Finite Group Databasebeta

← #13149451

Subgroups — #13149451

Order 1728.

825 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721760C211yesmaximal, characteristic, complemented
2#64106⋊C3C211yesmaximal, characteristic, complemented
3#64099⋊C3C211yesmaximal, characteristic, complemented
4C4×#6200C211yesmaximal, characteristic
5#11721130C212yesmaximal
6#11721129C211yesmaximal, characteristic, complemented
7#11721121C212yesmaximal, complemented
8#11721095C212yesmaximal, complemented
9#11721094C211yesmaximal, characteristic, complemented
10#11721088C212yesmaximal, complemented
11#11721082C211yesmaximal, characteristic, complemented
12#10594584—31nomaximal
13#10594456—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17C2×#6200C2211yescharacteristic
18C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
19C4⋊C4×H(3,𝔽3)C2211yescharacteristic
20#88542C2212yes—
21C4⋊#1084C2211yescharacteristic
22C4×#1084C2212yes—
23#3752⋊C3C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2212yes—
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26C4⋊((C3⋊Dic3)⋊C3)C2212yes—
27#3750⋊C3C2212yes—
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29#88451C2211yescharacteristic
30#88451C2212yes—
31#88450C2211yescharacteristic
32#88448C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88448C2211yescharacteristic
36#88447C2212yes—
37#64106D311yescharacteristic, complemented
38#64099D311yescharacteristic, complemented
39C3×#897—31no—
40C3×#892—31no—
41C3×#892—31no—
42C3×#892—61no—
43C3×#890—31no—
44C4×#626—31no—
45C4×#625—31no—
46#63900—31no—
47#63891—32no—
48D3×C4⋊Dic3—31no—
49#63877—32no—
50#63869—32no—
51#63869—32no—
52#63861—32no—
53#63858—32no—
54#63857—31no—
55#63856—31no—
56#63854—32no—
57#63844—31no—
58#63843—31no—
59#63836—32no—
60C22×#272C2311yescharacteristic
61C22×(C3⋊D3)⋊C3C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2312yes—
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2312yes—
65C2×#1084C2311yescharacteristic
66C4×#272—22no—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4×(C3⋊D3)⋊C3—22no—
73#634⋊C3—22no—
74#6200—22no—
75((C3⋊D3)⋊C3)⋊C4—24no—
76#5694—91no—
77#5642—91no—
78#5529—91no—
79C2×C4×C3⋊D3C2×D311yescharacteristic
80C2×#634C2×D311yescharacteristic
81C2×C3×C4×D3—31no—
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—61no—
85C2×C3×Dic6—31no—
86C2×#626—31no—
87C2×#625—31no—
88C2×#624—32no—
89C2×D3×Dic3—32no—
90C32×C4⋊C4—21no—
91C32×C4⋊C4C2×D311yescharacteristic
92C32×C4⋊C4C2×D311yescharacteristic
93#3752C2×D312yes—
94C4⋊(C3⋊Dic3)C2×D311yescharacteristic
95C4⋊(C3⋊Dic3)C2×D312yes—
96C4⋊(C3⋊Dic3)C2×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—62no—
103C3×C4⋊Dic3—32no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—31no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—32no—
108C3×C4⋊Dic3—61no—
109C3×Dic3⋊C4—32no—
110C3×C4×Dic3—31no—
111C3×C4×Dic3—32no—
112C3×C4×Dic3—32no—
113C3×C4×Dic3—62no—
