—32no—
59C22×C3×C3⋊D3C2312yes—
60C33×Q8—41no—
61C2×C33×C4C2311yescharacteristic
62C2×C33×C4C2312yes—
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2311yescharacteristic
65C2×C3×C3⋊Dic3C2311yescharacteristic
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C2×C3×C3⋊Dic3C2312yes—
69C3×C4×C3⋊D3—42no—
70C3×#634—41no—
71#6296—41no—
72#6295—41no—
73#5735—91no—
74#5646—92no—
75C2×C32×Q8—21no—
76C2×C32×Q8—22no—
77C2×C32×Q8—41no—
78C2×C32×Q8C2×D312yes—
79C2×C32×Q8C2×D311yescharacteristic
80C2×C4×C3⋊D3C2×D312yes—
81C2×#634C2×D311yescharacteristic
82C2×C3×C4×D3—32no—
83C2×C3×C4×D3—32no—
84C2×C3×C4×D3—62no—
85C2×C3×Dic6—32no—
86C2×C3×Dic6—61no—
87C2×#627—32no—
88C2×#627—31no—
89C2×#626—32no—
90C2×#625—31no—
91#3752C2×D312yes—
92#3752C2×D312yes—
93C4⋊(C3⋊Dic3)C2×D312yes—
94#3750C2×D312yes—
95#3750C2×D312yes—
96#3750C2×D312yes—
97#3750C2×D312yes—
98C4×C3⋊Dic3C2×D311yescharacteristic
99C3×(C2×D3)⋊C4—32no—
100C3×(C2×D3)⋊C4—32no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×(C2×D3)⋊C4—62no—
104C3×(C2×D3)⋊C4—62no—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3—32no—
107C3×C4⋊Dic3—62no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—32no—
116C3×Dic3⋊C4—62no—
117C3×Dic3⋊C4—62no—
118C3×Dic3⋊C4—62no—
119C3×Dic3⋊C4—62no—
120C3×C4×Dic3—32no—
121C3×C4×Dic3—61no—
122#3724—32no—
123#3724—32no—
124#3724—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—32no—
129Dic3⋊Dic3—32no—
130Dic3⋊Dic3—32no—
131#3722—32no—
132#3722—32no—
133#3722—32no—
134(C2×D3)⋊Dic3—32no—
135(C2×D3)⋊Dic3—32no—
136(C2×D3)⋊Dic3—32no—
137(C2×D3)⋊Dic3—32no—
138(C2×D3)⋊Dic3—32no—
139(C2×D3)⋊Dic3—32no—
140Dic32—32no—
141Dic32—31no—
142Dic32—32no—
143Dic32—31no—
144C22×C33C2411yescharacteristic
145C2×C3×C3⋊D3—22no—
146C2×C3×C3⋊D3—22no—
147C2×C3×C3⋊D3—22no—
148C33×C4—21no—
149C33×C4—22no—
150C3×C3⋊Dic3—21no—
151C3×C3⋊Dic3—21no—
152C3×C3⋊Dic3—22no—
153C3×C3⋊Dic3—22no—
154C3×C3⋊Dic3—22no—
155C3×C3⋊Dic3—22no—
156C3×C4⋊Q8—91no—
157C3×C22⋊Q8—92no—
158#947—91no—
159#946—32no—
160#946—32no—
161#946—62no—
162#944—32no—
163#944—61no—
164#944—91no—
165#929—92no—
166#924—92no—
167#898—92no—
168#898—92no—
169#898—92no—
170#896—92no—
171#896—92no—
172#894—92no—
173#890—92no—
174#889—92no—
175#889—92no—
176#888—92no—
177#885—92no—
178#883—92no—
179#879—92no—
180#879—92no—
181#876—92no—
182C22×C3⋊D3C22×D312yes—
183C22×C3×D3—32no—
184C22×C3×D3—32no—
185C22×C3×D3—62no—
186C32×Q8—41no—
187C32×Q8—42no—
188C32×Q8—42no—
189C32×Q8—44no—
190C32×Q8—44no—
191C2×C32×C4—21no—
192C2×C32×C4—22no—
193C2×C32×C4—22no—
194C2×C32×C4—22no—
195C2×C32×C4—22no—
196C2×C32×C4—41no—
197C2×C32×C4—42no—
198C2×C32×C4C22×D312yes—
199C2×C32×C4C22×D312yes—
200C2×C32×C4C22×D312yes—
201C2×C32×C4C22×D311yescharacteristic
202C2×C32×C4C22×D312yes—
203C2×C3⋊Dic3C22×D311yescharacteristic
204C2×C3⋊Dic3C22×D312yes—
205C2×C3⋊Dic3C22×D312yes—
206C2×C3⋊Dic3C22×D311yescharacteristic
207C2×C3⋊Dic3C22×D312yes—
208C2×C3⋊Dic3C22×D312yes—
209C4×C3⋊D3—42no—
210#634—41no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—32no—
218C2×C3×Dic3—32no—
219C2×C3×Dic3—32no—
220C2×C3×Dic3—32no—
221C2×C3×Dic3—62no—
222C2×C3×Dic3—61no—
223C2×C3×Dic3—61no—
224C2×C3×Dic3—62no—
225C2×C3×Dic3—62no—
226C2×C3×Dic3—62no—
227C3×C4×D3—122no—
228C3×C4×D3—122no—
