Finite Group Databasebeta

← #13169337

Subgroups — #13169337

Order 1728.

1006 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723665C211yesmaximal, characteristic
2#11723535C211yesmaximal, characteristic
3#11723400C212yesmaximal
4C3×#63846C212yesmaximal
5C3×#63838C211yesmaximal, characteristic
6#11723036C212yesmaximal
7#11723017C212yesmaximal
8#11722942C212yesmaximal
9#11722940C212yesmaximal
10#10594321—32nomaximal
11#10594315—31nomaximal
12C2×#6296C2211yescharacteristic
13C2×#6293C2212yes—
14C2×C3×#627C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×#3750C2212yes—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×Dic3⋊C4C2212yes—
20#88815C2211yescharacteristic
21#88813C2212yes—
22#88807C2212yes—
23#88806C2212yes—
24Dic3⋊(C3⋊Dic3)C2212yes—
25Dic3⋊(C3⋊Dic3)C2212yes—
26Dic3×C3⋊Dic3C2212yes—
27C3×#3724C2211yescharacteristic
28C3×Dic3⋊Dic3C2212yes—
29C3×Dic3⋊Dic3C2212yes—
30C3×Dic3⋊Dic3C2212yes—
31C3×Dic32C2212yes—
32C3×Dic32C2212yes—
33#64099—32no—
34#64097—31no—
35C3×#890—32no—
36C3×#889—32no—
37C3×#887—32no—
38#63918—32no—
39#63918—32no—
40#63918—32no—
41#63918—61no—
42C4×#627—32no—
43#63847—32no—
44#63846—32no—
45#63846D312yes—
46#63845—32no—
47#63845—32no—
48#63845—32no—
49#63845—62no—
50#63844—32no—
51#63843—32no—
52Dic3×Dic6—32no—
53#63840—32no—
54#63838—31no—
55#63838D311yescharacteristic
56#63837—32no—
57#63837—32no—
58#63837—32no—
59C2×C33×C4C2311yescharacteristic
60C2×C3⋊(C3⋊Dic3)C2312yes—
61C2×C3×C3⋊Dic3C2312yes—
62C2×C3×C3⋊Dic3C2312yes—
63C2×C3×C3⋊Dic3C2311yescharacteristic
64C2×C3×C3⋊Dic3C2311yescharacteristic
65C2×C32×Dic3C2312yes—
66C2×C32×Dic3C2312yes—
67C2×C32×Dic3C2312yes—
68#6296—22no—
69#6293—24no—
70C3×#627Q814yescomplemented
71#5685—92no—
72#5570—91no—
73C2×#634—32no—
74C2×C3×Dic6—32no—
75C2×#627—32no—
76C2×#627—31no—
77C2×#627—32no—
78C2×#627—32no—
79C2×#627—62no—
80C2×#627C2×D311yescharacteristic
81C32×C4⋊C4—32no—
82C32×C4⋊C4—31no—
83C4⋊(C3⋊Dic3)—31no—
84C4⋊(C3⋊Dic3)—32no—
85C4⋊(C3⋊Dic3)—32no—
86C4⋊(C3⋊Dic3)—32no—
87C4⋊(C3⋊Dic3)—62no—
88C4⋊(C3⋊Dic3)—61no—
89C4⋊(C3⋊Dic3)—121no—
90#3750—32no—
91#3750—32no—
92#3750C2×D312yes—
93C4×C3⋊Dic3—32no—
94C4×C3⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3—21no—
96C3×C4⋊Dic3—32no—
97C3×C4⋊Dic3—32no—
98C3×C4⋊Dic3—32no—
99C3×C4⋊Dic3C2×D312yes—
100C3×Dic3⋊C4—22no—
101C3×Dic3⋊C4—32no—
102C3×Dic3⋊C4—32no—
103C3×Dic3⋊C4—32no—
104C3×Dic3⋊C4—32no—
105C3×Dic3⋊C4—32no—
106C3×Dic3⋊C4—32no—
107C3×Dic3⋊C4—62no—
108C3×Dic3⋊C4C2×D312yes—
109C3×Dic3⋊C4C2×D312yes—
110C3×C4×Dic3—32no—
111C3×C4×Dic3—32no—
112C3×C4×Dic3—32no—
113C3×C4×Dic3—32no—
114C3×C4×Dic3—32no—
115C3×C4×Dic3—61no—
116#3724—31no—
117#3724—32no—