114#3724—32no—
115#3724—32no—
116#3724—32no—
117#3724—31no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—31no—
122#3722—31no—
123(C2×D3)⋊Dic3—32no—
124(C2×D3)⋊Dic3—32no—
125(C2×D3)⋊Dic3—31no—
126(C2×D3)⋊Dic3—31no—
127(C2×D3)⋊Dic3—31no—
128Dic32—32no—
129Dic32—32no—
130C22×H(3,𝔽3)C2411yescharacteristic
131C2×#272—21no—
132C2×#272—21no—
133C2×#272—21no—
134C2×(C3⋊D3)⋊C3—21no—
135C2×(C3⋊D3)⋊C3—21no—
136C2×(C3⋊D3)⋊C3#4112yes—
137C4×H(3,𝔽3)—22no—
138C4×H(3,𝔽3)#4112yes—
139#1084—21no—
140#1084—22no—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3—22no—
144(C3⋊Dic3)⋊C3—21no—
145(C3⋊Dic3)⋊C3—21no—
146C3×#125—91no—
147C3×C22⋊Q8—91no—
148C3×#117—91no—
149#946—91no—
150Q8×Dic3—91no—
151C4×#67—91no—
152C4×#67—91no—
153C2×C4⋊Dic3—91no—
154#924—91no—
155#898—92no—
156#898—92no—
157#897—31no—
158#897—91no—
159#897—91no—
160#894—92no—
161#892—31no—
162#892—31no—
163#892—61no—
164#892—92no—
165D3×C4⋊C4—91no—
166D3×C4⋊C4—91no—
167#890—31no—
168#890—91no—
169#890—92no—
170#890—92no—
171#887—91no—
172#886—92no—
173#883—92no—
174#879—92no—
175#877—92no—
176C4×D12—91no—
177#872—92no—
178#870—92no—
179C4×Dic6—91no—
180C22×C3⋊D3C22×D311yescharacteristic
181C22×C3×D3—31no—
182C22×C3×D3—31no—
183C22×C3×D3—31no—
184C22×C3×D3—61no—
185C2×C32×C4—21no—
186C2×C32×C4—22no—
187C2×C32×C4C22×D312yes—
188C2×C32×C4C22×D312yes—
189C2×C32×C4C22×D311yescharacteristic
190C2×C32×C4C22×D311yescharacteristic
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D312yes—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C4×C3⋊D3—22no—
197#634—22no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—31no—
206C2×C3×Dic3—31no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—62no—
209C3×C4×D3—64no—
210C3×C4×D3—62no—
211C3×C4×D3—62no—
212C3×C4×D3—62no—
213C3×Dic6—62no—
214#626—62no—
215#625—62no—
216#624—64no—
217D3×Dic3—64no—
218#552—271noSylow
219C2×H(3,𝔽3)C2×#4111yescharacteristic
220C2×H(3,𝔽3)C2×#4111yescharacteristic
221C2×H(3,𝔽3)2−1+411yescharacteristic
222#272—41no—
223(C3⋊D3)⋊C3—22no—
224C2×C3×Q8—91no—
225C22×C3×C4—91no—
226C22×C3×C4—91no—
227C2×#67—91no—
228C2×#67—91no—
229C22×Dic3—92no—
230C2×D12—91no—
231C2×C4×D3—31no—
232C2×C4×D3—31no—
233C2×C4×D3—31no—
234C2×C4×D3—61no—
235C2×C4×D3—91no—
236C2×C4×D3—92no—
237C2×C4×D3—92no—
238C2×Dic6—31no—
239C2×Dic6—91no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—92no—
247C3×C4⋊C4—91no—
248C3×C4⋊C4D3211yescharacteristic
249C3×C22⋊C4—92no—
250C3×C22⋊C4—92no—
251C3×C42—91no—
252C3×C42—92no—
253#217—91no—
254#217—92no—
255#217—91no—
256(C2×D3)⋊C4—32no—
257(C2×D3)⋊C4—32no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—62no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—91no—
262(C2×D3)⋊C4—91no—
263(C2×D3)⋊C4—91no—
264(C2×D3)⋊C4—92no—
265(C2×D3)⋊C4—92no—
266(C2×D3)⋊C4—91no—
267C4⋊Dic3—31no—
268C4⋊Dic3—31no—
269C4⋊Dic3—32no—
270C4⋊Dic3—32no—
271C4⋊Dic3—31no—
272C4⋊Dic3—61no—
273C4⋊Dic3—92no—
