Finite Group Databasebeta

← #13150631

Subgroups — #13150631

Order 1728.

841 conjugacy classes of subgroups.

The subgroup lattice has 563 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊#1084C211yesmaximal, characteristic, complemented
2C4×#638⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic, complemented
4#11721307C212yesmaximal
5#11721298C212yesmaximal, complemented
6#11721111C212yesmaximal, complemented
7#11721106C211yesmaximal, characteristic, complemented
8#11721093C211yesmaximal, characteristic, complemented
9#11721085C212yesmaximal, complemented
10#11721082C211yesmaximal, characteristic, complemented
11#11721081C211yesmaximal, characteristic, complemented
12#10594849—31nomaximal
13#10594825—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C22×#1084C2212yes—
16C2×#638⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20C4⋊#1084C2212yes—
21C4⋊#1084C2211yescharacteristic
22#3757⋊C3C2211yescharacteristic
23#3757⋊C3C2212yes—
24#3752⋊C3C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#3750⋊C3C2212yes—
27#3750⋊C3C2211yescharacteristic
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29#88450C2212yes—
30#88450C2212yes—
31#88449C2212yes—
32#88449C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#88447C2212yes—
37C4×#638D311yescharacteristic
38#64133D311yescharacteristic, complemented
39C3×C4×#67—31no—
40C2×C3×C4⋊Dic3—31no—
41C2×C3×C4⋊Dic3—31no—
42C2×C3×C4⋊Dic3—61no—
43C3×#924—31no—
44#63966—32no—
45#63961—32no—
46#63955—32no—
47#63955—32no—
48#63872—31no—
49#63871—31no—
50#63869—31no—
51#63858—32no—
52#63857—31no—
53#63849—31no—
54#63848—32no—
55#63847—32no—
56#63846—32no—
57#63844—31no—
58#63843—31no—
59#63841—31no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)—22no—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×C4×H(3,𝔽3)C2311yescharacteristic
65C2×#1084—24no—
66C2×#1084C2312yes—
67C2×#1084C2312yes—
68#638⋊C3—22no—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2312yes—
73C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
74C4×(C3⋊D3)⋊C3—22no—
75#634⋊C3—22no—
76#5642—91no—
77#5541—91no—
78#5513—91no—
79C22×C32×C4—21no—
80C22×C32×C4C2×D311yescharacteristic
81C22×C32×C4C2×D311yescharacteristic
82C2×#638C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D311yescharacteristic
84C2×#634C2×D311yescharacteristic
85C2×C3×#67—31no—
86C22×C3×Dic3—32no—
87C22×C3×Dic3—32no—
88C22×C3×Dic3—62no—
89C2×C3×C4×D3—31no—
90C2×C3×Dic6—31no—
91#3757C2×D312yes—
92#3757C2×D311yescharacteristic
93#3752C2×D311yescharacteristic
94C4⋊(C3⋊Dic3)C2×D311yescharacteristic
95#3750C2×D311yescharacteristic
96#3750C2×D312yes—
97C4×C3⋊Dic3C2×D311yescharacteristic
98C3×#217—32no—
99C3×#217—31no—
100C3×(C2×D3)⋊C4—31no—
101C3×C4⋊Dic3—31no—
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—32no—
105C3×C4⋊Dic3—32no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—31no—
108C3×C4⋊Dic3—61no—
109C3×C4⋊Dic3—62no—
110C3×C4⋊Dic3—61no—
111C3×Dic3⋊C4—32no—
112C3×Dic3⋊C4—31no—
