Finite Group Databasebeta

← #13149579

Subgroups — #13149579

Order 1728.

851 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#11721765C211yesmaximal, characteristic
2#64097⋊C3C211yesmaximal, characteristic, complemented
3#64096⋊C3C211yesmaximal, characteristic, complemented
4#11721148C212yesmaximal
5C4×#6199C211yesmaximal, characteristic
6#11721094C211yesmaximal, characteristic
7#11721090C211yesmaximal, characteristic, complemented
8#11721085C212yesmaximal, complemented
9#11721083C211yesmaximal, characteristic, complemented
10#11721082C212yesmaximal, complemented
11#11721076C212yesmaximal
12#10594387—31nomaximal
13#10594385—31nomaximal
14C2×C4×#272C2211yescharacteristic
15C2×#6229C2211yescharacteristic
16C2×#634⋊C3C2212yes—
17C2×#634⋊C3C2211yescharacteristic
18C2×#6199C2211yescharacteristic
19C2×#6199C2211yescharacteristic
20C4⋊C4×H(3,𝔽3)C2211yescharacteristic
21#88542C2212yes—
22C4⋊#1084C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24#3750⋊C3C2212yes—
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29#88451C2212yes—
30#88449C2212yes—
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88447C2212yes—
35#88447C2211yescharacteristic
36#88447C2211yescharacteristic
37#64097D311yescharacteristic, complemented
38#64096D311yescharacteristic, complemented
39C3×#897—31no—
40C3×#897—31no—
41C3×#897—61no—
42C3×#888—31no—
43C3×#887—31no—
44#63896—32no—
45#63887—32no—
46#63878—32no—
47#63877—32no—
48C4×D3×Dic3—31no—
49C4×D3×Dic3—31no—
50#63873—31no—
51#63866—31no—
52#63857—31no—
53#63856—31no—
54#63847—32no—
55#63846—32no—
56#63844—32no—
57#63843—32no—
58#63843—31no—
59Dic3×Dic6—31no—
60C22×#272C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2311yescharacteristic
64C2×#1084C2312yes—
65C4×#272—22no—
66#6229—22no—
67C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73#634⋊C3—22no—
74#634⋊C3—24no—
75#6199—22no—
76#6199Q814yescomplemented
77#5728—91no—
78#5622—91no—
79#5578—91no—
80C2×#634C2×D311yescharacteristic
81C2×#634C2×D312yes—
82C2×C3×C4×D3—31no—
83C2×C3×C4×D3—31no—
84C2×C3×C4×D3—61no—
85C2×C3×Dic6—31no—
86C2×C3×Dic6—32no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—61no—
90C2×D3×Dic3—31no—
91C2×D3×Dic3—31no—
92C2×D3×Dic3—31no—
93C2×D3×Dic3—31no—
94C32×C4⋊C4—21no—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C4⋊C4C2×D311yescharacteristic
97C4⋊(C3⋊Dic3)C2×D311yescharacteristic
98#3750C2×D312yes—
99#3750C2×D312yes—
100C4×C3⋊Dic3C2×D311yescharacteristic
101C4×C3⋊Dic3C2×D312yes—
102C4×C3⋊Dic3C2×D311yescharacteristic
103C3×(C2×D3)⋊C4—32no—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—62no—
106C3×C4⋊Dic3—31no—
107C3×C4⋊Dic3—32no—
108C3×C4⋊Dic3—32no—
109C3×C4⋊Dic3—62no—
110C3×Dic3⋊C4—32no—
111C3×Dic3⋊C4—32no—
112C3×C4×Dic3—31no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—31no—
115#3724—32no—
116#3724—32no—
