Finite Group Databasebeta

← #13147848

Subgroups — #13147848

Order 1728.

875 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721716C211yesmaximal, characteristic, complemented
2#64084⋊C3C211yesmaximal, characteristic, complemented
3C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
4#11721141C211yesmaximal, characteristic
5#11721139C212yesmaximal, complemented
6#11721132C211yesmaximal, characteristic, complemented
7#11721122C212yesmaximal, complemented
8#11721120C212yesmaximal, complemented
9#11721109C212yesmaximal, complemented
10#11721106C211yesmaximal, characteristic, complemented
11#11721082C211yesmaximal, characteristic, complemented
12#10593976—31nomaximal
13#10593943—31nomaximal
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C2×#6198C2212yes—
20C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
21C42×H(3,𝔽3)C2211yescharacteristic
22C4⋊#1084C2211yescharacteristic
23C4⋊#1084C2211yescharacteristic
24#88540C2212yes—
25#88540C2212yes—
26#3752⋊C3C2212yes—
27#3752⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
31#88450C2212yes—
32#88450C2212yes—
33#88450C2212yes—
34#88449C2212yes—
35#88448C2212yes—
36#88447C2212yes—
37#64084D311yescharacteristic, complemented
38C42×C3⋊D3D311yescharacteristic, complemented
39C3×#875—31no—
40C3×C42×D3—31no—
41C3×#870—31no—
42C3×#870—31no—
43C3×#870—61no—
44#63916—31no—
45#63909—31no—
46#63904—32no—
47#63898—31no—
48#63894—31no—
49#63881—32no—
50#63878—32no—
51#63871—31no—
52#63868—32no—
53#63862—32no—
54#63857—31no—
55#63853—32no—
56#63850—32no—
57#63844—31no—
58#63843—31no—
59#63836—32no—
60C22×(C3⋊D3)⋊C3C2312yes—
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084C2312yes—
66C2×#1084C2312yes—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C4⋊((C3⋊D3)⋊C3)—22no—
72C4×(C3⋊D3)⋊C3—22no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3—22no—
75#634⋊C3—22no—
76#6198—24no—
77((C3⋊D3)⋊C3)⋊C4—24no—
78#5691—91no—
79#5680—91no—
80#5479—91no—
81C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×C4×C3⋊D3C2×D311yescharacteristic
85C2×#634C2×D311yescharacteristic
86C2×C3×D12—31no—
87C2×C3×C4×D3—31no—
88C2×C3×C4×D3—31no—
89C2×C3×C4×D3—31no—
90C2×C3×Dic6—31no—
91C2×#626—32no—
92C2×#626—32no—
93C2×#624—32no—
94C2×D3×Dic3—32no—
95C32×C42—21no—
96C32×C42C2×D311yescharacteristic
97C32×C42C2×D311yescharacteristic
98#3752C2×D312yes—
99#3752C2×D312yes—
100C4×C3⋊Dic3C2×D311yescharacteristic
101C4×C3⋊Dic3C2×D311yescharacteristic
102C4×C3⋊Dic3C2×D311yescharacteristic
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—31no—
109C3×C4⋊Dic3—61no—
110C3×C4⋊Dic3—61no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—32no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—32no—
115C3×Dic3⋊C4—62no—
116C3×Dic3⋊C4—62no—
117C3×C4×Dic3—31no—
118C3×C4×Dic3—31no—
119C3×C4×Dic3—31no—
120#3724—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124#3722—32no—
125#3722—32no—
126#3722—32no—
127(C2×D3)⋊Dic3—32no—
