Finite Group Databasebeta

← #13149467

Subgroups — #13149467

Order 1728.

789 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721756C211yesmaximal, characteristic, complemented
2#64101⋊C3C211yesmaximal, characteristic
3#64097⋊C3C211yesmaximal, characteristic, complemented
4#11721135C212yesmaximal
5#11721129C211yesmaximal, characteristic
6#11721128C212yesmaximal
7#11721105C211yesmaximal, characteristic
8#11721095C211yesmaximal, characteristic, complemented
9#11721093C211yesmaximal, characteristic, complemented
10#11721091C212yesmaximal, complemented
11#11721090C212yesmaximal, complemented
12#10594502—31nomaximal
13#10594351—31nomaximal
14C2×#6229C2212yes—
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#634⋊C3C2212yes—
17C2×((C3⋊D3)⋊C3)⋊C4C2212yes—
18C4⋊C4×H(3,𝔽3)C2211yescharacteristic
19C4⋊#1084C2211yescharacteristic
20#88540C2212yes—
21C4×#1084C2211yescharacteristic
22#3752⋊C3C2212yes—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25#3750⋊C3C2212yes—
26C4×(C3⋊Dic3)⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28#88450C2211yescharacteristic
29#88450C2211yescharacteristic
30#88449C2212yes—
31#88448C2211yescharacteristic
32#88448C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88448C2211yescharacteristic
36#88448C2212yes—
37#64101D311yescharacteristic
38#64097D311yescharacteristic, complemented
39C3×#892—31no—
40C3×#888—31no—
41C3×#888—31no—
42C3×#888—31no—
43C3×#888—61no—
44#63900—31no—
45#63896—32no—
46D3×C4⋊Dic3—31no—
47#63888—32no—
48#63888—32no—
49#63887—32no—
50#63874—31no—
51#63873—32no—
52#63872—31no—
53#63870—32no—
54#63869—31no—
55#63869—31no—
56#63869—31no—
57#63867—32no—
58#63866—32no—
59#63855—31no—
60C22×(C3⋊D3)⋊C3C2311yescharacteristic
61C2×C4×H(3,𝔽3)C2312yes—
62C2×C4×H(3,𝔽3)C2311yescharacteristic
63C2×#1084C2312yes—
64C2×#1084C2311yescharacteristic
65C2×#1084C2311yescharacteristic
66#6229—42no—
67C2×(C3⋊Dic3)⋊C3C2312yes—
68C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2312yes—
70C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C4×(C3⋊D3)⋊C3Q814yescomplemented
73#634⋊C3—42no—
74((C3⋊D3)⋊C3)⋊C4—42no—
75#5729—91no—
76#5729—91no—
77#5519—91no—
78C2×C4×C3⋊D3C2×D311yescharacteristic
79C2×#634C2×D312yes—
80C2×C3×C4×D3—31no—
81C2×C3×Dic6—32no—
82C2×C3×Dic6—32no—
83C2×C3×Dic6—32no—
84C2×C3×Dic6—62no—
85C2×#624—32no—
86C2×D3×Dic3—32no—
87C32×C4⋊C4—21no—
88C32×C4⋊C4C2×D311yescharacteristic
89C32×C4⋊C4C2×D311yescharacteristic
90#3752C2×D312yes—
91C4⋊(C3⋊Dic3)C2×D311yescharacteristic
92C4⋊(C3⋊Dic3)C2×D311yescharacteristic
93#3750C2×D312yes—
94C4×C3⋊Dic3C2×D311yescharacteristic
95C4×C3⋊Dic3C2×D312yes—
96C3×(C2×D3)⋊C4—32no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—61no—
102C3×Dic3⋊C4—32no—
103C3×Dic3⋊C4—32no—
104C3×Dic3⋊C4—32no—
105C3×Dic3⋊C4—62no—
