Finite Group Databasebeta

← #13150740

Subgroups — #13150740

Order 1728.

921 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721824C211yesmaximal, characteristic, complemented
2C2×#3752⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721310C212yesmaximal
5#11721299C212yesmaximal, complemented
6#11721135C212yesmaximal, complemented
7#11721134C212yesmaximal, complemented
8#11721133C212yesmaximal, complemented
9#11721128C212yesmaximal, complemented
10#10594981—31nomaximal
11#10594971—31nomaximal
12C23×(C3⋊D3)⋊C3C2211yescharacteristic
13C22×C4×H(3,𝔽3)C2211yescharacteristic
14C2×#6229C2211yescharacteristic, complemented
15C22×(C3⋊Dic3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic, complemented
17C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C2×#6196C2212yescomplemented
20#88552C2212yes—
21C4⋊#1084C2211yescharacteristic, complemented
22#88540C2212yescomplemented
23#3757⋊C3C2212yes—
24#3752⋊C3C2212yes—
25#3752⋊C3C2212yes—
26C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
27#3750⋊C3C2212yescomplemented
28#88450C2212yes—
29#88450C2212yes—
30#88449C2212yescomplemented
31#88448C2212yescomplemented
32#88448C2212yescomplemented
33C2×#3752D311yescharacteristic
34#64133D311yescharacteristic, complemented
35C2×C3×(C2×D3)⋊C4—31no—
36C3×#924—31no—
37C3×#924—31no—
38C3×#924—31no—
39C3×#924—61no—
40#63982—32no—
41#63974—32no—
42D3×#217—32no—
43#63964—32no—
44#63899—32no—
45#63896—32no—
46#63895—32no—
47#63895—32no—
48#63893—32no—
49#63888—32no—
50#63888—32no—
51#63887—32no—
52C23×H(3,𝔽3)C2311yescharacteristic
53C22×(C3⋊D3)⋊C3—41no—
54C22×(C3⋊D3)⋊C3—41no—
55C22×(C3⋊D3)⋊C3Q814yescomplemented
56C22×(C3⋊D3)⋊C3C2312yes—
57C2×C4×H(3,𝔽3)—41no—
58C2×C4×H(3,𝔽3)C2311yescharacteristic
59C2×C4×H(3,𝔽3)C2311yescharacteristic
60C2×#1084C2312yes—
61C2×#1084C2312yes—
62#6229—41no—
63C2×(C3⋊Dic3)⋊C3—41no—
64C2×(C3⋊Dic3)⋊C3C2312yes—
65C2×(C3⋊Dic3)⋊C3C2312yes—
66C2×(C3⋊Dic3)⋊C3C2312yes—
67#634⋊C3—41no—
68((C3⋊D3)⋊C3)⋊C4—42no—
69((C3⋊D3)⋊C3)⋊C4—42no—
70#6196—42no—
71#5634—91no—
72#5634—91no—
73#5483—91no—
74C23×C3⋊D3C2×D311yescharacteristic
75C23×C3×D3—31no—
76C22×C32×C4—21no—
77C22×C32×C4C2×D311yescharacteristic
78C22×C32×C4C2×D311yescharacteristic
79C22×C3⋊Dic3C2×D311yescharacteristic
80C2×#634C2×D311yescharacteristic, complemented
81C22×C3×Dic3—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C2×#627—32no—
87C2×#627—32no—
88C2×#624—32no—
89C2×#624—32no—
90C2×D3×Dic3—32no—
91C2×D3×Dic3—32no—
92#3757C2×D312yes—
93#3752C2×D312yes—
94#3752C2×D312yes—
95C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
96#3750C2×D312yescomplemented
97C3×#217—32no—
98C3×#217—32no—
99C3×#217—32no—
100C3×#217—62no—
101C3×(C2×D3)⋊C4—32no—
102C3×(C2×D3)⋊C4—32no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—61no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—32no—
110C3×Dic3⋊C4—62no—
111#3724—32no—
112#3724—32no—
113Dic3⋊Dic3—32no—
114Dic3⋊Dic3—32no—
115Dic3⋊Dic3—32no—
116Dic3⋊Dic3—32no—
117#3722—32no—
118#3722—32no—