113C3×C4×Dic3—31no—
114#3724—32no—
115#3724—32no—
116Dic3⋊Dic3—32no—
117Dic3⋊Dic3—32no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122#3722—32no—
123#3722—32no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—32no—
126Dic32—32no—
127Dic32—32no—
128Dic32—32no—
129Dic32—32no—
130C22×H(3,𝔽3)—21no—
131C22×H(3,𝔽3)—21no—
132C22×H(3,𝔽3)#4112yes—
133C22×H(3,𝔽3)C2411yescharacteristic
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3—21no—
137C4×H(3,𝔽3)—21no—
138C4×H(3,𝔽3)#4112yes—
139#1084—22no—
140#1084—22no—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3—21no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—22no—
146C3×C22⋊Q8—91no—
147C3×C4×D4—91no—
148C2×C3×C4⋊C4—91no—
149#946—91no—
150Q8×Dic3—91no—
151#935—92no—
152#929—92no—
153C4×#67—31no—
154C4×#67—91no—
155#926—91no—
156#926—92no—
157C2×C4⋊Dic3—31no—
158C2×C4⋊Dic3—31no—
159C2×C4⋊Dic3—61no—
160#924—31no—
161#924—91no—
162#924—91no—
163#898—92no—
164#898—92no—
165#897—91no—
166#892—91no—
167#892—91no—
168#890—91no—
169#890—92no—
170#890—91no—
171#889—92no—
172#889—92no—
173#887—91no—
174#886—92no—
175#879—92no—
176#877—92no—
177#877—92no—
178#870—91no—
179C4×Dic6—91no—
180C23×C32—21no—
181C23×C32C22×D311yescharacteristic
182C23×C32C22×D311yescharacteristic
183C22×C3⋊D3C22×D311yescharacteristic
184C22×C3×D3—31no—
185C2×C32×C4—24no—
186C2×C32×C4—22no—
187C2×C32×C4—22no—
188C2×C32×C4—21no—
189C2×C32×C4—21no—
190C2×C32×C4C22×D311yescharacteristic
191C2×C32×C4C22×D311yescharacteristic
192C2×C32×C4C22×D311yescharacteristic
193C2×C32×C4C22×D311yescharacteristic
194#638—22no—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D312yes—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D311yescharacteristic
200C4×C3⋊D3—22no—
201#634—22no—
202C3×#67—62no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—31no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
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—
214C2×C3×Dic3—64no—
215C2×C3×Dic3—64no—
216C2×C3×Dic3—124no—
217C3×C4×D3—62no—
218C3×Dic6—62no—
219#552—271noSylow
220C2×H(3,𝔽3)—22no—
221C2×H(3,𝔽3)C2×#4111yescharacteristic
222C2×H(3,𝔽3)C2×#4111yescharacteristic
223C2×H(3,𝔽3)2−1+411yescharacteristic
224(C3⋊D3)⋊C3—41no—
225C2×C3×Q8—91no—
226C2×C3×D4—91no—
227C22×C3×C4—31no—
228C22×C3×C4—31no—
229C22×C3×C4—61no—
230C22×C3×C4—91no—
231C22×C3×C4—92no—
232C22×C3×C4D3211yescharacteristic
233C2×#67—31no—
234C2×#67—91no—
235C22×Dic3—32no—
236C22×Dic3—32no—
237C22×Dic3—62no—
238C2×C4×D3—31no—
239C2×C4×D3—91no—
240C2×Dic6—31no—
241C2×Dic6—91no—
242C3×C4⋊C4—91no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—91no—
248C3×C22⋊C4—91no—
249C3×C22⋊C4—91no—
250C3×C22⋊C4—92no—
251C3×C42—91no—
252#217—32no—
253#217—31no—
254#217—91no—
255#217—92no—
256#217—92no—
257#217—92no—
258(C2×D3)⋊C4—31no—
259(C2×D3)⋊C4—91no—
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—32no—
267C4⋊Dic3—32no—
268C4⋊Dic3—31no—
269C4⋊Dic3—31no—
270C4⋊Dic3—31no—
271C4⋊Dic3—61no—
272C4⋊Dic3—61no—
