Finite Group Databasebeta

← #13168674

Subgroups — #13168674

Order 1728.

1108 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1108 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6296C211yesmaximal, characteristic, complemented
2D12×C3⋊Dic3C211yesmaximal, characteristic
3Dic6×C3⋊Dic3C212yesmaximal, complemented
4C3×#63873C211yesmaximal, characteristic, complemented
5C3×#63863C212yesmaximal
6#11723022C214yesmaximal, complemented
7#11722981C212yesmaximal, complemented
8#11722917C212yesmaximal, complemented
9#10595384—32nomaximal
10#10595119—31nomaximal
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2212yes—
13C2×#6296C2211yescharacteristic
14C2×D3×C3⋊Dic3C2212yes—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2212yes—
17C3×C4×C3⋊Dic3C2211yescharacteristic
18#88815C2212yes—
19#88815C2211yescharacteristic
20Dic3⋊(C3⋊Dic3)C2214yes—
21(C2×D3)⋊(C3⋊Dic3)C2212yes—
22Dic3×C3⋊Dic3C2214yes—
23C3×#3724C2212yes—
24C3×#3724C2212yes—
25C3×(C2×D3)⋊Dic3C2214yes—
26C3×(C2×D3)⋊Dic3C2214yes—
27Q8×C3⋊Dic3—32no—
28D4×C3⋊Dic3—31no—
29C3×#947—32no—
30C3×#944—32no—
31C3×#936—32no—
32#63982—32no—
33#63957—32no—
34#63956—34no—
35C4×#627—31no—
36C4×#627—32no—
37#63893—34no—
38Dic3×D12—32no—
39Dic3×D12—61no—
40#63878—34no—
41#63873D311yescharacteristic, complemented
42#63863D312yes—
43#63862—34no—
44#63861—34no—
45Dic3×Dic6—32no—
46Dic3×Dic6—32no—
47Dic3×Dic6—62no—
48C22×C32×D3C2312yes—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)C2312yes—
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C2×C32×Dic3C2314yes—
56C32×D12—22no—
57C32×Dic6—24no—
58#6296—22no—
59D3×C3⋊Dic3—24no—
60#5732—91no—
61#5671—92no—
62C2×C32×Q8—32no—
63C2×C32×D4—31no—
64C22×C3⋊Dic3—32no—
65C2×C3×D12—21no—
66C2×C3×D12C2×D312yes—
67C2×C3×Dic6—22no—
68C2×C3×Dic6C2×D312yes—
69C2×C3×Dic6C2×D312yes—
70C2×#627—32no—
71C2×#627—31no—
72C2×D3×Dic3—34no—
73C2×D3×Dic3—62no—
74#3757—32no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—32no—
77C4⋊(C3⋊Dic3)—34no—
78C4⋊(C3⋊Dic3)—62no—
79C4⋊(C3⋊Dic3)—61no—
80C4⋊(C3⋊Dic3)—121no—
81C4×C3⋊Dic3—34no—
82C4×C3⋊Dic3C2×D312yes—
83C4×C3⋊Dic3C2×D311yescharacteristic
84C3×#217—34no—
85C3×#217—34no—
86C3×(C2×D3)⋊C4—34no—
87C3×(C2×D3)⋊C4—34no—
88C3×Dic3⋊C4—34no—
89C3×Dic3⋊C4—34no—
90C3×C4×Dic3—32no—
91C3×C4×Dic3—32no—
92C3×C4×Dic3—32no—
93C3×C4×Dic3—62no—
94C3×C4×Dic3—61no—
95#3724—32no—
96#3724—31no—
97#3724C2×D312yes—
98#3724C2×D312yes—
99Dic3⋊Dic3—32no—
100Dic3⋊Dic3—32no—
101Dic3⋊Dic3—32no—
102Dic3⋊Dic3—34no—
103Dic3⋊Dic3—34no—
104Dic3⋊Dic3—64no—
105(C2×D3)⋊Dic3—34no—
106(C2×D3)⋊Dic3—62no—
107(C2×D3)⋊Dic3C2×D314yes—
108(C2×D3)⋊Dic3C2×D314yes—
109Dic32—34no—
110Dic32—34no—
111Dic32—64no—
112C22×C33C2411yescharacteristic
113C2×C32×D3—22no—
114C2×C32×D3—22no—
115C2×C32×D3—22no—
116C33×C4#4112yes—