117#3724—32no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123(C2×D3)⋊Dic3—32no—
124(C2×D3)⋊Dic3—32no—
125Dic32—32no—
126Dic32—31no—
127Dic32—32no—
128Dic32—31no—
129Dic32—31no—
130Dic32—31no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C2×#272—21no—
133C2×#272—21no—
134C2×#272C2×Q812yes—
135C4×H(3,𝔽3)—22no—
136C4×H(3,𝔽3)#4112yes—
137#1084—22no—
138#1084—21no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3C2×Q812yes—
143(C3⋊Dic3)⋊C3C2×Q812yes—
144(C3⋊Dic3)⋊C3#4112yes—
145C3×C4⋊Q8—91no—
146C3×C22⋊Q8—91no—
147C3×C4×Q8—91no—
148#946—92no—
149#946—91no—
150Q8×Dic3—91no—
151Q8×Dic3—92no—
152#944—91no—
153#944—91no—
154#924—91no—
155C2×C4×Dic3—91no—
156#897—31no—
157#897—31no—
158#897—61no—
159#897—91no—
160#896—92no—
161#892—92no—
162#890—92no—
163#889—92no—
164#889—92no—
165#888—31no—
166#888—91no—
167#887—31no—
168#887—91no—
169#887—92no—
170#887—91no—
171#878—92no—
172#877—92no—
173#872—92no—
174C42×D3—91no—
175C42×D3—91no—
176#869—91no—
177C4×Dic6—92no—
178C4×Dic6—91no—
179C22×C3×D3—31no—
180C22×C3×D3—31no—
181C22×C3×D3—61no—
182C2×C32×C4—21no—
183C2×C32×C4—22no—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D312yes—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D312yes—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D312yes—
191C2×C3⋊Dic3C22×D311yescharacteristic
192C2×C3⋊Dic3C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D311yescharacteristic
194#634—22no—
195#634—24no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—32no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—31no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—61no—
207C2×C3×Dic3—62no—
208C3×C4×D3—62no—
209C3×C4×D3—62no—
210C3×C4×D3—122no—
211C3×Dic6—64no—
212C3×Dic6—62no—
213C3×Dic6—62no—
214C3×Dic6—62no—
215C3×Dic6—122no—
216D3×Dic3—34no—
217D3×Dic3—34no—
218D3×Dic3—62no—
219D3×Dic3—62no—
220#533—271noSylow
221C2×H(3,𝔽3)C22×Q811yescharacteristic
222C2×H(3,𝔽3)C2×#4111yescharacteristic
223C2×H(3,𝔽3)C2×#4111yescharacteristic
224#272—22no—
225C2×C3×Q8—91no—
226C2×C3×Q8—92no—
227C2×C3×Q8—91no—
228C22×C3×C4—91no—
229C22×Dic3—91no—
230C22×Dic3—91no—
231C2×C4×D3—31no—
232C2×C4×D3—31no—
233C2×C4×D3—61no—
234C2×C4×D3—91no—
235C2×C4×D3—91no—
236C2×C4×D3—91no—
237C2×C4×D3—91no—
238C2×Dic6—31no—
239C2×Dic6—31no—
240C2×Dic6—32no—
241C2×Dic6—31no—
242C2×Dic6—61no—
243C2×Dic6—92no—
244C2×Dic6—91no—
245C3×C4⋊C4—31no—
246C3×C4⋊C4—31no—
247C3×C4⋊C4—61no—
248C3×C4⋊C4—91no—
249C3×C4⋊C4—92no—
250C3×C4⋊C4—92no—
251C3×C4⋊C4—92no—
252C3×C4⋊C4D3211yescharacteristic
253C3×C22⋊C4—92no—
254C3×C42—91no—
255C3×C42—92no—
256C3×C42—91no—
257#217—92no—
258(C2×D3)⋊C4—32no—
259(C2×D3)⋊C4—32no—
260(C2×D3)⋊C4—62no—
261(C2×D3)⋊C4—92no—
262(C2×D3)⋊C4—92no—
263C4⋊Dic3—32no—
264C4⋊Dic3—32no—
265C4⋊Dic3—31no—
266C4⋊Dic3—62no—
267C4⋊Dic3—91no—
268C4⋊Dic3—92no—
269C4⋊Dic3—92no—
270C4⋊Dic3—92no—
271C4⋊Dic3—92no—