274C4⋊Dic3—92no—
275C4⋊Dic3—91no—
276C4⋊Dic3—91no—
277C4⋊Dic3—92no—
278C4⋊Dic3—92no—
279C4⋊Dic3—92no—
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—91no—
289Dic3⋊C4—92no—
290C4×Dic3—32no—
291C4×Dic3—32no—
292C4×Dic3—31no—
293C4×Dic3—62no—
294C4×Dic3—91no—
295C4×Dic3—92no—
296C4×Dic3—92no—
297C4×Dic3—92no—
298C22×C32—21no—
299C22×C32C23×D311yescharacteristic
300C22×C32C23×D311yescharacteristic
301C2×C3⋊D3—21no—
302C2×C3⋊D3—21no—
303C2×C3⋊D3#23912yes—
304C2×C3×D3—32no—
305C2×C3×D3—61no—
306C2×C3×D3—61no—
307C2×C3×D3—62no—
308C2×C3×D3—61no—
309C2×C3×D3—61no—
310C2×C3×D3—61no—
311C2×C3×D3—61no—
312C2×C3×D3—61no—
313C2×C3×D3—61no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C32×C4—22no—
317C32×C4—22no—
318C32×C4—22no—
319C32×C4—42no—
320C32×C4#23712yes—
321C32×C4#23912yes—
322C3⋊Dic3—22no—
323C3⋊Dic3—21no—
324C3⋊Dic3—22no—
325C3⋊Dic3—21no—
326C3⋊Dic3—21no—
327C3×Dic3—62no—
328C3×Dic3—61no—
329C3×Dic3—62no—
330C3×Dic3—61no—
331C3×Dic3—62no—
332C3×Dic3—61no—
333C3×Dic3—62no—
334C3×Dic3—61no—
335C3×Dic3—62no—
336C3×Dic3—61no—
337C3×Dic3—122no—
338#126—272no—
339#125—91no—
340#125—272no—
341#123—272no—
342C22⋊Q8—91no—
343C22⋊Q8—271no—
344C4×Q8—271no—
345C4×D4—271no—
346#117—91no—
347#117—272no—
348C2×C4⋊C4—271no—
349H(3,𝔽3)#55211yescharacteristic, complemented, Sylow
350C23×C3—91no—
351C23×C3—91no—
352C22×D3—31no—
353C22×D3—31no—
354C22×D3—31no—
355C22×D3—61no—
356C22×D3—91no—
357C3×Q8—182no—
358C2×C3×C4—31no—
359C2×C3×C4—31no—
360C2×C3×C4—32no—
361C2×C3×C4—32no—
362C2×C3×C4—61no—
363C2×C3×C4—62no—
364C2×C3×C4—91no—
365C2×C3×C4—92no—
366C2×C3×C4—91no—
367C2×C3×C4—91no—
368C2×C3×C4—92no—
369C2×C3×C4—92no—
370C2×C3×C4—91no—
371C2×C3×C4—182no—
372C2×C3×C4—182no—
373C2×C3×C4C2×D3212yes—
374C2×C3×C4C2×D3211yescharacteristic
375#67—182no—
376#67—182no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—32no—
380C2×Dic3—32no—
381C2×Dic3—31no—
382C2×Dic3—31no—
383C2×Dic3—32no—
384C2×Dic3—32no—
385C2×Dic3—31no—
386C2×Dic3—61no—
387C2×Dic3—62no—
388C2×Dic3—91no—
389C2×Dic3—91no—
390C2×Dic3—92no—
391C2×Dic3—91no—
392C2×Dic3—92no—
393C2×Dic3—184no—
394D12—182no—
395C4×D3—62no—
396C4×D3—62no—
397C4×D3—64no—
398C4×D3—62no—
399C4×D3—182no—
400C4×D3—184no—
401C4×D3—184no—
402Dic6—62no—
403Dic6—182no—
404C2×C32—21no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32C2×#23711yescharacteristic
408C2×C32C2×#23911yescharacteristic
409C2×C32C2×#23911yescharacteristic
410C2×C32D3×#4111yescharacteristic
411C2×C32#101011yescharacteristic
412C2×C32#101011yescharacteristic
413C3⋊D3—22no—
414C3×D3—62no—
415C3×D3—122no—
416C3×D3—121no—
417C3×D3—121no—
418C2×Q8—91no—
419C2×D4—271no—
420C22×C4—91no—
421C22×C4—91no—
422C22×C4—272no—
423C4⋊C4—91no—
424C4⋊C4—91no—
425C4⋊C4—92no—
426C4⋊C4—92no—
427C4⋊C4—92no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C4⋊C4—271no—
431C4⋊C4—272no—
432C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
433C22⋊C4—92no—