118#3724—32no—
119#3724—62no—
120#3724C2×D311yescharacteristic
121Dic3⋊Dic3—32no—
122Dic3⋊Dic3—32no—
123Dic3⋊Dic3—32no—
124Dic3⋊Dic3—32no—
125Dic3⋊Dic3—32no—
126Dic3⋊Dic3—32no—
127Dic3⋊Dic3—32no—
128Dic3⋊Dic3—62no—
129Dic3⋊Dic3—62no—
130Dic3⋊Dic3—62no—
131Dic3⋊Dic3C2×D312yes—
132Dic3⋊Dic3C2×D312yes—
133Dic3⋊Dic3C2×D312yes—
134Dic32—32no—
135Dic32—32no—
136Dic32—32no—
137Dic32—32no—
138Dic32—32no—
139Dic32—62no—
140Dic32C2×D312yes—
141Dic32C2×D312yes—
142C22×C33C2411yescharacteristic
143C33×C4—21no—
144C3⋊(C3⋊Dic3)—22no—
145C3×C3⋊Dic3—22no—
146C3×C3⋊Dic3—22no—
147C3×C3⋊Dic3C2×Q812yes—
148C3×C3⋊Dic3#4112yes—
149C32×Dic3—22no—
150C32×Dic3—22no—
151C32×Dic3C2×Q814yes—
152C3×#125—92no—
153C3×C4×Q8—91no—
154Q8×Dic3—92no—
155#890—32no—
156#890—92no—
157#890—92no—
158#890—92no—
159#890—182no—
160#889—32no—
161#889—92no—
162#889—92no—
163#889—92no—
164#888—92no—
165#888—92no—
166#888—92no—
167#888—181no—
168#887—32no—
169#887—92no—
170#887—92no—
171#887—92no—
172#869—92no—
173#869—91no—
174C4×Dic6—92no—
175C4×Dic6—92no—
176C4×Dic6—91no—
177C4×Dic6—92no—
178C2×C32×C4—21no—
179C2×C32×C4—22no—
180C2×C32×C4—32no—
181C2×C32×C4—32no—
182C2×C32×C4—32no—
183C2×C32×C4—41no—
184C2×C32×C4C22×D312yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—32no—
188C2×C3⋊Dic3—32no—
189C2×C3⋊Dic3—62no—
190C2×C3⋊Dic3—62no—
191C2×C3⋊Dic3—62no—
192C2×C3⋊Dic3—122no—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197#634—64no—
198C2×C3×Dic3—22no—
199C2×C3×Dic3—22no—
200C2×C3×Dic3—22no—
201C2×C3×Dic3—32no—
202C2×C3×Dic3—32no—
203C2×C3×Dic3—32no—
204C2×C3×Dic3—32no—
205C2×C3×Dic3—32no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—61no—
208C2×C3×Dic3—61no—
209C2×C3×Dic3—62no—
210C2×C3×Dic3—62no—
211C2×C3×Dic3C22×D312yes—
212C2×C3×Dic3C22×D312yes—
213C2×C3×Dic3C22×D312yes—
214C2×C3×Dic3C22×D312yes—
215C2×C3×Dic3C22×D312yes—
216C2×C3×Dic3C22×D312yes—
217C3×Dic6—38no—
218#627—64no—
219#627—64no—
220#627—64no—
221#627—62no—
222#627—124no—
223#627Dic614yescomplemented
224#557—271noSylow
225C2×C33C22×Q811yescharacteristic
226C2×C33C2×#4111yescharacteristic
227C2×C332−1+411yescharacteristic
228C2×C3×Q8—91no—
229C2×Dic6—32no—
230C2×Dic6—91no—
231C2×Dic6—92no—
232C2×Dic6—92no—
233C2×Dic6—92no—
234C2×Dic6—92no—
235C2×Dic6—182no—
236C3×C4⋊C4—32no—
237C3×C4⋊C4—32no—
238C3×C4⋊C4—32no—
239C3×C4⋊C4—61no—
240C3×C4⋊C4—62no—
241C3×C4⋊C4—92no—
242C3×C4⋊C4—92no—
243C3×C4⋊C4—92no—
244C3×C4⋊C4—91no—
245C3×C4⋊C4—92no—
246C3×C42—92no—
247C3×C42—92no—
248C3×C42—91no—
249C4⋊Dic3—32no—
250C4⋊Dic3—32no—
251C4⋊Dic3—32no—
252C4⋊Dic3—91no—
253C4⋊Dic3—92no—
254C4⋊Dic3—92no—
255C4⋊Dic3—92no—
256C4⋊Dic3—92no—
257C4⋊Dic3—91no—