106C3×C4×Dic3—31no—
107C3×C4×Dic3—31no—
108C3×C4×Dic3—31no—
109C3×C4×Dic3—32no—
110C3×C4×Dic3—61no—
111#3724—32no—
112#3724—32no—
113#3724—32no—
114#3724—31no—
115#3724—32no—
116#3724—31no—
117Dic3⋊Dic3—31no—
118Dic3⋊Dic3—31no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125#3722—31no—
126#3722—31no—
127(C2×D3)⋊Dic3—31no—
128(C2×D3)⋊Dic3—31no—
129C22×H(3,𝔽3)C2411yescharacteristic
130C2×(C3⋊D3)⋊C3—21no—
131C2×(C3⋊D3)⋊C3—21no—
132C2×(C3⋊D3)⋊C3C2×Q812yes—
133C4×H(3,𝔽3)—22no—
134C4×H(3,𝔽3)C2×Q812yes—
135#1084—21no—
136#1084—22no—
137#1084—21no—
138(C3⋊Dic3)⋊C3—22no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—21no—
142(C3⋊Dic3)⋊C3C2×Q812yes—
143C3×C4⋊Q8—91no—
144C3×C4⋊Q8—91no—
145C3×#117—91no—
146#946—92no—
147#946—92no—
148#944—92no—
149#944—92no—
150#926—91no—
151C2×C4⋊Dic3—91no—
152#896—92no—
153#896—92no—
154#892—31no—
155#892—91no—
156D3×C4⋊C4—91no—
157D3×C4⋊C4—91no—
158#890—91no—
159#890—91no—
160#890—91no—
161#890—91no—
162#889—92no—
163#889—92no—
164#888—31no—
165#888—31no—
166#888—31no—
167#888—61no—
168#888—91no—
169#878—92no—
170#878—92no—
171#872—91no—
172#872—91no—
173#870—92no—
174#870—91no—
175#870—91no—
176#869—92no—
177C22×C3⋊D3C22×D311yescharacteristic
178C22×C3×D3—31no—
179C2×C32×C4—21no—
180C2×C32×C4—22no—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D312yes—
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3C22×D311yescharacteristic
186C2×C3⋊Dic3C22×D311yescharacteristic
187C2×C3⋊Dic3C22×D312yes—
188C2×C3⋊Dic3C22×D312yes—
189C2×C3⋊Dic3C22×D311yescharacteristic
190C4×C3⋊D3Dic614yescomplemented
191#634—42no—
192C2×C3×Dic3—31no—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—62no—
206C3×C4×D3—34no—
207C3×Dic6—122no—
208C3×Dic6—122no—
209C3×Dic6—122no—
210C3×Dic6—124no—
211#624—122no—
212D3×Dic3—122no—
213#544—271noSylow
214C2×H(3,𝔽3)C22×Q811yescharacteristic
215C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
216C2×H(3,𝔽3)2−1+411yescharacteristic
217(C3⋊D3)⋊C3—22no—
218C2×C3×Q8—92no—
219C2×C3×Q8—92no—
220C22×C3×C4—91no—
221C22×Dic3—92no—
222C2×C4×D3—31no—
223C2×C4×D3—91no—
224C2×C4×D3—92no—
225C2×C4×D3—92no—
226C2×Dic6—32no—
227C2×Dic6—32no—
228C2×Dic6—32no—
229C2×Dic6—62no—
230C2×Dic6—92no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—31no—
233C3×C4⋊C4—61no—
234C3×C4⋊C4—92no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—91no—
237C3×C4⋊C4—92no—
238C3×C4⋊C4—91no—
239C3×C4⋊C4D3211yescharacteristic
240C3×C22⋊C4—92no—
241C3×C42—92no—
242C3×C42—91no—
243C3×C42—91no—
244#217—91no—
245#217—91no—
246(C2×D3)⋊C4—32no—
247(C2×D3)⋊C4—92no—
248(C2×D3)⋊C4—91no—
249(C2×D3)⋊C4—91no—
250(C2×D3)⋊C4—91no—
251(C2×D3)⋊C4—91no—
252C4⋊Dic3—31no—
253C4⋊Dic3—31no—
254C4⋊Dic3—31no—
255C4⋊Dic3—31no—
256C4⋊Dic3—61no—
257C4⋊Dic3—91no—
258C4⋊Dic3—91no—
259C4⋊Dic3—92no—