434C22⋊C4—92no—
435C22⋊C4—271no—
436C22⋊C4—271no—
437C22⋊C4—272no—
438C42—92no—
439C42—91no—
440C42—272no—
441C22×C3—31no—
442C22×C3—31no—
443C22×C3—61no—
444C22×C3—92no—
445C22×C3—181no—
446C22×C3—181no—
447C22×C3—181no—
448C22×C3—181no—
449C22×C3—181no—
450C22×C3C22×D3211yescharacteristic
451C2×D3—32no—
452C2×D3—61no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—61no—
460C2×D3—62no—
461C2×D3—92no—
462C2×D3—121no—
463C2×D3—121no—
464C2×D3—181no—
465C2×D3—181no—
466C3×C4—22no—
467C3×C4—32no—
468C3×C4—32no—
469C3×C4—62no—
470C3×C4—62no—
471C3×C4—62no—
472C3×C4—122no—
473C3×C4—182no—
474C3×C4—181no—
475C3×C4—181no—
476C3×C4—182no—
477C3×C4—181no—
478C3×C4—181no—
479C3×C4—182no—
480C3×C4#379612yes—
481Dic3—61no—
482Dic3—61no—
483Dic3—62no—
484Dic3—62no—
485Dic3—62no—
486Dic3—62no—
487Dic3—61no—
488Dic3—61no—
489Dic3—62no—
490Dic3—61no—
491Dic3—122no—
492Dic3—182no—
493Dic3—181no—
494Dic3—182no—
495Dic3—181no—
496Dic3—181no—
497C32—21no—
498C32#564211yescharacteristic, complemented
499C32#569411yescharacteristic, complemented
500C23—91no—
501C23—91no—
502Q8—182no—
503D4—542no—
504C2×C4—91no—
505C2×C4—91no—
506C2×C4—92no—
507C2×C4—91no—
508C2×C4—92no—
509C2×C4—92no—
510C2×C4—91no—
511C2×C4—182no—
512C2×C4—182no—
513C2×C4—544no—
514C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
515C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
516C2×C3—31no—
517C2×C3—31no—
518C2×C3—31no—
519C2×C3—31no—
520C2×C3—31no—
521C2×C3—31no—
522C2×C3—61no—
523C2×C3—61no—
524C2×C3—61no—
525C2×C3—182no—
526C2×C3—361no—
527C2×C3C2×#379611yescharacteristic
528C2×C3#6430911yescharacteristic
529C2×C3D3×#23911yescharacteristic
530D3—62no—
531D3—121no—
532D3—121no—
533D3—122no—
534D3—182no—
535C22—92no—
536C22—181no—
537C22—181no—
538C22—181no—
539C22—181no—
540C22—181no—
541C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
542C4—22no—
543C4—181no—
544C4—182no—
545C4—182no—
546C4—181no—
547C4—181no—
548C4—181no—
549C4—182no—
550C4#8864912yes—
551C3—31no—
552C3—31no—
553C3—61no—
554C3#1059445711yescharacteristic
555C2—182no—
556C2—361no—
557C2C2×#8864911yescharacteristic
558C2#1172250211yescharacteristic
559C2#1172251911yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC2×#6200C22
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#552
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11721760C2semidirect
2. #64106⋊C3C2semidirect
3. #64099⋊C3C2semidirect
6. #11721129C2semidirect
7. #11721121C2semidirect
8. #11721095C2semidirect
9. #11721094C2semidirect
10. #11721088C2semidirect
11. #11721082C2semidirect
37. #64106D3semidirect
38. #64099D3semidirect
349. H(3,𝔽3)#552semidirect
498. C32#5642semidirect
499. C32#5694semidirect

Maximal subgroups

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

subgroupindex
#117217602
#64106⋊C32
#64099⋊C32
C4×#62002
#117211302
#117211292
#117211212
#117210952
#117210942
#117210882
#117210822
#105945843
#105944563