Finite Group Databasebeta

← #13147764

Subgroups — #13147764

Order 1728.

847 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6229C211yesmaximal, characteristic, complemented
2C42×(C3⋊D3)⋊C3C211yesmaximal, characteristic
3C4×#634⋊C3C211yesmaximal, characteristic, complemented
4#11721116C211yesmaximal, characteristic, complemented
5C4×#6196C212yesmaximal, complemented
6#11721107C211yesmaximal, characteristic
7#11721106C211yesmaximal, characteristic
8#11721105C212yesmaximal
9#11721101C212yesmaximal
10#11721091C212yesmaximal, complemented
11#11721090C211yesmaximal, characteristic, complemented
12#10593882—31nomaximal
13#10593873—31nomaximal
14C2×#6229C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#634⋊C3C2211yescharacteristic
19C2×#6196C2212yes—
20C42×H(3,𝔽3)C2211yescharacteristic
21C4⋊#1084C2211yescharacteristic
22#88540C2212yes—
23C4×#1084C2212yes—
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28C4×(C3⋊Dic3)⋊C3C2212yes—
29C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
30#88450C2212yes—
31#88450C2212yes—
32#88449C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88448C2212yes—
36#88448C2212yes—
37C42×C3⋊D3D311yescharacteristic
38C4×#634D311yescharacteristic, complemented
39C3×C42×D3—31no—
40C3×C4×Dic6—31no—
41C3×C4×Dic6—31no—
42C3×C4×Dic6—31no—
43C3×C4×Dic6—61no—
44#63919—31no—
45#63918—31no—
46C4×#627—32no—
47C4×#627—32no—
48#63874—32no—
49#63873—31no—
50#63871—31no—
51#63871—31no—
52#63870—32no—
53#63867—32no—
54#63866—31no—
55#63865—32no—
56#63857—31no—
57#63856—31no—
58#63855—32no—
59#63851—32no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2311yescharacteristic
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×C4×H(3,𝔽3)C2311yescharacteristic
64C2×#1084C2312yes—
65C2×#1084C2312yes—
66#6229—22no—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2312yes—
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4×(C3⋊D3)⋊C3—22no—
73C4×(C3⋊D3)⋊C3—22no—
74C4×(C3⋊D3)⋊C3Q814yescomplemented
75#634⋊C3—22no—
76#6196—24no—
77#5578—91no—
78#5578—91no—
79#5470—91no—
80C2×C4×C3⋊D3C2×D311yescharacteristic
81C2×C4×C3⋊D3C2×D311yescharacteristic
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×#634C2×D311yescharacteristic
84C2×C3×C4×D3—31no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×Dic6—31no—
88C2×C3×Dic6—31no—
89C2×C3×Dic6—31no—
90C2×C3×Dic6—61no—
91C2×#627—32no—
92C2×#627—32no—
93C32×C42—21no—
94C32×C42C2×D311yescharacteristic
95C32×C42C2×D311yescharacteristic
96C4⋊(C3⋊Dic3)C2×D311yescharacteristic
97#3750C2×D312yes—
98C4×C3⋊Dic3C2×D312yes—
99C4×C3⋊Dic3C2×D311yescharacteristic
100C4×C3⋊Dic3C2×D311yescharacteristic
101C4×C3⋊Dic3C2×D311yescharacteristic
102C3×C4⋊Dic3—31no—
103C3×C4⋊Dic3—31no—
104C3×C4⋊Dic3—31no—
105C3×C4⋊Dic3—61no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—32no—
108C3×Dic3⋊C4—32no—