128(C2×D3)⋊Dic3—32no—
129(C2×D3)⋊Dic3—32no—
130Dic32—32no—
131Dic32—32no—
132C22×H(3,𝔽3)C2411yescharacteristic
133C2×(C3⋊D3)⋊C3—21no—
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—22no—
136C2×(C3⋊D3)⋊C3—22no—
137C2×(C3⋊D3)⋊C3—22no—
138C2×(C3⋊D3)⋊C3#4112yes—
139C4×H(3,𝔽3)—21no—
140C4×H(3,𝔽3)—21no—
141C4×H(3,𝔽3)#4112yes—
142#1084—22no—
143#1084—22no—
144(C3⋊Dic3)⋊C3—22no—
145(C3⋊Dic3)⋊C3—21no—
146(C3⋊Dic3)⋊C3—21no—
147(C3⋊Dic3)⋊C3#4112yes—
148C3×#125—91no—
149C3×#124—91no—
150C2×C3×C42—91no—
151#946—91no—
152Q8×Dic3—91no—
153#938—91no—
154D4×Dic3—91no—
155#930—91no—
156#929—92no—
157C4×#67—92no—
158#924—91no—
159#898—92no—
160#897—91no—
161#895—91no—
162#894—92no—
163#893—91no—
164#893—92no—
165#892—92no—
166#887—91no—
167#883—92no—
168#67⋊C4—92no—
169#879—92no—
170#877—92no—
171#876—92no—
172#875—31no—
173#875—91no—
174C4×D12—91no—
175#872—92no—
176C42×D3—31no—
177C42×D3—91no—
178#870—31no—
179#870—31no—
180#870—61no—
181C4×Dic6—91no—
182C22×C3⋊D3C22×D312yes—
183C22×C3⋊D3C22×D311yescharacteristic
184C22×C3×D3—31no—
185C22×C3×D3—32no—
186C2×C32×C4—21no—
187C2×C32×C4—21no—
188C2×C32×C4—21no—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194C2×C32×C4C22×D311yescharacteristic
195C2×C3⋊Dic3C22×D312yes—
196C2×C3⋊Dic3C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D311yescharacteristic
198C2×C3⋊Dic3C22×D311yescharacteristic
199C4⋊(C3⋊D3)—22no—
200C4×C3⋊D3—22no—
201C4×C3⋊D3—22no—
202C4×C3⋊D3—22no—
203#634—22no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—31no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—62no—
213C2×C3×Dic3—62no—
214C3×D12—62no—
215C3×C4×D3—62no—
216C3×C4×D3—62no—
217C3×C4×D3—62no—
218C3×Dic6—62no—
219#626—64no—
220#626—64no—
221#624—64no—
222D3×Dic3—64no—
223#529—271noSylow
224C2×H(3,𝔽3)C2×#4111yescharacteristic
225C2×H(3,𝔽3)C2×#4111yescharacteristic
226C2×H(3,𝔽3)C2×#4111yescharacteristic
227(C3⋊D3)⋊C3—22no—
228(C3⋊D3)⋊C3—42no—
229C2×C3×Q8—91no—
230C2×C3×D4—91no—
231C22×C3×C4—91no—
232C22×C3×C4—91no—
233C22×C3×C4—91no—
234C2×#67—92no—
235C2×#67—92no—
236C22×Dic3—92no—
237C2×D12—31no—
238C2×D12—91no—
239C2×D12—92no—
240C2×C4×D3—31no—
241C2×C4×D3—31no—
242C2×C4×D3—31no—
243C2×C4×D3—91no—
244C2×C4×D3—91no—
245C2×C4×D3—91no—
246C2×C4×D3—92no—
247C2×C4×D3—92no—
248C2×Dic6—31no—
249C2×Dic6—91no—
250C3×C4⋊C4—91no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4—91no—
253C3×C4⋊C4—92no—
254C3×C22⋊C4—92no—
255C3×C22⋊C4—92no—
256C3×C42—31no—
257C3×C42—31no—
258C3×C42—61no—
259C3×C42—91no—
260C3×C42—91no—
261C3×C42—91no—
262C3×C42D3211yescharacteristic
263#217—92no—
264#217—92no—
265#217—92no—
266(C2×D3)⋊C4—32no—
267(C2×D3)⋊C4—32no—
268(C2×D3)⋊C4—92no—
269(C2×D3)⋊C4—92no—
270(C2×D3)⋊C4—92no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—92no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275(C2×D3)⋊C4—92no—
276C4⋊Dic3—31no—
277C4⋊Dic3—31no—
278C4⋊Dic3—31no—
279C4⋊Dic3—31no—
280C4⋊Dic3—61no—
281C4⋊Dic3—61no—
282C4⋊Dic3—92no—