273C4⋊Dic3—62no—
274C4⋊Dic3—91no—
275C4⋊Dic3—92no—
276C4⋊Dic3—92no—
277C4⋊Dic3—92no—
278C4⋊Dic3—92no—
279Dic3⋊C4—31no—
280Dic3⋊C4—32no—
281Dic3⋊C4—92no—
282Dic3⋊C4—91no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287Dic3⋊C4—92no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—92no—
291C4×Dic3—31no—
292C4×Dic3—91no—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C4×Dic3—92no—
298C4×Dic3—92no—
299C22×C32—22no—
300C22×C32—22no—
301C22×C32—21no—
302C22×C32—21no—
303C22×C32—21no—
304C22×C32—21no—
305C22×C32—21no—
306C22×C32—22no—
307C22×C32#23512yes—
308C22×C32#23912yes—
309C22×C32C23×D311yescharacteristic
310C22×C32C23×D311yescharacteristic
311C2×C3⋊D3—21no—
312C2×C3⋊D3—21no—
313C2×C3⋊D3—21no—
314C2×C3×D3—61no—
315C2×C3×D3—61no—
316C2×C3×D3—61no—
317C32×C4—22no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—22no—
321C32×C4#23712yes—
322C32×C4#23712yes—
323C3⋊Dic3—21no—
324C3⋊Dic3—22no—
325C3⋊Dic3—21no—
326C3⋊Dic3—22no—
327C3⋊Dic3—21no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331C3×Dic3—61no—
332C3×Dic3—61no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335C3×Dic3—61no—
336C3×Dic3—62no—
337C3×Dic3—122no—
338C3×Dic3—122no—
339#126—272no—
340#125—271no—
341#125—272no—
342#123—272no—
343C22⋊Q8—91no—
344C22⋊Q8—271no—
345C4×Q8—271no—
346C4×D4—91no—
347#117—272no—
348#117—271no—
349C2×C4⋊C4—91no—
350H(3,𝔽3)#55211yescharacteristic, complemented, Sylow
351C23×C3—31no—
352C23×C3—31no—
353C23×C3—61no—
354C23×C3—91no—
355C23×C3C2×D3211yescharacteristic
356C22×D3—31no—
357C22×D3—91no—
358C3×Q8—182no—
359C3×D4—182no—
360C2×C3×C4—22no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—64no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—91no—
371C2×C3×C4—91no—
372C2×C3×C4—92no—
373C2×C3×C4—92no—
374C2×C3×C4—91no—
375C2×C3×C4—92no—
376C2×C3×C4—92no—
377C2×C3×C4—182no—
378C2×C3×C4—184no—
379C2×C3×C4C2×D3211yescharacteristic
380C2×C3×C4C2×D3211yescharacteristic
381#67—62no—
382#67—182no—
383C2×Dic3—32no—
384C2×Dic3—32no—
385C2×Dic3—32no—
386C2×Dic3—32no—
387C2×Dic3—31no—
388C2×Dic3—31no—
389C2×Dic3—31no—
390C2×Dic3—32no—
391C2×Dic3—32no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—64no—
395C2×Dic3—64no—
396C2×Dic3—91no—
397C2×Dic3—91no—
398C2×Dic3—92no—
399C2×Dic3—92no—
400C2×Dic3—91no—
401C2×Dic3—124no—
402C4×D3—62no—
403C4×D3—182no—
404Dic6—62no—
405Dic6—182no—
406C2×C32—24no—
407C2×C32—22no—
408C2×C32—22no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—21no—
412C2×C32C2×#23511yescharacteristic
413C2×C32C2×#23711yescharacteristic
414C2×C32C2×#23711yescharacteristic
415C2×C32C2×#23911yescharacteristic
416C2×C32#101011yescharacteristic
417C2×C32#101011yescharacteristic
418C3⋊D3—41no—
419C3×D3—121no—
420C2×Q8—91no—
421C2×D4—91no—
422C22×C4—92no—
423C22×C4—91no—
424C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
425C4⋊C4—91no—
426C4⋊C4—91no—
427C4⋊C4—92no—
428C4⋊C4—91no—
429C4⋊C4—91no—
430C4⋊C4—92no—
431C4⋊C4—272no—
432C4⋊C4—272no—
433C4⋊C4—272no—
434C4⋊C4—272no—