117C3⋊(C3⋊Dic3)—22no—
118C3×C3⋊Dic3—22no—
119C3×C3⋊Dic3—22no—
120C3×C3⋊Dic3—21no—
121C3×C3⋊Dic3#4112yes—
122C32×Dic3—24no—
123C3×C4⋊Q8—91no—
124C3×#124—92no—
125#947—32no—
126Q8×Dic3—92no—
127Q8×Dic3—92no—
128Q8×Dic3—182no—
129#944—32no—
130#936—32no—
131D4×Dic3—92no—
132D4×Dic3—181no—
133#898—94no—
134#897—92no—
135#892—92no—
136#879—94no—
137#879—94no—
138#878—94no—
139#877—94no—
140C4×D12—91no—
141C4×Dic6—92no—
142C4×Dic6—91no—
143C4×Dic6—92no—
144C23×C32—32no—
145C22×C3×D3—22no—
146C22×C3×D3C22×D314yes—
147C32×Q8—64no—
148C32×D4—62no—
149C2×C32×C4—22no—
150C2×C32×C4—21no—
151C2×C32×C4—34no—
152C2×C32×C4—41no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D312yes—
155C2×C3⋊Dic3—32no—
156C2×C3⋊Dic3—34no—
157C2×C3⋊Dic3—64no—
158C2×C3⋊Dic3—62no—
159C2×C3⋊Dic3—64no—
160C2×C3⋊Dic3—122no—
161C2×C3⋊Dic3C22×D312yes—
162C2×C3⋊Dic3C22×D312yes—
163C2×C3⋊Dic3C22×D311yescharacteristic
164C2×C3⋊Dic3C22×D311yescharacteristic
165C2×C3×Dic3—24no—
166C2×C3×Dic3—32no—
167C2×C3×Dic3—32no—
168C2×C3×Dic3—32no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—62no—
176C2×C3×Dic3C22×D314yes—
177C2×C3×Dic3C22×D314yes—
178C3×D12—24no—
179C3×D12—24no—
180C3×Dic6—28no—
181C3×Dic6—24no—
182C3×Dic6—24no—
183#627—64no—
184#627—62no—
185D3×Dic3—68no—
186D3×Dic3—124no—
187#556—271noSylow
188C2×C33C2×#4111yescharacteristic
189C2×C33C2×#4111yescharacteristic
190C2×C332−1+411yescharacteristic
191C32×D3—42no—
192C2×C3×Q8—32no—
193C2×C3×Q8—32no—
194C2×C3×Q8—62no—
195C2×C3×D4—32no—
196C2×C3×D4—61no—
197C22×Dic3—94no—
198C22×Dic3—182no—
199C2×D12D3211yescharacteristic
200C2×C4×D3—92no—
201C2×Dic6—92no—
202C2×Dic6—91no—
203C2×Dic6D3212yes—
204C3×C4⋊C4—92no—
205C3×C4⋊C4—92no—
206C3×C22⋊C4—94no—
207C3×C22⋊C4—94no—
208C3×C42—92no—
209C3×C42—91no—
210#217—34no—
211#217—34no—
212#217—94no—
213#217—182no—
214(C2×D3)⋊C4—34no—
215(C2×D3)⋊C4—34no—
216(C2×D3)⋊C4—92no—
217C4⋊Dic3—92no—
218C4⋊Dic3—91no—
219C4⋊Dic3—94no—
220C4⋊Dic3—94no—
221C4⋊Dic3—92no—
222C4⋊Dic3—91no—
223C4⋊Dic3—184no—
224C4⋊Dic3—181no—
225C4⋊Dic3—182no—
226C4⋊Dic3—361no—
227Dic3⋊C4—34no—
228Dic3⋊C4—34no—
229Dic3⋊C4—92no—
230Dic3⋊C4—92no—
231Dic3⋊C4—92no—
232Dic3⋊C4—94no—
233C4×Dic3—32no—
234C4×Dic3—32no—
235C4×Dic3—32no—
236C4×Dic3—61no—
237C4×Dic3—62no—
238C4×Dic3—94no—
239C4×Dic3—94no—
240C4×Dic3—94no—
241C4×Dic3—184no—
242C22×C32—22no—
243C22×C32—21no—
244C22×C32—41no—
245C22×C32—62no—
246C22×C32—62no—
247C22×C32—62no—
248C22×C32C23×D311yescharacteristic
249C22×C32C23×D312yes—
250C2×C3×D3—24no—
251C2×C3×D3—24no—
252C2×C3×D3—24no—
253C2×C3×D3—24no—
254C2×C3×D3—42no—
255C2×C3×D3—42no—
256C32×C4—24no—
257C32×C4—22no—
258C32×C4—42no—
259C32×C4—64no—
260C32×C4#23712yes—
261C32×C4#23714yes—
262C3⋊Dic3—22no—
263C3⋊Dic3—21no—
264C3⋊Dic3—22no—
265C3⋊Dic3—62no—
266C3⋊Dic3—64no—
