Finite Group Databasebeta

← #13150609

Subgroups — #13150609

Order 1728.

746 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11721824C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic
3#64133⋊C3C211yesmaximal, characteristic
4#11721297C212yesmaximal
5#11721297C212yesmaximal
6#11721093C211yesmaximal, characteristic, complemented
7#11721090C211yesmaximal, characteristic, complemented
8#11721087C212yesmaximal, complemented
9#11721086C212yesmaximal, complemented
10#11721083C211yesmaximal, characteristic, complemented
11#11721082C211yesmaximal, characteristic, complemented
12#10594826—31nomaximal
13#10594824—31nomaximal
14C22×C4×H(3,𝔽3)C2211yescharacteristic
15C2×#6229C2211yescharacteristic
16C2×#634⋊C3C2211yescharacteristic
17#88552C2212yes—
18C4⋊#1084C2211yescharacteristic
19#88540C2212yes—
20#3757⋊C3C2212yes—
21#3757⋊C3C2211yescharacteristic
22#3757⋊C3C2211yescharacteristic
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
24C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
25C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic
26#3750⋊C3C2212yes—
27C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
28C4×(C3⋊Dic3)⋊C3C2211yescharacteristic
29#88449C2212yes—
30#88449C2212yes—
31#88448C2212yes—
32#88448C2212yes—
33#88448C2212yes—
34#88448C2212yes—
35#88447C2212yes—
36#88447C2212yes—
37#64135D311yescharacteristic
38#64133D311yescharacteristic
39C3×#926—31no—
40C3×#924—31no—
41C3×#924—31no—
42C3×#924—31no—
43C3×#924—61no—
44#63956—32no—
45#63956—32no—
46#63956—32no—
47#63956—32no—
48#63873—31no—
49#63872—31no—
50#63869—31no—
51#63866—31no—
52#63847—32no—
53#63846—32no—
54#63845—32no—
55#63845—32no—
56#63844—31no—
57#63843—31no—
58#63843—31no—
59Dic3×Dic6—31no—
60C23×H(3,𝔽3)C2311yescharacteristic
61C2×C4×H(3,𝔽3)—22no—
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)⋊C3C2311yescharacteristic
69C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
70C2×(C3⋊Dic3)⋊C3C2312yes—
71C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
72C2×(C3⋊Dic3)⋊C3C2311yescharacteristic
73#634⋊C3—22no—
74#5625—91no—
75#5625—91no—
76#5518—91no—
77C22×C32×C4—21no—
78C22×C32×C4C2×D311yescharacteristic
79C22×C32×C4C2×D311yescharacteristic
80C2×#634C2×D311yescharacteristic
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—61no—
85#3757C2×D312yes—
86#3757C2×D311yescharacteristic
87#3757C2×D311yescharacteristic
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic
89C4⋊(C3⋊Dic3)C2×D311yescharacteristic
90C4⋊(C3⋊Dic3)C2×D311yescharacteristic
91#3750C2×D312yes—
92C4×C3⋊Dic3C2×D311yescharacteristic
93C4×C3⋊Dic3C2×D311yescharacteristic
94C3×#217—32no—
95C3×#217—32no—
96C3×#217—32no—
97C3×#217—31no—
98C3×#217—31no—
99C3×#217—62no—
100C3×C4⋊Dic3—31no—
101C3×C4⋊Dic3—31no—
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—31no—
111C3×C4×Dic3—31no—
112#3724—32no—
113#3724—32no—
114#3724—32no—
115#3724—32no—
116Dic3⋊Dic3—32no—
117Dic3⋊Dic3—32no—
118Dic3⋊Dic3—32no—
119Dic3⋊Dic3—32no—
120Dic3⋊Dic3—32no—
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic32—32no—
125Dic32—32no—
126Dic32—32no—
127Dic32—32no—
128C22×H(3,𝔽3)—21no—
129C22×H(3,𝔽3)—21no—
130C22×H(3,𝔽3)—21no—
131C22×H(3,𝔽3)C2411yescharacteristic
132C4×H(3,𝔽3)—21no—
133C4×H(3,𝔽3)#4112yes—
134#1084—22no—
135#1084—22no—
136(C3⋊Dic3)⋊C3—22no—
137(C3⋊Dic3)⋊C3—21no—
138(C3⋊Dic3)⋊C3—21no—
139(C3⋊Dic3)⋊C3—21no—
140(C3⋊Dic3)⋊C3—22no—
141(C3⋊Dic3)⋊C3—21no—
142C3×C22⋊Q8—91no—
143C3×C22⋊Q8—91no—
144C3×#117—91no—
145Q8×Dic3—91no—
146Q8×Dic3—91no—
147#944—91no—
148#944—91no—
149#926—31no—
150#926—91no—
151#924—31no—
152#924—31no—
153#924—31no—
154#924—61no—
155#924—91no—
156#890—91no—
157#890—92no—