435C22⋊C4—91no—
436C22⋊C4—92no—
437C22⋊C4—91no—
438C22⋊C4—272no—
439C22⋊C4—272no—
440C42—91no—
441C42—272no—
442C42—272no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—31no—
446C22×C3—31no—
447C22×C3—32no—
448C22×C3—32no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—62no—
454C22×C3—62no—
455C22×C3—61no—
456C22×C3—62no—
457C22×C3—181no—
458C22×C3—181no—
459C22×C3—181no—
460C22×C3#379912yes—
461C22×C3C22×D3211yescharacteristic
462C2×D3—61no—
463C2×D3—61no—
464C2×D3—61no—
465C2×D3—181no—
466C2×D3—181no—
467C2×D3—181no—
468C3×C4—21no—
469C3×C4—32no—
470C3×C4—32no—
471C3×C4—61no—
472C3×C4—61no—
473C3×C4—62no—
474C3×C4—62no—
475C3×C4—181no—
476C3×C4—181no—
477C3×C4—182no—
478C3×C4—181no—
479C3×C4—182no—
480C3×C4—182no—
481C3×C4—182no—
482C3×C4#380512yes—
483Dic3—62no—
484Dic3—62no—
485Dic3—62no—
486Dic3—62no—
487Dic3—61no—
488Dic3—62no—
489Dic3—61no—
490Dic3—62no—
491Dic3—61no—
492Dic3—122no—
493Dic3—122no—
494Dic3—182no—
495Dic3—181no—
496Dic3—181no—
497Dic3—182no—
498Dic3—181no—
499C32—21no—
500C32#554111yescharacteristic, complemented
501C32#564211yescharacteristic, complemented
502C23—91no—
503C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
504Q8—182no—
505D4—182no—
506C2×C4—22no—
507C2×C4—91no—
508C2×C4—91no—
509C2×C4—91no—
510C2×C4—92no—
511C2×C4—92no—
512C2×C4—92no—
513C2×C4—92no—
514C2×C4—184no—
515C2×C4—182no—
516C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
517C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
518C2×C3—22no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—31no—
523C2×C3—31no—
524C2×C3—31no—
525C2×C3—62no—
526C2×C3—62no—
527C2×C3—64no—
528C2×C3—61no—
529C2×C3—61no—
530C2×C3—61no—
531C2×C3—361no—
532C2×C3#6429811yescharacteristic
533C2×C3C2×#379911yescharacteristic
534C2×C3C2×#380511yescharacteristic
535D3—121no—
536D3—361no—
537C22—21no—
538C22—21no—
539C22—181no—
540C22—181no—
541C22—181no—
542C22#8865212yes—
543C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
544C4—21no—
545C4—181no—
546C4—182no—
547C4—181no—
548C4—182no—
549C4—182no—
550C4—182no—
551C4—181no—
552C4#8867212yes—
553C3—31no—
554C3—31no—
555C3—61no—
556C3#1059485311yescharacteristic
557C2—22no—
558C2—361no—
559C2#1172248311yescharacteristic
560C2C2×#8865211yescharacteristic
561C2C2×#8867211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC2×#638⋊C3C22
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#552
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. C2×C4⋊#1084C2semidirect
3. #64133⋊C3C2semidirect
5. #11721298C2semidirect
6. #11721111C2semidirect
7. #11721106C2semidirect
8. #11721093C2semidirect
9. #11721085C2semidirect
10. #11721082C2semidirect
11. #11721081C2semidirect
38. #64133D3semidirect
350. H(3,𝔽3)#552semidirect
500. C32#5541semidirect
501. C32#5642semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#10842
C4×#638⋊C32
#64133⋊C32
#117213072
#117212982
#117211112
#117211062
#117210932
#117210852
#117210822
#117210812
#105948493
#105948253