283C4⋊Dic3—92no—
284Dic3⋊C4—32no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—62no—
289Dic3⋊C4—62no—
290Dic3⋊C4—92no—
291Dic3⋊C4—92no—
292Dic3⋊C4—92no—
293Dic3⋊C4—92no—
294C4×Dic3—31no—
295C4×Dic3—31no—
296C4×Dic3—31no—
297C4×Dic3—91no—
298C4×Dic3—91no—
299C4×Dic3—91no—
300C4×Dic3—92no—
301C4×Dic3—92no—
302C4×Dic3—92no—
303C22×C32—21no—
304C22×C32C23×D311yescharacteristic
305C22×C32C23×D311yescharacteristic
306C2×C3⋊D3—21no—
307C2×C3⋊D3—21no—
308C2×C3⋊D3—22no—
309C2×C3⋊D3—22no—
310C2×C3⋊D3—22no—
311C2×C3⋊D3#23512yes—
312C2×C3×D3—32no—
313C2×C3×D3—61no—
314C2×C3×D3—61no—
315C2×C3×D3—62no—
316C2×C3×D3—62no—
317C2×C3×D3—62no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—22no—
323C32×C4—41no—
324C32×C4—41no—
325C32×C4#23512yes—
326C32×C4#23712yes—
327C3⋊Dic3—21no—
328C3⋊Dic3—21no—
329C3⋊Dic3—22no—
330C3⋊Dic3#23512yes—
331C3×Dic3—32no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335C3×Dic3—61no—
336C3×Dic3—61no—
337C3×Dic3—62no—
338C3×Dic3—62no—
339C3×Dic3—122no—
340C3×Dic3—122no—
341#126—272no—
342#125—91no—
343#124—91no—
344#123—272no—
345C22⋊Q8—271no—
346#121—271no—
347C4×Q8—271no—
348C4×D4—271no—
349C4×D4—272no—
350#117—272no—
351C2×C42—91no—
352H(3,𝔽3)#52911yescharacteristic, complemented, Sylow
353C23×C3—91no—
354C23×C3—92no—
355C22×D3—31no—
356C22×D3—32no—
357C22×D3—91no—
358C22×D3—92no—
359C3×Q8—182no—
360C3×D4—182no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—31no—
366C2×C3×C4—31no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—91no—
371C2×C3×C4—92no—
372C2×C3×C4—91no—
373C2×C3×C4—91no—
374C2×C3×C4—92no—
375C2×C3×C4—92no—
376C2×C3×C4—182no—
377C2×C3×C4—182no—
378C2×C3×C4—182no—
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381C2×C3×C4C2×D3211yescharacteristic
382#67—184no—
383#67—184no—
384C2×Dic3—31no—
385C2×Dic3—32no—
386C2×Dic3—32no—
387C2×Dic3—32no—
388C2×Dic3—32no—
389C2×Dic3—31no—
390C2×Dic3—31no—
391C2×Dic3—32no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—91no—
395C2×Dic3—91no—
396C2×Dic3—91no—
397C2×Dic3—92no—
398C2×Dic3—184no—
399D12—62no—
400D12—182no—
401D12—184no—
402C4×D3—62no—
403C4×D3—62no—
404C4×D3—62no—
405C4×D3—182no—
406C4×D3—182no—
407C4×D3—182no—
408C4×D3—184no—
409C4×D3—184no—
410Dic6—62no—
411Dic6—182no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32C2×#23511yescharacteristic
416C2×C32C2×#23511yescharacteristic
417C2×C32C2×#23511yescharacteristic
418C2×C32C2×#23711yescharacteristic
419C2×C32D3×#4111yescharacteristic
420C2×C32D3×#4111yescharacteristic
421C3⋊D3—22no—
422C3⋊D3—42no—
423C3×D3—62no—
424C3×D3—122no—
425C2×Q8—91no—
426C2×D4—91no—
427C2×D4—272no—
428C22×C4—91no—
429C22×C4—91no—
430C22×C4—91no—
431C22×C4—272no—
432C4⋊C4—92no—
433C4⋊C4—91no—
434C4⋊C4—92no—
435C4⋊C4—91no—
436C4⋊C4—272no—
437C4⋊C4—272no—
438C22⋊C4—92no—
439C22⋊C4—92no—
440C22⋊C4—272no—
441C22⋊C4—272no—
442C22⋊C4—272no—
443C42—91no—
444C42—91no—
445C42—91no—
446C42—272no—
447C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