258C4⋊Dic3—92no—
259C4⋊Dic3—92no—
260C4⋊Dic3—182no—
261C4⋊Dic3—182no—
262C4⋊Dic3—181no—
263C4⋊Dic3—182no—
264C4⋊Dic3—361no—
265C4⋊Dic3D3211yescharacteristic
266Dic3⋊C4—32no—
267Dic3⋊C4—32no—
268Dic3⋊C4—32no—
269Dic3⋊C4—32no—
270Dic3⋊C4—32no—
271Dic3⋊C4—32no—
272Dic3⋊C4—62no—
273Dic3⋊C4—92no—
274Dic3⋊C4—92no—
275Dic3⋊C4—92no—
276Dic3⋊C4—92no—
277Dic3⋊C4—92no—
278Dic3⋊C4—92no—
279Dic3⋊C4—92no—
280Dic3⋊C4—92no—
281Dic3⋊C4—182no—
282Dic3⋊C4—182no—
283Dic3⋊C4D3212yes—
284C4×Dic3—32no—
285C4×Dic3—32no—
286C4×Dic3—32no—
287C4×Dic3—32no—
288C4×Dic3—32no—
289C4×Dic3—61no—
290C4×Dic3—92no—
291C4×Dic3—92no—
292C4×Dic3—92no—
293C4×Dic3—92no—
294C4×Dic3—92no—
295C4×Dic3—92no—
296C4×Dic3—182no—
297C22×C32—22no—
298C22×C32—21no—
299C22×C32—41no—
300C22×C32C23×D311yescharacteristic
301C22×C32C23×D312yes—
302C32×C4—21no—
303C32×C4—22no—
304C32×C4—22no—
305C32×C4—34no—
306C32×C4—42no—
307C32×C4—42no—
308C32×C4—62no—
309C32×C4—62no—
310C3⋊Dic3—22no—
311C3⋊Dic3—22no—
312C3⋊Dic3—62no—
313C3⋊Dic3—62no—
314C3⋊Dic3—62no—
315C3⋊Dic3—122no—
316C3⋊Dic3—122no—
317C3⋊Dic3—122no—
318C3⋊Dic3—242no—
319C3⋊Dic3C2×Dic612yes—
320C3⋊Dic3#23912yes—
321C3×Dic3—22no—
322C3×Dic3—22no—
323C3×Dic3—22no—
324C3×Dic3—22no—
325C3×Dic3—24no—
326C3×Dic3—34no—
327C3×Dic3—34no—
328C3×Dic3—42no—
329C3×Dic3—42no—
330C3×Dic3—62no—
331C3×Dic3—62no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—62no—
335C3×Dic3—62no—
336C3×Dic3—122no—
337C3×Dic3—122no—
338C3×Dic3C2×Dic614yes—
339C3×Dic3D3×Q814yes—
340C4⋊Q8—271no—
341C4⋊Q8—272no—
342#125—92no—
343#125—272no—
344#125—272no—
345C4×Q8—91no—
346C4×Q8—272no—
347C4×Q8—271no—
348C4×Q8—272no—
349C33#55711yescharacteristic, complemented, Sylow
350C3×Q8—94no—
351C2×C3×C4—21no—
352C2×C3×C4—22no—
353C2×C3×C4—32no—
354C2×C3×C4—32no—
355C2×C3×C4—32no—
356C2×C3×C4—32no—
357C2×C3×C4—32no—
358C2×C3×C4—32no—
359C2×C3×C4—41no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4—91no—
364C2×C3×C4—91no—
365C2×C3×C4—92no—
366C2×C3×C4—92no—
367C2×C3×C4C2×D3212yes—
368C2×C3×C4C2×D3211yescharacteristic
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—62no—
378C2×Dic3—62no—
379C2×Dic3—92no—
380C2×Dic3—92no—
381C2×Dic3—92no—
382C2×Dic3—182no—
383C2×Dic3—182no—
384C2×Dic3—182no—
385C2×Dic3—362no—
386C2×Dic3C2×D3212yes—
387C2×Dic3C2×D3212yes—
388C2×Dic3C2×D3212yes—
389Dic6—38no—
390Dic6—184no—
391Dic6—184no—
392Dic6—184no—
393Dic6—184no—
394Dic6—182no—
395Dic6—364no—
396C2×C32—22no—
397C2×C32—21no—
398C2×C32—22no—
399C2×C32—21no—
400C2×C32—21no—
401C2×C32—22no—
402C2×C32—41no—
403C2×C32—41no—
404C2×C32—41no—
405C2×C32C22×Dic611yescharacteristic
406C2×C32C2×D3×Q812yes—
407C2×C32C2×#23911yescharacteristic