109C3×Dic3⋊C4—62no—
110C3×C4×Dic3—32no—
111C3×C4×Dic3—31no—
112C3×C4×Dic3—31no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—31no—
116C3×C4×Dic3—62no—
117#3724—32no—
118#3724—32no—
119#3724—32no—
120#3724—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127#3722—32no—
128#3722—32no—
129(C2×D3)⋊Dic3—32no—
130(C2×D3)⋊Dic3—32no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C2×(C3⋊D3)⋊C3—21no—
133C2×(C3⋊D3)⋊C3—21no—
134C2×(C3⋊D3)⋊C3C2×Q812yes—
135C4×H(3,𝔽3)—21no—
136C4×H(3,𝔽3)C2×Q812yes—
137C4×H(3,𝔽3)#4112yes—
138#1084—22no—
139#1084—22no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—22no—
142(C3⋊Dic3)⋊C3—21no—
143(C3⋊Dic3)⋊C3C2×Q812yes—
144(C3⋊Dic3)⋊C3#4112yes—
145C3×C4×Q8—91no—
146C3×C4×Q8—91no—
147C2×C3×C42—91no—
148#946—91no—
149#946—91no—
150#944—91no—
151#944—91no—
152#926—92no—
153#924—92no—
154#924—91no—
155#924—91no—
156#897—91no—
157#897—91no—
158#896—92no—
159#896—92no—
160#889—92no—
161#889—92no—
162#888—91no—
163#888—91no—
164#872—92no—
165#872—92no—
166C42×D3—31no—
167C42×D3—91no—
168#870—92no—
169#869—91no—
170#869—91no—
171C4×Dic6—31no—
172C4×Dic6—31no—
173C4×Dic6—31no—
174C4×Dic6—61no—
175C4×Dic6—91no—
176C4×Dic6—92no—
177C4×Dic6—92no—
178C4×Dic6—92no—
179C22×C3⋊D3C22×D311yescharacteristic
180C22×C3×D3—31no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—21no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C32×C4C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D312yes—
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C4×C3⋊D3—22no—
196C4×C3⋊D3—22no—
197C4×C3⋊D3Dic614yescomplemented
198#634—22no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—31no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—62no—
209C2×C3×Dic3—62no—
210C3×C4×D3—34no—
211C3×C4×D3—62no—
212C3×C4×D3—62no—
213C3×Dic6—64no—
214C3×Dic6—62no—
215C3×Dic6—62no—
216C3×Dic6—62no—
217#627—64no—
218#627—64no—
219#533—271noSylow
220C2×H(3,𝔽3)C22×Q811yescharacteristic
221C2×H(3,𝔽3)C2×#4111yescharacteristic
222C2×H(3,𝔽3)C2×#4111yescharacteristic
223(C3⋊D3)⋊C3—22no—
224C2×C3×Q8—91no—
225C2×C3×Q8—91no—
226C22×C3×C4—91no—
227C22×C3×C4—91no—
228C22×C3×C4—91no—
229C2×C4×D3—31no—
230C2×C4×D3—31no—
231C2×C4×D3—31no—
232C2×C4×D3—91no—
233C2×C4×D3—91no—
234C2×C4×D3—91no—
235C2×Dic6—31no—
236C2×Dic6—31no—
237C2×Dic6—31no—
238C2×Dic6—61no—
239C2×Dic6—91no—
240C2×Dic6—92no—
241C2×Dic6—92no—
242C2×Dic6—92no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—92no—
245C3×C4⋊C4—91no—
246C3×C4⋊C4—92no—
247C3×C42—31no—
248C3×C42—31no—
249C3×C42—61no—
250C3×C42—91no—
251C3×C42—92no—
252C3×C42—91no—
253C3×C42—91no—
254C3×C42—92no—
255C3×C42D3211yescharacteristic
256#217—92no—
257#217—92no—
258(C2×D3)⋊C4—92no—
259(C2×D3)⋊C4—92no—
260(C2×D3)⋊C4—92no—
261(C2×D3)⋊C4—92no—
262C4⋊Dic3—31no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266C4⋊Dic3—91no—