158#890—92no—
159#890—91no—
160#889—92no—
161#889—92no—
162#889—92no—
163#889—92no—
164#887—91no—
165#887—91no—
166#879—92no—
167#879—92no—
168#879—92no—
169#879—92no—
170#879—92no—
171#879—92no—
172#870—91no—
173#869—91no—
174C4×Dic6—91no—
175C4×Dic6—91no—
176C23×C32—21no—
177C23×C32C22×D311yescharacteristic
178C23×C32C22×D311yescharacteristic
179C2×C32×C4—22no—
180C2×C32×C4—22no—
181C2×C32×C4—21no—
182C2×C32×C4—21no—
183C2×C32×C4—42no—
184C2×C32×C4C22×D311yescharacteristic
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3C22×D311yescharacteristic
189C2×C3⋊Dic3C22×D311yescharacteristic
190C2×C3⋊Dic3C22×D312yes—
191C2×C3⋊Dic3C22×D312yes—
192C2×C3⋊Dic3C22×D311yescharacteristic
193C2×C3⋊Dic3C22×D311yescharacteristic
194#634—22no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—62no—
206C2×C3×Dic3—62no—
207C3×Dic6—62no—
208C3×Dic6—62no—
209C3×Dic6—62no—
210C3×Dic6—122no—
211#541—271noSylow
212C2×H(3,𝔽3)—41no—
213C2×H(3,𝔽3)C2×#4111yescharacteristic
214C2×H(3,𝔽3)2−1+411yescharacteristic
215C2×H(3,𝔽3)2−1+411yescharacteristic
216C2×C3×Q8—91no—
217C2×C3×Q8—91no—
218C22×C3×C4—31no—
219C22×C3×C4—31no—
220C22×C3×C4—61no—
221C22×C3×C4D3211yescharacteristic
222C2×Dic6—31no—
223C2×Dic6—31no—
224C2×Dic6—31no—
225C2×Dic6—61no—
226C2×Dic6—91no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—91no—
229C3×C4⋊C4—91no—
230C3×C4⋊C4—92no—
231C3×C4⋊C4—91no—
232C3×C4⋊C4—92no—
233C3×C22⋊C4—91no—
234C3×C22⋊C4—91no—
235C3×C22⋊C4—92no—
236C3×C22⋊C4—92no—
237C3×C42—91no—
238C3×C42—91no—
239#217—32no—
240#217—32no—
241#217—32no—
242#217—31no—
243#217—31no—
244#217—62no—
245#217—92no—
246#217—91no—
247#217—91no—
248C4⋊Dic3—31no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—31no—
252C4⋊Dic3—31no—
253C4⋊Dic3—61no—
254C4⋊Dic3—91no—
255C4⋊Dic3—91no—
256C4⋊Dic3—91no—
257C4⋊Dic3—92no—
258C4⋊Dic3—92no—
259C4⋊Dic3—92no—
260C4⋊Dic3—92no—
261C4⋊Dic3—92no—
262C4⋊Dic3—92no—
263Dic3⋊C4—32no—
264Dic3⋊C4—32no—
265Dic3⋊C4—32no—
266Dic3⋊C4—62no—
267Dic3⋊C4—92no—
268Dic3⋊C4—92no—
269Dic3⋊C4—92no—
270Dic3⋊C4—92no—
271Dic3⋊C4—92no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280C4×Dic3—31no—
281C4×Dic3—31no—
282C4×Dic3—91no—
283C4×Dic3—91no—
284C4×Dic3—92no—
285C4×Dic3—92no—
286C4×Dic3—92no—
287C4×Dic3—92no—
288C4×Dic3—92no—
289C4×Dic3—92no—
290C22×C32—22no—
291C22×C32—21no—
292C22×C32—21no—
293C22×C32—21no—
294C22×C32—21no—
295C22×C32—21no—
296C22×C32—21no—
297C22×C32—21no—
298C22×C32—41no—
299C22×C32—41no—
300C22×C32C23×D311yescharacteristic
301C22×C32C23×D311yescharacteristic
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—22no—
305C32×C4—41no—
306C32×C4#23512yes—
307C32×C4#23712yes—
308C3⋊Dic3—22no—
309C3⋊Dic3—21no—
310C3⋊Dic3—21no—
311C3⋊Dic3—21no—
312C3⋊Dic3—21no—
313C3⋊Dic3—22no—
314C3×Dic3—62no—
315C3×Dic3—62no—
316C3×Dic3—62no—
317C3×Dic3—61no—
318C3×Dic3—61no—
319C3×Dic3—62no—
320C3×Dic3—62no—
321C3×Dic3—61no—
322C3×Dic3—61no—
323C3×Dic3—62no—
324C3×Dic3—122no—
325C3×Dic3—122no—
326C4⋊Q8—271no—
327#126—272no—
328#126—272no—
329#125—271no—
330#125—272no—
331#125—272no—
332C22⋊Q8—91no—
333C22⋊Q8—91no—
334C4×Q8—271no—
335C4×Q8—271no—
336#117—91no—
337H(3,𝔽3)#54111yescharacteristic, complemented, Sylow
338C23×C3—31no—
339C23×C3—31no—
340C23×C3—61no—
341C23×C3C2×D3211yescharacteristic
342C3×Q8—182no—
343C3×Q8—182no—
344C2×C3×C4—22no—
345C2×C3×C4—31no—
346C2×C3×C4—31no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—62no—
350C2×C3×C4—62no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—91no—
354C2×C3×C4—91no—
355C2×C3×C4—92no—
356C2×C3×C4—92no—
357C2×C3×C4—91no—
358C2×C3×C4—91no—
359C2×C3×C4—92no—
360C2×C3×C4—92no—
361C2×C3×C4—122no—
362C2×C3×C4C2×D3211yescharacteristic