408C2×C32D3×#4112yes—
409C2×C32#101012yes—
410C2×C32#101011yescharacteristic
411C2×Q8—91no—
412C2×Q8—272no—
413C2×Q8—271no—
414C4⋊C4—32no—
415C4⋊C4—31no—
416C4⋊C4—91no—
417C4⋊C4—92no—
418C4⋊C4—92no—
419C4⋊C4—92no—
420C4⋊C4—92no—
421C4⋊C4—272no—
422C4⋊C4—272no—
423C4⋊C4—272no—
424C4⋊C4—272no—
425C4⋊C4—271no—
426C4⋊C4—271no—
427C42—92no—
428C42—92no—
429C42—91no—
430C42—272no—
431C42—272no—
432C22×C3—21no—
433C22×C3—22no—
434C22×C3—41no—
435C22×C3C22×D3212yes—
436C22×C3C22×D3211yescharacteristic
437C3×C4—22no—
438C3×C4—21no—
439C3×C4—22no—
440C3×C4—34no—
441C3×C4—34no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—64no—
445C3×C4—62no—
446C3×C4—62no—
447C3×C4—62no—
448C3×C4—62no—
449C3×C4—92no—
450C3×C4—92no—
451C3×C4—122no—
452C3×C4—122no—
453C3×C4—182no—
454C3×C4—182no—
455Dic3—22no—
456Dic3—22no—
457Dic3—34no—
458Dic3—34no—
459Dic3—62no—
460Dic3—62no—
461Dic3—62no—
462Dic3—62no—
463Dic3—62no—
464Dic3—62no—
465Dic3—122no—
466Dic3—122no—
467Dic3—182no—
468Dic3—182no—
469Dic3—182no—
470Dic3—362no—
471Dic3—362no—
472Dic3—362no—
473Dic3—722no—
474Dic3D3×Dic614yes—
475C32—21no—
476C32—22no—
477C32—41no—
478C32#557011yescharacteristic, complemented
479C32#568512yescomplemented
480Q8—94no—
481Q8—544no—
482Q8—542no—
483C2×C4—32no—
484C2×C4—32no—
485C2×C4—32no—
486C2×C4—91no—
487C2×C4—91no—
488C2×C4—92no—
489C2×C4—92no—
490C2×C4—272no—
491C2×C4D3311yescharacteristic
492C2×C3—21no—
493C2×C3—22no—
494C2×C3—21no—
495C2×C3—22no—
496C2×C3—22no—
497C2×C3—21no—
498C2×C3—41no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3C2×D3×Dic612yes—
502C2×C3#6429811yescharacteristic
503C2×C3#6430912yes—
504C2×C3#6431211yescharacteristic
505C2×C3D32×Q811yescharacteristic
506C2×C3D3×#23912yes—
507C22C2×D3311yescharacteristic
508C4—21no—
509C4—34no—
510C4—62no—
511C4—62no—
512C4—92no—
513C4—92no—
514C4—182no—
515C4—182no—
516C4—542no—
517C3—22no—
518C3—21no—
519C3—41no—
520C3#1059431511yescharacteristic, complemented
521C3#1059432112yescomplemented
522C2D32×Dic611yescharacteristic
523C2#1172437711yescharacteristic
524C2#1172442711yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, prime radical, cosocleC22×C33C24
FittingC2×C33×C4C23
hypercenterC2×C4D33
hypocenterC33#557
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
70. C3×#627Q8semidirect
223. #627Dic6semidirect
349. C33#557semidirect
478. C32#5570semidirect
479. C32#5685semidirect
520. C3#10594315semidirect
521. C3#10594321semidirect

Maximal subgroups

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

subgroupindex
#117236652
#117235352
#117234002
C3×#638462
C3×#638382
#117230362
#117230172
#117229422
#117229402
#105943213
#105943153