267C4⋊Dic3—92no—
268C4⋊Dic3—92no—
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—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285Dic3⋊C4—92no—
286Dic3⋊C4—92no—
287C4×Dic3—32no—
288C4×Dic3—32no—
289C4×Dic3—32no—
290C4×Dic3—31no—
291C4×Dic3—31no—
292C4×Dic3—31no—
293C4×Dic3—62no—
294C4×Dic3—91no—
295C4×Dic3—92no—
296C4×Dic3—91no—
297C4×Dic3—91no—
298C22×C32—21no—
299C22×C32C23×D311yescharacteristic
300C22×C32C23×D311yescharacteristic
301C2×C3⋊D3—21no—
302C2×C3⋊D3—21no—
303C2×C3⋊D3C2×Dic612yes—
304C2×C3×D3—32no—
305C2×C3×D3—61no—
306C2×C3×D3—61no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—22no—
310C32×C4—22no—
311C32×C4—41no—
312C32×C4C2×Dic612yes—
313C32×C4#23512yes—
314C32×C4#23512yes—
315C32×C4D3×Q812yes—
316C3⋊Dic3—22no—
317C3⋊Dic3—21no—
318C3⋊Dic3—22no—
319C3⋊Dic3C2×Dic612yes—
320C3⋊Dic3#23512yes—
321C3×Dic3—32no—
322C3×Dic3—32no—
323C3×Dic3—64no—
324C3×Dic3—62no—
325C3×Dic3—62no—
326C3×Dic3—61no—
327C3×Dic3—62no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331C3×Dic3—122no—
332C4⋊Q8—271no—
333C4⋊Q8—271no—
334#125—272no—
335C22⋊Q8—272no—
336C22⋊Q8—271no—
337C22⋊Q8—271no—
338C4×Q8—91no—
339C4×Q8—91no—
340C4×Q8—272no—
341#117—272no—
342C2×C42—91no—
343H(3,𝔽3)#53311yescharacteristic, complemented, Sylow
344C23×C3—91no—
345C22×D3—31no—
346C22×D3—91no—
347C3×Q8—182no—
348C3×Q8—182no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—31no—
352C2×C3×C4—31no—
353C2×C3×C4—31no—
354C2×C3×C4—31no—
355C2×C3×C4—61no—
356C2×C3×C4—61no—
357C2×C3×C4—61no—
358C2×C3×C4—94no—
359C2×C3×C4—91no—
360C2×C3×C4—91no—
361C2×C3×C4—91no—
362C2×C3×C4—92no—
363C2×C3×C4—92no—
364C2×C3×C4—92no—
365C2×C3×C4—92no—
366C2×C3×C4—182no—
367C2×C3×C4—182no—
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×Dic3—31no—
372C2×Dic3—31no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—32no—
377C2×Dic3—31no—
378C2×Dic3—32no—
379C2×Dic3—32no—
380C2×Dic3—62no—
381C2×Dic3—62no—
382C2×Dic3—91no—
383C2×Dic3—91no—
384C2×Dic3—91no—
385C2×Dic3—92no—
386C2×Dic3—92no—
387C4×D3—34no—
388C4×D3—62no—
389C4×D3—62no—
390C4×D3—94no—
391C4×D3—182no—
392C4×D3—182no—
393Dic6—62no—
394Dic6—62no—
395Dic6—64no—
396Dic6—62no—
397Dic6—182no—
398Dic6—184no—
399Dic6—184no—
400Dic6—184no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32C22×Dic611yescharacteristic
405C2×C32C2×#23511yescharacteristic
406C2×C32C2×#23511yescharacteristic
407C2×C32C2×#23511yescharacteristic
408C2×C32C2×D3×Q811yescharacteristic
409C2×C32D3×#4111yescharacteristic
410C3⋊D3—22no—
411C3×D3—62no—
412C2×Q8—91no—
413C2×Q8—91no—
414C2×Q8—272no—
415C22×C4—91no—
416C22×C4—91no—
417C22×C4—91no—
418C4⋊C4—92no—
419C4⋊C4—92no—
420C4⋊C4—91no—
421C4⋊C4—91no—
422C4⋊C4—272no—
423C4⋊C4—272no—
424C4⋊C4—272no—
425C4⋊C4—272no—
426C4⋊C4—272no—
427C22⋊C4—272no—
428C22⋊C4—272no—
429C42—91no—
430C42—91no—
431C42—91no—