272Dic3⋊C4—32no—
273Dic3⋊C4—32no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—92no—
282Dic3⋊C4—92no—
283Dic3⋊C4—92no—
284C4×Dic3—32no—
285C4×Dic3—31no—
286C4×Dic3—31no—
287C4×Dic3—91no—
288C4×Dic3—91no—
289C4×Dic3—92no—
290C4×Dic3—92no—
291C4×Dic3—91no—
292C4×Dic3—91no—
293C4×Dic3—91no—
294C4×Dic3—92no—
295C4×Dic3—91no—
296C4×Dic3—91no—
297C4×Dic3—91no—
298C4×Dic3—92no—
299C22×C32—21no—
300C22×C32C23×D311yescharacteristic
301C22×C32C23×D311yescharacteristic
302C2×C3×D3—32no—
303C2×C3×D3—32no—
304C2×C3×D3—62no—
305C2×C3×D3—61no—
306C2×C3×D3—61no—
307C2×C3×D3—61no—
308C2×C3×D3—61no—
309C2×C3×D3—121no—
310C2×C3×D3—121no—
311C32×C4—22no—
312C32×C4—22no—
313C32×C4—22no—
314C32×C4—42no—
315C32×C4#23512yes—
316C32×C4#23912yes—
317C3⋊Dic3—22no—
318C3⋊Dic3—22no—
319C3⋊Dic3—21no—
320C3⋊Dic3#23712yes—
321C3⋊Dic3D3×Q812yes—
322C3⋊Dic3D3×Q812yes—
323C3×Dic3—32no—
324C3×Dic3—32no—
325C3×Dic3—32no—
326C3×Dic3—62no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331C3×Dic3—61no—
332C3×Dic3—62no—
333C3×Dic3—122no—
334C3×Dic3—121no—
335C4⋊Q8—91no—
336C4⋊Q8—271no—
337#125—272no—
338C22⋊Q8—91no—
339C22⋊Q8—272no—
340C22⋊Q8—271no—
341C4×Q8—91no—
342C4×Q8—271no—
343C4×Q8—272no—
344#117—272no—
345C2×C42—271no—
346H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
347C23×C3—91no—
348C22×D3—31no—
349C22×D3—31no—
350C22×D3—61no—
351C3×Q8—182no—
352C3×Q8—184no—
353C3×Q8—182no—
354C2×C3×C4—31no—
355C2×C3×C4—31no—
356C2×C3×C4—32no—
357C2×C3×C4—32no—
358C2×C3×C4—61no—
359C2×C3×C4—62no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—91no—
363C2×C3×C4—91no—
364C2×C3×C4—92no—
365C2×C3×C4—92no—
366C2×C3×C4—91no—
367C2×C3×C4—92no—
368C2×C3×C4—182no—
369C2×C3×C4C2×D3212yes—
370C2×C3×C4C2×D3211yescharacteristic
371C2×Dic3—31no—
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—31no—
375C2×Dic3—31no—
376C2×Dic3—32no—
377C2×Dic3—32no—
378C2×Dic3—31no—
379C2×Dic3—32no—
380C2×Dic3—32no—
381C2×Dic3—61no—
382C2×Dic3—62no—
383C2×Dic3—91no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—91no—
387C2×Dic3—92no—
388C2×Dic3—92no—
389C2×Dic3—94no—
390C2×Dic3—182no—
391C4×D3—62no—
392C4×D3—62no—
393C4×D3—94no—
394C4×D3—94no—
395C4×D3—122no—
396C4×D3—182no—
397C4×D3—182no—
398Dic6—64no—
399Dic6—62no—
400Dic6—62no—
401Dic6—62no—
402Dic6—122no—
403Dic6—184no—
404Dic6—182no—
405C2×C32—21no—
406C2×C32—21no—
407C2×C32—21no—
408C2×C32C2×#23511yescharacteristic
409C2×C32C2×#23711yescharacteristic
410C2×C32C2×D3×Q811yescharacteristic
411C2×C32C2×D3×Q811yescharacteristic
412C2×C32C2×#23911yescharacteristic
413C2×C32D3×#4111yescharacteristic
414C3×D3—62no—
415C3×D3—62no—
416C3×D3—122no—
417C2×Q8—91no—
418C2×Q8—91no—
419C2×Q8—92no—
420C22×C4—91no—
421C22×C4—271no—
422C22×C4—271no—
423C4⋊C4—92no—
424C4⋊C4—92no—
425C4⋊C4—92no—
426C4⋊C4—91no—
427C4⋊C4—272no—
428C4⋊C4—272no—
429C4⋊C4—272no—
430C4⋊C4—272no—