229C3×C4×D3—124no—
230C3×Dic6—122no—
231C3×Dic6—122no—
232#627—122no—
233#627—121no—
234#626—122no—
235#625—121no—
236#563—271noSylow
237C2×C33H(5,𝔽2)11yescharacteristic
238C2×C332−1+412yes—
239C3×C3⋊D3—42no—
240C2×C3×Q8—21no—
241C2×C3×Q8—22no—
242C2×C3×Q8—41no—
243C2×C3×Q8—91no—
244C2×C3×Q8D3212yes—
245C2×C3×Q8D3211yescharacteristic
246C22×C3×C4—92no—
247C2×#67—92no—
248C2×D12—91no—
249C2×C4×D3—32no—
250C2×C4×D3—32no—
251C2×C4×D3—62no—
252C2×Dic6—32no—
253C2×Dic6—61no—
254C2×Dic6—92no—
255C2×Dic6—91no—
256C3×C4⋊C4—92no—
257C3×C4⋊C4—92no—
258C3×C4⋊C4—92no—
259C3×C4⋊C4—92no—
260C3×C4⋊C4—92no—
261C3×C22⋊C4—92no—
262C3×C22⋊C4—92no—
263C3×C42—91no—
264#217—92no—
265#217—92no—
266#217—92no—
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—32no—
269(C2×D3)⋊C4—32no—
270(C2×D3)⋊C4—32no—
271(C2×D3)⋊C4—62no—
272(C2×D3)⋊C4—62no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276(C2×D3)⋊C4—92no—
277(C2×D3)⋊C4—92no—
278(C2×D3)⋊C4—92no—
279C4⋊Dic3—32no—
280C4⋊Dic3—32no—
281C4⋊Dic3—62no—
282C4⋊Dic3—92no—
283C4⋊Dic3—92no—
284C4⋊Dic3—92no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—32no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—32no—
292Dic3⋊C4—32no—
293Dic3⋊C4—62no—
294Dic3⋊C4—62no—
295Dic3⋊C4—62no—
296Dic3⋊C4—62no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303C4×Dic3—32no—
304C4×Dic3—61no—
305C4×Dic3—92no—
306C4×Dic3—92no—
307C4×Dic3—91no—
308C4×Dic3—91no—
309C22×C32—21no—
310C22×C32—22no—
311C22×C32—41no—
312C22×C32C23×D311yescharacteristic
313C22×C32C23×D312yes—
314C2×C3⋊D3—22no—
315C2×C3⋊D3—22no—
316C2×C3⋊D3—22no—
317C2×C3×D3—64no—
318C2×C3×D3—62no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—62no—
323C2×C3×D3—62no—
324C2×C3×D3—122no—
325C2×C3×D3—122no—
326C32×C4—21no—
327C32×C4—22no—
328C32×C4—22no—
329C32×C4—24no—
330C32×C4—22no—
331C32×C4—22no—
332C32×C4—22no—
333C32×C4—42no—
334C32×C4—42no—
335C32×C4—42no—
336C32×C4—42no—
337C32×C4—44no—
338C3⋊Dic3—22no—
339C3⋊Dic3—21no—
340C3⋊Dic3—21no—
341C3⋊Dic3—22no—
342C3⋊Dic3—22no—
343C3⋊Dic3—22no—
344C3×Dic3—64no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—62no—
355C3×Dic3—121no—
356C3×Dic3—121no—
357C3×Dic3—122no—
358C3×Dic3—122no—
359C3×Dic3—122no—
360C4⋊Q8—91no—
361C4⋊Q8—271no—
362#126—272no—
363#126—272no—
364#125—272no—
365#124—271no—
366#123—272no—
367C22⋊Q8—92no—
368C22⋊Q8—272no—
369C33#56311yescharacteristic, complemented, Sylow
370C23×C3—92no—
371C22×D3—32no—
372C22×D3—32no—
373C22×D3—62no—
374C3×Q8—42no—
375C3×Q8—41no—
376C3×Q8—42no—
377C3×Q8—44no—
378C3×Q8—44no—
379C3×Q8—361no—
380C2×C3×C4—21no—
381C2×C3×C4—22no—
382C2×C3×C4—22no—
383C2×C3×C4—22no—
384C2×C3×C4—22no—
385C2×C3×C4—41no—
386C2×C3×C4—42no—
387C2×C3×C4—91no—
388C2×C3×C4—92no—
389C2×C3×C4—92no—
390C2×C3×C4—91no—
391C2×C3×C4—92no—
392C2×C3×C4—92no—
393C2×C3×C4—362no—
394C2×C3×C4C2×D3212yes—
395C2×C3×C4C2×D3212yes—
396C2×C3×C4C2×D3212yes—
397C2×C3×C4C2×D3211yescharacteristic
398C2×C3×C4C2×D3212yes—
399#67—362no—
400C2×Dic3—32no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—32no—
404C2×Dic3—32no—
405C2×Dic3—32no—
406C2×Dic3—32no—
407C2×Dic3—32no—
408C2×Dic3—32no—
409C2×Dic3—32no—
410C2×Dic3—62no—
411C2×Dic3—61no—
412C2×Dic3—61no—
413C2×Dic3—62no—
414C2×Dic3—62no—
415C2×Dic3—62no—
416D12—361no—
