Finite Group Databasebeta

← #13178753

Subgroups — #13178753

Order 1728.

1044 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×Q8×C3⋊Dic3C211yesmaximal, characteristic
2C3×#64153C212yesmaximal
3C4×#6296C211yesmaximal, characteristic
4#11723679C212yesmaximal
5#11723678C211yesmaximal, characteristic
6#11723669C214yesmaximal
7#11723668C212yesmaximal
8#11723665C212yesmaximal
9#10595420—32nomaximal
10#10595338—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×#634C2212yes—
13C2×#6296C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2212yes—
16C3×#3750C2214yes—
17C3×#3750C2214yes—
18C3×C4×C3⋊Dic3C2212yes—
19C3×C4×C3⋊Dic3C2212yes—
20C3×C4×C3⋊Dic3C2211yescharacteristic
21#88815C2212yes—
22#88815C2214yes—
23#88815C2212yes—
24#88815C2212yes—
25#88815C2211yescharacteristic
26#88813C2214yes—
27Q8×C3⋊Dic3D311yescharacteristic
28#64153D312yes—
29C3×Q8×Dic3—32no—
30C3×Q8×Dic3—61no—
31C3×#944—32no—
32C3×#944—32no—
33C3×#944—62no—
34C4×#627—31no—
35C4×#627—32no—
36#63873—31no—
37#63870—32no—
38#63867—34no—
39#63866—32no—
40#63847—34no—
41#63846—34no—
42#63845—34no—
43#63844—32no—
44#63843—32no—
45Dic3×Dic6—32no—
46#63838—32no—
47#63837—34no—
48C33×Q8—22no—
49C2×C33×C4C2312yes—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2314yes—
55C2×C3×C3⋊Dic3C2311yescharacteristic
56C2×C3×C3⋊Dic3C2312yes—
57C3×#634—24no—
58#6296Q814yescomplemented
59#5721—92no—
60#5567—91no—
61C2×C32×Q8—21no—
62C2×C32×Q8—22no—
63C2×C32×Q8—41no—
64C2×C32×Q8C2×D311yescharacteristic
65C2×C32×Q8C2×D312yes—
66C2×#634C2×D312yes—
67C2×C3×Dic6—32no—
68C2×C3×Dic6—32no—
69C2×C3×Dic6—62no—
70C2×#627—32no—
71C2×#627—31no—
72C4⋊(C3⋊Dic3)C2×D311yescharacteristic
73C4⋊(C3⋊Dic3)C2×D312yes—
74#3750C2×D314yes—
75#3750C2×D314yes—
76C4×C3⋊Dic3C2×D312yes—
77C4×C3⋊Dic3C2×D312yes—
78C4×C3⋊Dic3C2×D311yescharacteristic
79C3×C4⋊Dic3—32no—
80C3×C4⋊Dic3—34no—
81C3×C4⋊Dic3—61no—
82C3×C4⋊Dic3—62no—
83C3×Dic3⋊C4—34no—
84C3×Dic3⋊C4—34no—
85C3×Dic3⋊C4—34no—
86C3×Dic3⋊C4—34no—
87C3×Dic3⋊C4—64no—
88C3×Dic3⋊C4—64no—
89C3×C4×Dic3—32no—
90C3×C4×Dic3—32no—
91C3×C4×Dic3—34no—
92C3×C4×Dic3—32no—
93C3×C4×Dic3—62no—
94C3×C4×Dic3—61no—
95C3×C4×Dic3—62no—
96#3724—32no—
97#3724—34no—
98#3724—32no—
99#3724—32no—
100#3724—31no—
101Dic3⋊Dic3—34no—
102Dic3⋊Dic3—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—32no—
105Dic3⋊Dic3—32no—
106Dic3⋊Dic3—34no—
107Dic3⋊Dic3—32no—
108Dic32—34no—
109Dic32—34no—
110Dic32—34no—
111C22×C33C2411yescharacteristic
112C33×C4—22no—
113C33×C4#4112yes—
114C3×C3⋊Dic3—24no—
115C3×C3⋊Dic3—22no—
116C3×C3⋊Dic3—22no—
117C3×C3⋊Dic3—21no—
118C3×C3⋊Dic3C2×Q812yes—
119C3×C3⋊Dic3C2×Q814yes—
120C3×C4⋊Q8—92no—
121C3×C4×Q8—91no—
122Q8×Dic3—32no—
123Q8×Dic3—61no—
124Q8×Dic3—92no—
125#944—32no—
126#944—32no—
127#944—62no—
128#944—92no—
129#890—94no—
130#889—94no—
131#889—94no—
132#889—94no—
133#887—94no—
134#887—92no—
135#870—92no—
136#869—91no—
137C4×Dic6—92no—
138C4×Dic6—91no—
139C4×Dic6—92no—
140C4×Dic6—92no—
141C32×Q8—22no—
142C32×Q8—24no—
143C32×Q8—24no—