119(C2×D3)⋊Dic3—32no—
120(C2×D3)⋊Dic3—32no—
121C22×H(3,𝔽3)—21no—
122C22×H(3,𝔽3)—21no—
123C22×H(3,𝔽3)C2×Q812yes—
124C22×H(3,𝔽3)C2411yescharacteristic
125C2×(C3⋊D3)⋊C3—22no—
126C2×(C3⋊D3)⋊C3—22no—
127C2×(C3⋊D3)⋊C3—44no—
128C2×(C3⋊D3)⋊C3C2×Q814yes—
129C4×H(3,𝔽3)—21no—
130C4×H(3,𝔽3)—21no—
131#1084—22no—
132#1084—22no—
133(C3⋊Dic3)⋊C3—22no—
134(C3⋊Dic3)⋊C3—22no—
135(C3⋊Dic3)⋊C3—22no—
136C3×C22⋊Q8—91no—
137C3×C22⋊Q8—91no—
138C2×C3×C22⋊C4—91no—
139C2×#217—92no—
140#946—92no—
141#946—92no—
142C2×(C2×D3)⋊C4—31no—
143C2×(C2×D3)⋊C4—91no—
144#924—31no—
145#924—31no—
146#924—31no—
147#924—61no—
148#924—91no—
149#924—92no—
150#897—92no—
151#897—92no—
152#896—92no—
153#896—92no—
154#896—92no—
155#896—92no—
156D3×C22⋊C4—92no—
157D3×C22⋊C4—92no—
158#878—92no—
159#878—92no—
160#878—92no—
161#878—92no—
162#878—92no—
163#878—92no—
164C23×C32—21no—
165C23×C32C22×D311yescharacteristic
166C23×C32C22×D311yescharacteristic
167C22×C3⋊D3—41no—
168C22×C3⋊D3—41no—
169C22×C3⋊D3Dic614yescomplemented
170C22×C3⋊D3C22×D312yes—
171C22×C3×D3—32no—
172C22×C3×D3—34no—
173C22×C3×D3—121no—
174C22×C3×D3—121no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—41no—
178C2×C32×C4—41no—
179C2×C32×C4—42no—
180C2×C32×C4C22×D311yescharacteristic
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—41no—
185C2×C3⋊Dic3C22×D312yes—
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D312yes—
188#634—41no—
189C2×C3×Dic3—32no—
190C2×C3×Dic3—32no—
191C2×C3×Dic3—32no—
192C2×C3×Dic3—32no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—62no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3—121no—
199C3×Dic6—121no—
200C3×Dic6—121no—
201C3×Dic6—121no—
202C3×Dic6—122no—
203#627—122no—
204#627—122no—
205#624—122no—
206#624—122no—
207D3×Dic3—122no—
208D3×Dic3—122no—
209#543—271noSylow
210C2×H(3,𝔽3)—22no—
211C2×H(3,𝔽3)C22×Q811yescharacteristic
212C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
213C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
214(C3⋊D3)⋊C3—24no—
215C24×C3—91no—
216C23×D3—31no—
217C23×D3—91no—
218C2×C3×Q8—91no—
219C2×C3×Q8—91no—
220C22×C3×C4—31no—
221C22×C3×C4—31no—
222C22×C3×C4—61no—
223C22×C3×C4—91no—
224C22×C3×C4D3211yescharacteristic
225C22×Dic3—31no—
226C22×Dic3—91no—
227C22×Dic3—92no—
228C22×Dic3—92no—
229C2×C4×D3—92no—
230C2×C4×D3—92no—
231C2×C4×D3—92no—
232C2×C4×D3—92no—
233C2×Dic6—31no—
234C2×Dic6—31no—
235C2×Dic6—31no—
236C2×Dic6—61no—
237C2×Dic6—91no—
238C2×Dic6—92no—
239C2×Dic6—92no—
240C2×Dic6—92no—
241C3×C4⋊C4—91no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—92no—
245C3×C22⋊C4—92no—
246C3×C22⋊C4—92no—
247C3×C22⋊C4—92no—
248C3×C22⋊C4—92no—
249#217—32no—
250#217—32no—
251#217—32no—
252#217—62no—
253#217—92no—
254#217—92no—
255#217—92no—
256(C2×D3)⋊C4—32no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263(C2×D3)⋊C4—92no—
264C4⋊Dic3—31no—
265C4⋊Dic3—31no—
266C4⋊Dic3—31no—
267C4⋊Dic3—61no—
268C4⋊Dic3—91no—
269C4⋊Dic3—92no—
270C4⋊Dic3—92no—
271C4⋊Dic3—92no—
272Dic3⋊C4—32no—
273Dic3⋊C4—32no—
274Dic3⋊C4—32no—