363C2×C3×C4C2×D3211yescharacteristic
364C2×Dic3—32no—
365C2×Dic3—32no—
366C2×Dic3—32no—
367C2×Dic3—32no—
368C2×Dic3—31no—
369C2×Dic3—31no—
370C2×Dic3—31no—
371C2×Dic3—31no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—62no—
375C2×Dic3—62no—
376C2×Dic3—91no—
377C2×Dic3—91no—
378C2×Dic3—91no—
379C2×Dic3—92no—
380C2×Dic3—91no—
381C2×Dic3—92no—
382Dic6—62no—
383Dic6—62no—
384Dic6—62no—
385Dic6—122no—
386Dic6—182no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—42no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32C2×#23511yescharacteristic
394C2×C32C2×#23711yescharacteristic
395C2×C32#100711yescharacteristic
396C2×C32#101011yescharacteristic
397C2×C32#101011yescharacteristic
398C2×C32#101011yescharacteristic
399C2×Q8—91no—
400C2×Q8—91no—
401C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
402C4⋊C4—91no—
403C4⋊C4—91no—
404C4⋊C4—91no—
405C4⋊C4—91no—
406C4⋊C4—92no—
407C4⋊C4—92no—
408C4⋊C4—272no—
409C4⋊C4—272no—
410C4⋊C4—272no—
411C4⋊C4—272no—
412C4⋊C4—272no—
413C4⋊C4—272no—
414C22⋊C4—91no—
415C22⋊C4—91no—
416C22⋊C4—92no—
417C22⋊C4—92no—
418C42—91no—
419C42—91no—
420C42—272no—
421C42—272no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—31no—
426C22×C3—31no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3—61no—
430C22×C3—61no—
431C22×C3—61no—
432C22×C3—61no—
433C22×C3—62no—
434C22×C3—61no—
435C22×C3—121no—
436C22×C3—121no—
437C22×C3C22×D3211yescharacteristic
438C3×C4—21no—
439C3×C4—32no—
440C3×C4—32no—
441C3×C4—61no—
442C3×C4—61no—
443C3×C4—62no—
444C3×C4—121no—
445C3×C4—182no—
446C3×C4—181no—
447C3×C4—181no—
448C3×C4—181no—
449C3×C4—182no—
450C3×C4—181no—
451C3×C4—182no—
452C3×C4—182no—
453C3×C4#380412yes—
454Dic3—62no—
455Dic3—62no—
456Dic3—62no—
457Dic3—62no—
458Dic3—61no—
459Dic3—61no—
460Dic3—61no—
461Dic3—61no—
462Dic3—62no—
463Dic3—62no—
464Dic3—122no—
465Dic3—122no—
466Dic3—182no—
467Dic3—181no—
468Dic3—181no—
469Dic3—181no—
470Dic3—181no—
471Dic3—182no—
472C32—21no—
473C32#551811yescharacteristic, complemented
474C32#562511yescharacteristic, complemented
475C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
476Q8—182no—
477Q8—182no—
478C2×C4—22no—
479C2×C4—91no—
480C2×C4—91no—
481C2×C4—91no—
482C2×C4—91no—
483C2×C4—92no—
484C2×C4—92no—
485C2×C4—92no—
486C2×C4—92no—
487C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
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—41no—
496C2×C3—61no—
497C2×C3—61no—
498C2×C3—61no—
499C2×C3—121no—
500C2×C3—121no—
501C2×C3—122no—
502C2×C3#6429711yescharacteristic
503C2×C3#6429811yescharacteristic
504C2×C3C2×#380411yescharacteristic
505C22—21no—
506C22—21no—
507C22—21no—
508C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
509C4—21no—
510C4—182no—
511C4—182no—
512C4—182no—
513C4—181no—
514C4—181no—
515C4—181no—
516C4—181no—
517C4—182no—
518C4#8867112yes—
519C3—31no—
520C3—31no—
521C3—61no—
522C3#1059482411yescharacteristic
523C2—41no—
524C2#1172248211yescharacteristic
525C2#1172248311yescharacteristic
526C2C2×#8867111yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC22×C4×H(3,𝔽3)C22
prime radicalC23×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#541
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
6. #11721093C2semidirect
7. #11721090C2semidirect
8. #11721087C2semidirect
9. #11721086C2semidirect
10. #11721083C2semidirect
11. #11721082C2semidirect
337. H(3,𝔽3)#541semidirect
473. C32#5518semidirect
474. C32#5625semidirect

Maximal subgroups

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

subgroupindex
#117218242
#64135⋊C32
#64133⋊C32
#117212972
#117210932
#117210902
#117210872
#117210862
#117210832
#117210822
#105948263
#105948243