144C32×Q8—44no—
145C32×Q8—44no—
146C2×C32×C4—22no—
147C2×C32×C4—21no—
148C2×C32×C4—22no—
149C2×C32×C4—24no—
150C2×C32×C4—41no—
151C2×C32×C4—42no—
152C2×C32×C4C22×D314yes—
153C2×C32×C4C22×D312yes—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D312yes—
156C2×C3⋊Dic3C22×D312yes—
157C2×C3⋊Dic3C22×D314yes—
158C2×C3⋊Dic3C22×D312yes—
159C2×C3⋊Dic3C22×D311yescharacteristic
160C2×C3⋊Dic3C22×D312yes—
161C2×C3⋊Dic3C22×D311yescharacteristic
162#634—24no—
163C2×C3×Dic3—32no—
164C2×C3×Dic3—32no—
165C2×C3×Dic3—32no—
166C2×C3×Dic3—34no—
167C2×C3×Dic3—34no—
168C2×C3×Dic3—34no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—62no—
174C2×C3×Dic3—62no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—64no—
178C3×Dic6—64no—
179C3×Dic6—64no—
180C3×Dic6—124no—
181#627—38no—
182#627—34no—
183#557—271noSylow
184C2×C33C22×Q811yescharacteristic
185C2×C33C2×#4111yescharacteristic
186C2×C332−1+411yescharacteristic
187C2×C3×Q8—21no—
188C2×C3×Q8—22no—
189C2×C3×Q8—41no—
190C2×C3×Q8—92no—
191C2×C3×Q8D3211yescharacteristic
192C2×C3×Q8D3212yes—
193C2×Dic6—32no—
194C2×Dic6—32no—
195C2×Dic6—62no—
196C2×Dic6—91no—
197C2×Dic6—92no—
198C3×C4⋊C4—91no—
199C3×C4⋊C4—94no—
200C3×C4⋊C4—94no—
201C3×C4⋊C4—92no—
202C3×C42—92no—
203C3×C42—91no—
204C3×C42—92no—
205C4⋊Dic3—32no—
206C4⋊Dic3—34no—
207C4⋊Dic3—61no—
208C4⋊Dic3—62no—
209C4⋊Dic3—92no—
210C4⋊Dic3—91no—
211C4⋊Dic3—92no—
212C4⋊Dic3—92no—
213C4⋊Dic3—94no—
214Dic3⋊C4—34no—
215Dic3⋊C4—34no—
216Dic3⋊C4—34no—
217Dic3⋊C4—34no—
218Dic3⋊C4—64no—
219Dic3⋊C4—64no—
220Dic3⋊C4—92no—
221Dic3⋊C4—92no—
222Dic3⋊C4—92no—
223Dic3⋊C4—94no—
224Dic3⋊C4—94no—
225Dic3⋊C4—94no—
226Dic3⋊C4—94no—
227C4×Dic3—32no—
228C4×Dic3—32no—
229C4×Dic3—32no—
230C4×Dic3—34no—
231C4×Dic3—62no—
232C4×Dic3—62no—
233C4×Dic3—61no—
234C4×Dic3—94no—
235C4×Dic3—94no—
236C4×Dic3—94no—
237C22×C32—22no—
238C22×C32—21no—
239C22×C32—41no—
240C22×C32C23×D311yescharacteristic
241C22×C32C23×D312yes—
242C32×C4—22no—
243C32×C4—24no—
244C32×C4—24no—
245C32×C4—24no—
246C32×C4—22no—
247C32×C4—42no—
248C32×C4—44no—
249C32×C4—44no—
250C32×C4#23512yes—
251C32×C4#23714yes—
252C3⋊Dic3—22no—
253C3⋊Dic3—24no—
254C3⋊Dic3—22no—
255C3⋊Dic3—21no—
256C3⋊Dic3D3×Q814yes—
257C3⋊Dic3D3×Q812yes—
258C3×Dic3—34no—
259C3×Dic3—34no—
260C3×Dic3—34no—
261C3×Dic3—62no—
262C3×Dic3—64no—
263C3×Dic3—64no—
264C3×Dic3—64no—
265C3×Dic3—64no—
266C3×Dic3—64no—
267C3×Dic3—62no—
268C3×Dic3—62no—
269C3×Dic3—62no—
270C3×Dic3—124no—
271C3×Dic3—122no—
272C3×Dic3—121no—
273C4⋊Q8—92no—
274C4⋊Q8—271no—
275#125—272no—
276#125—274no—
277C4×Q8—91no—
278C4×Q8—271no—
279C4×Q8—272no—
280C4×Q8—272no—
281C33#55711yescharacteristic, complemented, Sylow
282C3×Q8—24no—
283C3×Q8—24no—
284C3×Q8—22no—
285C3×Q8—44no—
286C3×Q8—44no—
287C3×Q8—184no—
288C2×C3×C4—22no—
289C2×C3×C4—21no—
290C2×C3×C4—22no—
291C2×C3×C4—24no—
292C2×C3×C4—42no—
293C2×C3×C4—41no—
294C2×C3×C4—92no—
295C2×C3×C4—91no—
296C2×C3×C4—92no—
297C2×C3×C4—94no—
298C2×C3×C4—92no—
299C2×C3×C4—91no—
300C2×C3×C4C2×D3214yes—
301C2×C3×C4C2×D3212yes—