448C22×C3—31no—
449C22×C3—31no—
450C22×C3—61no—
451C22×C3—92no—
452C22×C3—181no—
453C22×C3—181no—
454C22×C3—182no—
455C22×C3—182no—
456C22×C3—182no—
457C22×C3C22×D3211yescharacteristic
458C2×D3—32no—
459C2×D3—61no—
460C2×D3—61no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—92no—
465C2×D3—181no—
466C2×D3—181no—
467C2×D3—182no—
468C2×D3—182no—
469C2×D3—182no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—32no—
473C3×C4—32no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—61no—
477C3×C4—61no—
478C3×C4—62no—
479C3×C4—92no—
480C3×C4—121no—
481C3×C4—121no—
482C3×C4—181no—
483C3×C4—181no—
484C3×C4—182no—
485C3×C4—182no—
486C3×C4—182no—
487C3×C4#379512yes—
488Dic3—32no—
489Dic3—62no—
490Dic3—62no—
491Dic3—62no—
492Dic3—62no—
493Dic3—62no—
494Dic3—61no—
495Dic3—61no—
496Dic3—92no—
497Dic3—122no—
498Dic3—122no—
499Dic3—182no—
500Dic3—181no—
501Dic3—181no—
502C32—21no—
503C32#547911yescharacteristic, complemented
504C32#568011yescharacteristic, complemented
505C23—91no—
506C23—92no—
507Q8—182no—
508D4—182no—
509D4—544no—
510C2×C4—91no—
511C2×C4—91no—
512C2×C4—91no—
513C2×C4—92no—
514C2×C4—92no—
515C2×C4—92no—
516C2×C4—182no—
517C2×C4—182no—
518C2×C4—182no—
519C2×C4—544no—
520C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
521C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
522C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—31no—
526C2×C3—31no—
527C2×C3—31no—
528C2×C3—31no—
529C2×C3—61no—
530C2×C3—61no—
531C2×C3—61no—
532C2×C3—182no—
533C2×C3—362no—
534C2×C3C2×#379511yescharacteristic
535C2×C3D3×#23511yescharacteristic
536C2×C3D3×#23511yescharacteristic
537D3—62no—
538D3—122no—
539D3—182no—
540D3—362no—
541C22—92no—
542C22—181no—
543C22—181no—
544C22—182no—
545C22—182no—
546C22—182no—
547C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
548C4—21no—
549C4—21no—
550C4—92no—
551C4—181no—
552C4—182no—
553C4—181no—
554C4—182no—
555C4—182no—
556C4#8864812yes—
557C3—31no—
558C3—31no—
559C3—61no—
560C3#1059392111yescharacteristic
561C2—182no—
562C2—362no—
563C2C2×#8864811yescharacteristic
564C2#1172249011yescharacteristic
565C2#1172249011yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radical#11721141C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#529
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721716C2semidirect
2. #64084⋊C3C2semidirect
3. C42×(C3⋊D3)⋊C3C2semidirect
5. #11721139C2semidirect
6. #11721132C2semidirect
7. #11721122C2semidirect
8. #11721120C2semidirect
9. #11721109C2semidirect
10. #11721106C2semidirect
11. #11721082C2semidirect
37. #64084D3semidirect
38. C42×C3⋊D3D3semidirect
352. H(3,𝔽3)#529semidirect
503. C32#5479semidirect
504. C32#5680semidirect

Maximal subgroups

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

subgroupindex
#117217162
#64084⋊C32
C42×(C3⋊D3)⋊C32
#117211412
#117211392
#117211322
#117211222
#117211202
#117211092
#117211062
#117210822
#105939763
#105939433