260C4⋊Dic3—91no—
261C4⋊Dic3—92no—
262C4⋊Dic3—92no—
263C4⋊Dic3—92no—
264C4⋊Dic3—91no—
265C4⋊Dic3—92no—
266Dic3⋊C4—32no—
267Dic3⋊C4—32no—
268Dic3⋊C4—32no—
269Dic3⋊C4—62no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—91no—
274Dic3⋊C4—91no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—91no—
282Dic3⋊C4—91no—
283Dic3⋊C4—92no—
284Dic3⋊C4—92no—
285C4×Dic3—31no—
286C4×Dic3—31no—
287C4×Dic3—31no—
288C4×Dic3—32no—
289C4×Dic3—61no—
290C4×Dic3—91no—
291C4×Dic3—92no—
292C22×C32—21no—
293C22×C32C23×D311yescharacteristic
294C22×C32C23×D311yescharacteristic
295C2×C3⋊D3—21no—
296C2×C3⋊D3—21no—
297C2×C3⋊D3C2×Dic612yes—
298C2×C3×D3—32no—
299C2×C3×D3—61no—
300C2×C3×D3—61no—
301C32×C4—22no—
302C32×C4—22no—
303C32×C4—22no—
304C32×C4—42no—
305C32×C4C2×Dic612yes—
306C32×C4D3×Q812yes—
307C3⋊Dic3—22no—
308C3⋊Dic3—21no—
309C3⋊Dic3—21no—
310C3⋊Dic3—22no—
311C3⋊Dic3C2×Dic612yes—
312C3×Dic3—32no—
313C3×Dic3—62no—
314C3×Dic3—62no—
315C3×Dic3—62no—
316C3×Dic3—61no—
317C3×Dic3—62no—
318C3×Dic3—61no—
319C3×Dic3—62no—
320C3×Dic3—61no—
321C3×Dic3—62no—
322C3×Dic3—61no—
323C3×Dic3—61no—
324C3×Dic3—61no—
325C3×Dic3—122no—
326C4⋊Q8—91no—
327C4⋊Q8—91no—
328C4⋊Q8—272no—
329#125—272no—
330#125—271no—
331#125—271no—
332C22⋊Q8—272no—
333C22⋊Q8—272no—
334#117—91no—
335#117—271no—
336C2×C4⋊C4—271no—
337H(3,𝔽3)#54411yescharacteristic, complemented, Sylow
338C23×C3—91no—
339C22×D3—31no—
340C22×D3—91no—
341C3×Q8—362no—
342C3×Q8—362no—
343C2×C3×C4—31no—
344C2×C3×C4—31no—
345C2×C3×C4—32no—
346C2×C3×C4—32no—
347C2×C3×C4—61no—
348C2×C3×C4—62no—
349C2×C3×C4—94no—
350C2×C3×C4—91no—
351C2×C3×C4—91no—
352C2×C3×C4—91no—
353C2×C3×C4—92no—
354C2×C3×C4—92no—
355C2×C3×C4—91no—
356C2×C3×C4—91no—
357C2×C3×C4—92no—
358C2×C3×C4C2×D3212yes—
359C2×C3×C4C2×D3211yescharacteristic
360C2×Dic3—31no—
361C2×Dic3—31no—
362C2×Dic3—31no—
363C2×Dic3—31no—
364C2×Dic3—31no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—31no—
368C2×Dic3—32no—
369C2×Dic3—31no—
370C2×Dic3—32no—
371C2×Dic3—61no—
372C2×Dic3—61no—
373C2×Dic3—62no—
374C2×Dic3—91no—
375C2×Dic3—91no—
376C2×Dic3—92no—
377C2×Dic3—91no—
378C2×Dic3—92no—
379C2×Dic3—362no—
380C4×D3—34no—
381C4×D3—94no—
382C4×D3—362no—
383C4×D3—362no—
384Dic6—122no—
385Dic6—122no—
386Dic6—122no—
387Dic6—124no—
388Dic6—362no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32C22×Dic611yescharacteristic
393C2×C32#100411yescharacteristic
394C2×C32C2×D3×Q811yescharacteristic
395C2×C32#100711yescharacteristic
396C2×C32#100911yescharacteristic
397C2×C32#101011yescharacteristic
398C3⋊D3—22no—
399C3×D3—62no—
400C2×Q8—92no—
401C2×Q8—92no—
402C22×C4—91no—
403C22×C4—272no—
404C4⋊C4—91no—
405C4⋊C4—91no—
406C4⋊C4—91no—
407C4⋊C4—92no—
408C4⋊C4—92no—
409C4⋊C4—272no—
410C4⋊C4—271no—
411C4⋊C4—271no—
412C4⋊C4—272no—
413C4⋊C4—272no—