275Dic3⋊C4—62no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—22no—
289C22×C32—41no—
290C22×C32—41no—
291C22×C32C2×Dic612yes—
292C22×C32D3×Q812yes—
293C22×C32C23×D311yescharacteristic
294C22×C32C23×D311yescharacteristic
295C2×C3⋊D3—22no—
296C2×C3⋊D3—22no—
297C2×C3⋊D3—44no—
298C2×C3⋊D3C2×Dic614yes—
299C2×C3×D3—34no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—124no—
303C32×C4—22no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—41no—
309C3⋊Dic3—22no—
310C3⋊Dic3—22no—
311C3⋊Dic3—22no—
312C3×Dic3—62no—
313C3×Dic3—62no—
314C3×Dic3—62no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C3×Dic3—62no—
318C3×Dic3—62no—
319C3×Dic3—122no—
320C3×Dic3—122no—
321C22⋊Q8—91no—
322C22⋊Q8—91no—
323C22⋊Q8—272no—
324C22⋊Q8—272no—
325C22⋊Q8—272no—
326C22⋊Q8—272no—
327C22⋊Q8—272no—
328C2×C22⋊C4—91no—
329C2×C22⋊C4—272no—
330H(3,𝔽3)#54311yescharacteristic, complemented, Sylow
331C23×C3—31no—
332C23×C3—31no—
333C23×C3—61no—
334C23×C3—92no—
335C23×C3—94no—
336C23×C3—361no—
337C23×C3—361no—
338C23×C3C2×D3211yescharacteristic
339C22×D3—32no—
340C22×D3—34no—
341C22×D3—92no—
342C22×D3—94no—
343C22×D3—121no—
344C22×D3—121no—
345C22×D3—361no—
346C22×D3—361no—
347C3×Q8—361no—
348C3×Q8—361no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—41no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—92no—
357C2×C3×C4—92no—
358C2×C3×C4—92no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4—121no—
362C2×C3×C4—121no—
363C2×C3×C4—122no—
364C2×C3×C4—361no—
365C2×C3×C4C2×D3211yescharacteristic
366C2×C3×C4C2×D3211yescharacteristic
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—62no—
375C2×Dic3—62no—
376C2×Dic3—92no—
377C2×Dic3—92no—
378C2×Dic3—92no—
379C2×Dic3—121no—
380C2×Dic3—361no—
381C2×Dic3—362no—
382C2×Dic3—362no—
383C4×D3—362no—
384C4×D3—362no—
385C4×D3—362no—
386C4×D3—362no—
387Dic6—121no—
388Dic6—121no—
389Dic6—121no—
390Dic6—122no—
391Dic6—361no—
392Dic6—362no—
393Dic6—362no—
394Dic6—362no—
395C2×C32—22no—
396C2×C32—22no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—21no—
400C2×C32—42no—
401C2×C32C22×Dic611yescharacteristic
402C2×C32#100411yescharacteristic
403C2×C32#100411yescharacteristic
404C2×C32#100411yescharacteristic
405C2×C32C2×D3×Q811yescharacteristic
406C2×C32#100911yescharacteristic
407C3⋊D3—24no—
408C3×D3—64no—
409C24—91no—
410C2×Q8—91no—
411C2×Q8—91no—
412C2×Q8—272no—
413C22×C4—91no—
414C22×C4—272no—
415C22×C4—272no—
416C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C4⋊C4—92no—
418C4⋊C4—92no—
419C4⋊C4—91no—
420C4⋊C4—91no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—272no—
424C22⋊C4—92no—
425C22⋊C4—92no—
426C22⋊C4—92no—
427C22⋊C4—92no—
428C22⋊C4—272no—
429C22⋊C4—272no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—31no—
433C22×C3—31no—
434C22×C3—32no—
435C22×C3—32no—
436C22×C3—61no—
437C22×C3—61no—
438C22×C3—61no—
439C22×C3—61no—
440C22×C3—61no—
441C22×C3—62no—
442C22×C3—94no—
443C22×C3—121no—
444C22×C3—121no—
445C22×C3—182no—
446C22×C3—182no—
447C22×C3—364no—
448C22×C3D3×Dic612yes—