432C42—92no—
433C42—92no—
434C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
435C22×C3—31no—
436C22×C3—31no—
437C22×C3—61no—
438C22×C3—92no—
439C22×C3—181no—
440C22×C3—181no—
441C22×C3C22×D3211yescharacteristic
442C2×D3—32no—
443C2×D3—61no—
444C2×D3—61no—
445C2×D3—92no—
446C2×D3—181no—
447C2×D3—181no—
448C3×C4—21no—
449C3×C4—32no—
450C3×C4—32no—
451C3×C4—32no—
452C3×C4—32no—
453C3×C4—61no—
454C3×C4—61no—
455C3×C4—62no—
456C3×C4—62no—
457C3×C4—92no—
458C3×C4—92no—
459C3×C4—121no—
460C3×C4—182no—
461C3×C4—181no—
462C3×C4—182no—
463C3×C4—182no—
464C3×C4—182no—
465C3×C4D3×Dic612yes—
466C3×C4#379812yes—
467Dic3—32no—
468Dic3—32no—
469Dic3—62no—
470Dic3—62no—
471Dic3—62no—
472Dic3—62no—
473Dic3—64no—
474Dic3—62no—
475Dic3—61no—
476Dic3—62no—
477Dic3—92no—
478Dic3—92no—
479Dic3—122no—
480Dic3—182no—
481Dic3—182no—
482Dic3—181no—
483C32—21no—
484C32#547011yescharacteristic, complemented
485C32#557811yescharacteristic, complemented
486C23—91no—
487Q8—182no—
488Q8—182no—
489Q8—544no—
490C2×C4—91no—
491C2×C4—91no—
492C2×C4—94no—
493C2×C4—91no—
494C2×C4—92no—
495C2×C4—92no—
496C2×C4—92no—
497C2×C4—92no—
498C2×C4—182no—
499C2×C4—182no—
500C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
501C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
502C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
503C2×C3—31no—
504C2×C3—31no—
505C2×C3—31no—
506C2×C3—31no—
507C2×C3—31no—
508C2×C3—31no—
509C2×C3—61no—
510C2×C3—61no—
511C2×C3—61no—
512C2×C3—182no—
513C2×C3C2×D3×Dic611yescharacteristic
514C2×C3C2×#379811yescharacteristic
515C2×C3D3×#23511yescharacteristic
516D3—62no—
517D3—182no—
518C22—92no—
519C22—181no—
520C22—181no—
521C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
522C4—21no—
523C4—92no—
524C4—92no—
525C4—181no—
526C4—182no—
527C4—182no—
528C4—182no—
529C4—182no—
530C4((C3⋊D3)⋊C3)⋊Q812yes—
531C4#8865012yes—
532C3—31no—
533C3—31no—
534C3—61no—
535C3#1059387311yescharacteristic
536C2—182no—
537C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
538C2C2×#8865011yescharacteristic
539C2#1172249011yescharacteristic
subgroupG/H
centerC2×C4C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC22×(C3⋊D3)⋊C3C23
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#533
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#6229C2semidirect
3. C4×#634⋊C3C2semidirect
4. #11721116C2semidirect
5. C4×#6196C2semidirect
10. #11721091C2semidirect
11. #11721090C2semidirect
38. C4×#634D3semidirect
74. C4×(C3⋊D3)⋊C3Q8semidirect
197. C4×C3⋊D3Dic6semidirect
343. H(3,𝔽3)#533semidirect
484. C32#5470semidirect
485. C32#5578semidirect

Maximal subgroups

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

subgroupindex
C4×#62292
C42×(C3⋊D3)⋊C32
C4×#634⋊C32
#117211162
C4×#61962
#117211072
#117211062
#117211052
#117211012
#117210912
#117210902
#105938823
#105938733