302C2×C3×C4C2×D3212yes—
303C2×C3×C4C2×D3211yescharacteristic
304C2×Dic3—32no—
305C2×Dic3—32no—
306C2×Dic3—32no—
307C2×Dic3—34no—
308C2×Dic3—32no—
309C2×Dic3—32no—
310C2×Dic3—32no—
311C2×Dic3—34no—
312C2×Dic3—34no—
313C2×Dic3—61no—
314C2×Dic3—62no—
315C2×Dic3—62no—
316C2×Dic3—62no—
317C2×Dic3—61no—
318C2×Dic3—64no—
319Dic6—64no—
320Dic6—64no—
321Dic6—94no—
322Dic6—98no—
323Dic6—124no—
324C2×C32—22no—
325C2×C32—21no—
326C2×C32—21no—
327C2×C32—22no—
328C2×C32—22no—
329C2×C32—21no—
330C2×C32—41no—
331C2×C32—41no—
332C2×C32—41no—
333C2×C32C2×#23511yescharacteristic
334C2×C32C2×#23712yes—
335C2×C32C2×D3×Q812yes—
336C2×C32C2×D3×Q811yescharacteristic
337C2×C32#100712yes—
338C2×C32#101011yescharacteristic
339C2×Q8—92no—
340C2×Q8—271no—
341C2×Q8#111311yescharacteristic
342C4⋊C4—91no—
343C4⋊C4—92no—
344C4⋊C4—94no—
345C4⋊C4—94no—
346C4⋊C4—271no—
347C4⋊C4—272no—
348C4⋊C4—274no—
349C4⋊C4—272no—
350C4⋊C4—272no—
351C42—92no—
352C42—91no—
353C42—92no—
354C42—274no—
355C22×C3—22no—
356C22×C3—21no—
357C22×C3—41no—
358C22×C3C22×D3212yes—
359C22×C3C22×D3211yescharacteristic
360C3×C4—24no—
361C3×C4—24no—
362C3×C4—22no—
363C3×C4—24no—
364C3×C4—22no—
365C3×C4—44no—
366C3×C4—44no—
367C3×C4—42no—
368C3×C4—92no—
369C3×C4—94no—
370C3×C4—182no—
371C3×C4—182no—
372C3×C4—181no—
373C3×C4—184no—
374C3×C4#380414yes—
375C3×C4#380512yes—
376Dic3—34no—
377Dic3—34no—
378Dic3—34no—
379Dic3—62no—
380Dic3—64no—
381Dic3—64no—
382Dic3—62no—
383Dic3—64no—
384Dic3—62no—
385Dic3—62no—
386Dic3—64no—
387Dic3—64no—
388Dic3—124no—
389Dic3—122no—
390Dic3—121no—
391C32—21no—
392C32—22no—
393C32—41no—
394C32#556711yescharacteristic, complemented
395C32#572112yescomplemented
396Q8—22no—
397Q8—184no—
398Q8—274no—
399C2×C4—91no—
400C2×C4—92no—
401C2×C4—92no—
402C2×C4—92no—
403C2×C4—91no—
404C2×C4—94no—
405C2×C4C2×#111312yes—
406C2×C4C2×#111311yescharacteristic
407C2×C3—22no—
408C2×C3—21no—
409C2×C3—22no—
410C2×C3—21no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3#6431512yes—
417C2×C3#6431611yescharacteristic
418C2×C3D32×Q811yescharacteristic
419C2×C3D32×Q812yes—
420C2×C3C2×#380412yes—
421C2×C3C2×#380511yescharacteristic
422C22C22×#111311yescharacteristic
423C4—22no—
424C4—92no—
425C4—94no—
426C4—182no—
427C4—182no—
428C4—181no—
429C4—184no—
430C4#8904612yes—
431C3—22no—
432C3—21no—
433C3—41no—
434C3#1059533811yescharacteristic, complemented
435C3#1059542012yescomplemented
436C2#1172462411yescharacteristic
437C2Q8×#111311yescharacteristic
438C2C2×#8904611yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, prime radical, cosocleC22×C33C24
FittingC2×C33×Q8C22
hypercenterC2×Q8#1113
hypocenterC33#557
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
58. #6296Q8semidirect
281. C33#557semidirect
394. C32#5567semidirect
395. C32#5721semidirect
434. C3#10595338semidirect
435. C3#10595420semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×C3⋊Dic32
C3×#641532
C4×#62962
#117236792
#117236782
#117236692
#117236682
#117236652
#105954203
#105953383