449C22×C3C22×D3211yescharacteristic
450C2×D3—34no—
451C2×D3—62no—
452C2×D3—62no—
453C2×D3—94no—
454C2×D3—124no—
455C2×D3—182no—
456C2×D3—182no—
457C2×D3—364no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—61no—
461C3×C4—61no—
462C3×C4—61no—
463C3×C4—61no—
464C3×C4—62no—
465C3×C4—121no—
466C3×C4—182no—
467C3×C4—182no—
468C3×C4—182no—
469C3×C4—182no—
470C3×C4—182no—
471Dic3—62no—
472Dic3—62no—
473Dic3—62no—
474Dic3—62no—
475Dic3—62no—
476Dic3—62no—
477Dic3—62no—
478Dic3—122no—
479Dic3—122no—
480Dic3—182no—
481Dic3—182no—
482Dic3—182no—
483C32—21no—
484C32#548311yescharacteristic, complemented
485C32#563411yescharacteristic, complemented
486C23—92no—
487C23—94no—
488C23—361no—
489C23—361no—
490C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
491Q8—361no—
492Q8—361no—
493Q8—1082no—
494C2×C4—41no—
495C2×C4—92no—
496C2×C4—92no—
497C2×C4—92no—
498C2×C4—92no—
499C2×C4—92no—
500C2×C4—361no—
501C2×C4—1082no—
502C2×C4—1082no—
503C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
504C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
505C2×C3—22no—
506C2×C3—31no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—31no—
510C2×C3—31no—
511C2×C3—31no—
512C2×C3—62no—
513C2×C3—62no—
514C2×C3—61no—
515C2×C3—61no—
516C2×C3—61no—
517C2×C3—122no—
518C2×C3—184no—
519C2×C3C2×D3×Dic611yescharacteristic
520C2×C3#6430711yescharacteristic
521C2×C3#6433011yescharacteristic
522D3—64no—
523D3—184no—
524C22—21no—
525C22—21no—
526C22—94no—
527C22—182no—
528C22—182no—
529C22—364no—
530C22((C3⋊D3)⋊C3)⋊Q812yes—
531C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
532C4—21no—
533C4—21no—
534C4—182no—
535C4—182no—
536C4—182no—
537C4—182no—
538C4—182no—
539C3—31no—
540C3—31no—
541C3—61no—
542C3#1059497111yescharacteristic
543C2—22no—
544C2—184no—
545C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
546C2#1172249211yescharacteristic
547C2#1172254511yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×(C3⋊D3)⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#543
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. #11721824C2semidirect
3. #64133⋊C3C2semidirect
5. #11721299C2semidirect
6. #11721135C2semidirect
7. #11721134C2semidirect
8. #11721133C2semidirect
9. #11721128C2semidirect
14. C2×#6229C22semidirect
16. C2×#634⋊C3C22semidirect
19. C2×#6196C22semidirect
21. C4⋊#1084C22semidirect
22. #88540C22semidirect
26. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
27. #3750⋊C3C22semidirect
30. #88449C22semidirect
31. #88448C22semidirect
32. #88448C22semidirect
34. #64133D3semidirect
55. C22×(C3⋊D3)⋊C3Q8semidirect
80. C2×#634C2×D3semidirect
95. C4⋊(C3⋊Dic3)C2×D3semidirect
96. #3750C2×D3semidirect
169. C22×C3⋊D3Dic6semidirect
330. H(3,𝔽3)#543semidirect
484. C32#5483semidirect
485. C32#5634semidirect

Maximal subgroups

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

subgroupindex
#117218242
C2×#3752⋊C32
#64133⋊C32
#117213102
#117212992
#117211352
#117211342
#117211332
#117211282
#105949813
#105949713