414C4⋊C4—272no—
415C4⋊C4—272no—
416C4⋊C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
417C22⋊C4—92no—
418C22⋊C4—271no—
419C22⋊C4—271no—
420C42—92no—
421C42—91no—
422C42—91no—
423C22×C3—31no—
424C22×C3—31no—
425C22×C3—61no—
426C22×C3—92no—
427C22×C3—181no—
428C22×C3—181no—
429C22×C3C22×D3211yescharacteristic
430C2×D3—32no—
431C2×D3—61no—
432C2×D3—61no—
433C2×D3—92no—
434C2×D3—181no—
435C2×D3—181no—
436C3×C4—22no—
437C3×C4—32no—
438C3×C4—32no—
439C3×C4—62no—
440C3×C4—62no—
441C3×C4—62no—
442C3×C4—92no—
443C3×C4—122no—
444C3×C4—182no—
445C3×C4—181no—
446C3×C4—181no—
447C3×C4—182no—
448C3×C4—182no—
449C3×C4—181no—
450C3×C4—181no—
451C3×C4D3×Dic612yes—
452Dic3—32no—
453Dic3—61no—
454Dic3—61no—
455Dic3—62no—
456Dic3—62no—
457Dic3—61no—
458Dic3—61no—
459Dic3—62no—
460Dic3—62no—
461Dic3—61no—
462Dic3—62no—
463Dic3—61no—
464Dic3—62no—
465Dic3—92no—
466Dic3—122no—
467Dic3—181no—
468Dic3—182no—
469Dic3—182no—
470Dic3—181no—
471C32—21no—
472C32#551911yescharacteristic, complemented
473C32#572911yescharacteristic, complemented
474C23—91no—
475Q8—362no—
476Q8—362no—
477C2×C4—91no—
478C2×C4—94no—
479C2×C4—92no—
480C2×C4—91no—
481C2×C4—91no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—92no—
485C2×C4—92no—
486C2×C4—1082no—
487C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
488C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—31no—
493C2×C3—31no—
494C2×C3—31no—
495C2×C3—61no—
496C2×C3—61no—
497C2×C3—61no—
498C2×C3—182no—
499C2×C3C2×D3×Dic611yescharacteristic
500C2×C3#6431411yescharacteristic
501C2×C3#6431511yescharacteristic
502D3—62no—
503D3—182no—
504C22—92no—
505C22—181no—
506C22—181no—
507C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
508C4—22no—
509C4—92no—
510C4—182no—
511C4—182no—
512C4—181no—
513C4—181no—
514C4—181no—
515C4—182no—
516C4—181no—
517C4((C3⋊D3)⋊C3)⋊Q812yes—
518C3—31no—
519C3—31no—
520C3—61no—
521C3#1059450211yescharacteristic
522C2—182no—
523C2C2×((C3⋊D3)⋊C3)⋊Q811yescharacteristic
524C2#1172250711yescharacteristic
525C2#1172251611yescharacteristic
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×(C3⋊D3)⋊C3C23
hypercenterC4⋊C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#544
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. #11721756C2semidirect
3. #64097⋊C3C2semidirect
8. #11721095C2semidirect
9. #11721093C2semidirect
10. #11721091C2semidirect
11. #11721090C2semidirect
38. #64097D3semidirect
72. C4×(C3⋊D3)⋊C3Q8semidirect
190. C4×C3⋊D3Dic6semidirect
337. H(3,𝔽3)#544semidirect
472. C32#5519semidirect
473. C32#5729semidirect

Maximal subgroups

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

subgroupindex
#117217562
#64101⋊C32
#64097⋊C32
#117211352
#117211292
#117211282
#117211052
#117210952
#117210932
#117210912
#117210902
#105945023
#105943513