417C4×D3—122no—
418C4×D3—122no—
419C4×D3—124no—
420Dic6—122no—
421Dic6—122no—
422Dic6—362no—
423Dic6—361no—
424C2×C32—22no—
425C2×C32—22no—
426C2×C32—22no—
427C2×C32—21no—
428C2×C32—22no—
429C2×C32—41no—
430C2×C32—42no—
431C2×C32#100412yes—
432C2×C32#100411yescharacteristic
433C2×C32#100712yes—
434C2×C32#100712yes—
435C2×C32#101012yes—
436C3⋊D3—42no—
437C3×D3—124no—
438C3×D3—122no—
439C3×D3—122no—
440C2×Q8—91no—
441C2×Q8—271no—
442C2×Q8#111311yescharacteristic
443C2×D4—271no—
444C22×C4—92no—
445C4⋊C4—92no—
446C4⋊C4—92no—
447C4⋊C4—92no—
448C4⋊C4—92no—
449C4⋊C4—92no—
450C4⋊C4—272no—
451C4⋊C4—272no—
452C4⋊C4—272no—
453C22⋊C4—92no—
454C22⋊C4—92no—
455C22⋊C4—272no—
456C22⋊C4—272no—
457C22⋊C4—272no—
458C42—91no—
459C42—271no—
460C42—271no—
461C22×C3—21no—
462C22×C3—22no—
463C22×C3—41no—
464C22×C3—182no—
465C22×C3—182no—
466C22×C3—182no—
467C22×C3C22×D3211yescharacteristic
468C22×C3C22×D3212yes—
469C2×D3—64no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—62no—
476C2×D3—122no—
477C2×D3—122no—
478C3×C4—22no—
479C3×C4—24no—
480C3×C4—22no—
481C3×C4—22no—
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—21no—
485C3×C4—42no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—44no—
489C3×C4—42no—
490C3×C4—182no—
491C3×C4—182no—
492C3×C4—182no—
493C3×C4—182no—
494C3×C4—181no—
495C3×C4—181no—
496Dic3—64no—
497Dic3—62no—
498Dic3—62no—
499Dic3—62no—
500Dic3—62no—
501Dic3—62no—
502Dic3—62no—
503Dic3—62no—
504Dic3—62no—
505Dic3—62no—
506Dic3—62no—
507Dic3—121no—
508Dic3—121no—
509Dic3—122no—
510Dic3—122no—
511Dic3—122no—
512C32—22no—
513C32—21no—
514C32—41no—
515C32#564612yescomplemented
516C32#573511yescharacteristic, complemented
517C23—92no—
518Q8—41no—
519Q8—361no—
520Q8—1081no—
521D4—1081no—
522C2×C4—92no—
523C2×C4—92no—
524C2×C4—92no—
525C2×C4—91no—
526C2×C4—91no—
527C2×C4—92no—
528C2×C4—362no—
529C2×C4C2×#111311yescharacteristic
530C2×C4C2×#111312yes—
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—22no—
534C2×C3—21no—
535C2×C3—22no—
536C2×C3—42no—
537C2×C3—41no—
538C2×C3—362no—
539C2×C3#6431512yes—
540C2×C3#6431512yes—
541C2×C3#6431612yes—
542C2×C3#6433012yes—
543C2×C3#6433011yescharacteristic
544D3—124no—
545D3—122no—
546D3—122no—
547C22—182no—
548C22—182no—
549C22—182no—
550C22C22×#111311yescharacteristic
551C4—22no—
552C4—21no—
553C4—182no—
554C4—182no—
555C4—182no—
556C4—182no—
557C4—181no—
558C4—181no—
559C3—22no—
560C3—21no—
561C3—41no—
562C3#1059534512yescomplemented
563C3#1059541611yescharacteristic, complemented
564C2—362no—
565C2#1172462412yes—
566C2#1172463311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC2×C33×Q8C22
prime radicalC2×#6295C22
hypercenterC2×Q8#1113
hypocenterC33#563
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C3×#64155C2semidirect
2. C3×#64153C2semidirect
4. #11723676C2semidirect
5. #11723675C2semidirect
6. #11723671C2semidirect
8. #11723669C2semidirect
9. #11723666C2semidirect
33. #64155D3semidirect
34. #64153D3semidirect
369. C33#563semidirect
515. C32#5646semidirect
516. C32#5735semidirect
562. C3#10595345semidirect
563. C3#10595416semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
C3×#641532
#117236942
#117236762
#117236752
#117236712
#117236702
#117236692
#117236662
#105954163
#105953453