431C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
432C22⋊C4—92no—
433C22⋊C4—272no—
434C42—91no—
435C42—91no—
436C42—92no—
437C42—271no—
438C42—271no—
439C42—272no—
440C22×C3—31no—
441C22×C3—31no—
442C22×C3—61no—
443C22×C3—92no—
444C22×C3—181no—
445C22×C3—181no—
446C22×C3C22×D3211yescharacteristic
447C2×D3—32no—
448C2×D3—32no—
449C2×D3—62no—
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—61no—
453C2×D3—61no—
454C2×D3—121no—
455C2×D3—121no—
456C3×C4—22no—
457C3×C4—32no—
458C3×C4—32no—
459C3×C4—62no—
460C3×C4—62no—
461C3×C4—62no—
462C3×C4—92no—
463C3×C4—92no—
464C3×C4—92no—
465C3×C4—122no—
466C3×C4—181no—
467C3×C4—182no—
468C3×C4—182no—
469C3×C4—181no—
470C3×C4—182no—
471C3×C4#379912yes—
472Dic3—32no—
473Dic3—32no—
474Dic3—32no—
475Dic3—62no—
476Dic3—62no—
477Dic3—61no—
478Dic3—61no—
479Dic3—62no—
480Dic3—62no—
481Dic3—61no—
482Dic3—92no—
483Dic3—92no—
484Dic3—92no—
485Dic3—122no—
486Dic3—121no—
487Dic3—182no—
488Dic3—182no—
489Dic3—181no—
490C32—21no—
491C32#557811yescharacteristic, complemented
492C32#572811yescharacteristic, complemented
493C23—91no—
494Q8—184no—
495Q8—182no—
496Q8—182no—
497C2×C4—91no—
498C2×C4—91no—
499C2×C4—91no—
500C2×C4—91no—
501C2×C4—92no—
502C2×C4—91no—
503C2×C4—92no—
504C2×C4—92no—
505C2×C4—182no—
506C2×C4—274no—
507C2×C4—542no—
508C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
509C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
510C2×C3—31no—
511C2×C3—31no—
512C2×C3—31no—
513C2×C3—31no—
514C2×C3—31no—
515C2×C3—31no—
516C2×C3—61no—
517C2×C3—61no—
518C2×C3—61no—
519C2×C3—182no—
520C2×C3C2×#379911yescharacteristic
521C2×C3D3×#23711yescharacteristic
522C2×C3D32×Q811yescharacteristic
523D3—62no—
524D3—62no—
525D3—122no—
526C22—92no—
527C22—181no—
528C22—181no—
529C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
530C4—22no—
531C4—92no—
532C4—92no—
533C4—92no—
534C4—181no—
535C4—182no—
536C4—181no—
537C4—182no—
538C4—182no—
539C4#8865212yes—
540C3—31no—
541C3—31no—
542C3—61no—
543C3#1059439711yescharacteristic
544C2—182no—
545C2C2×#8865211yescharacteristic
546C2#1172250411yescharacteristic
547C2Q8×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC4⋊C4×H(3,𝔽3)C22
prime radicalC22×#272C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#533
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
2. #64097⋊C3C2semidirect
3. #64096⋊C3C2semidirect
7. #11721090C2semidirect
8. #11721085C2semidirect
9. #11721083C2semidirect
10. #11721082C2semidirect
37. #64097D3semidirect
38. #64096D3semidirect
76. #6199Q8semidirect
346. H(3,𝔽3)#533semidirect
491. C32#5578semidirect
492. C32#5728semidirect

Maximal subgroups

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

subgroupindex
#117217652
#64097⋊C32
#64096⋊C32
#117211482
C4×#61992
#117210942
#117210902
#117210852
#117210832
#117210822
#117210762
#105943873
#105943853