267C3⋊Dic3—122no—
268C3⋊Dic3—124no—
269C3⋊Dic3—242no—
270C3⋊Dic3#23912yes—
271C3×Dic3—28no—
272C3×Dic3—24no—
273C3×Dic3—24no—
274C3×Dic3—34no—
275C3×Dic3—62no—
276C3×Dic3—64no—
277C3×Dic3—62no—
278C3×Dic3—62no—
279C3×Dic3—62no—
280C3×Dic3—62no—
281C3×Dic3—62no—
282C3×Dic3—121no—
283C3×Dic3—122no—
284C4⋊Q8—91no—
285#126—274no—
286#124—92no—
287C22⋊Q8—272no—
288C4×Q8—271no—
289C4×Q8—272no—
290C4×D4—271no—
291#117—272no—
292C33#55611yescharacteristic, complemented, Sylow
293C23×C3—34no—
294C23×C3—62no—
295C22×D3C2×D3212yes—
296C3×Q8—68no—
297C3×Q8—64no—
298C3×Q8—64no—
299C3×D4—64no—
300C3×D4—64no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—34no—
304C2×C3×C4—34no—
305C2×C3×C4—41no—
306C2×C3×C4—64no—
307C2×C3×C4—91no—
308C2×C3×C4—92no—
309C2×C3×C4—92no—
310C2×C3×C4—91no—
311C2×C3×C4C2×D3212yes—
312C2×C3×C4C2×D3211yescharacteristic
313C2×Dic3—32no—
314C2×Dic3—32no—
315C2×Dic3—32no—
316C2×Dic3—32no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C2×Dic3—62no—
320C2×Dic3—61no—
321C2×Dic3—62no—
322C2×Dic3—61no—
323C2×Dic3—94no—
324C2×Dic3—92no—
325C2×Dic3—184no—
326C2×Dic3—182no—
327C2×Dic3—188no—
328C2×Dic3—362no—
329C2×Dic3—364no—
330C2×Dic3C2×D3214yes—
331D12—22no—
332C4×D3—184no—
333Dic6—24no—
334Dic6—184no—
335Dic6—182no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—22no—
339C2×C32—22no—
340C2×C32—22no—
341C2×C32—21no—
342C2×C32—41no—
343C2×C32—41no—
344C2×C32—41no—
345C2×C32—122no—
346C2×C32C2×#23712yes—
347C2×C32C2×#23711yescharacteristic
348C2×C32C2×#23911yescharacteristic
349C2×C32#100711yescharacteristic
350C2×C32D3×#4112yes—
351C2×C32#101012yes—
352C3×D3—44no—
353C3×D3—44no—
354C2×Q8—32no—
355C2×Q8—271no—
356C2×D4—31no—
357C22×C4—272no—
358C4⋊C4—92no—
359C4⋊C4—92no—
360C4⋊C4—274no—
361C4⋊C4—271no—
362C4⋊C4—272no—
363C4⋊C4—271no—
364C22⋊C4—94no—
365C22⋊C4—94no—
366C22⋊C4—272no—
367C42—91no—
368C42—92no—
369C42—274no—
370C22×C3—22no—
371C22×C3—21no—
372C22×C3—41no—
373C22×C3—64no—
374C22×C3—64no—
375C22×C3—64no—
376C22×C3—64no—
377C22×C3—122no—
378C22×C3—122no—
379C22×C3C22×D3211yescharacteristic
380C22×C3C22×D3212yes—
381C2×D3—22no—
382C2×D3—22no—
383C2×D3—22no—
384C3×C4—22no—
385C3×C4—24no—
386C3×C4—42no—
387C3×C4—68no—
388C3×C4—64no—
389C3×C4—64no—
390C3×C4—92no—
391C3×C4—181no—
392C3×C4—182no—
393C3×C4—182no—
394C3×C4#380514yes—
395C3×C4#380512yes—
396Dic3—24no—
397Dic3—34no—
398Dic3—62no—
399Dic3—64no—
400Dic3—62no—
401Dic3—62no—
402Dic3—62no—
403Dic3—62no—
404Dic3—62no—
405Dic3—122no—
406Dic3—121no—
407Dic3—182no—
408Dic3—184no—
409Dic3—362no—
410Dic3—364no—
411Dic3—722no—
412C32—21no—
413C32—22no—
414C32—41no—
415C32#567112yescomplemented
416C32#573211yescharacteristic, complemented
417C23—32no—
418Q8—64no—
419Q8—542no—
420D4—62no—
421C2×C4—34no—
422C2×C4—91no—
423C2×C4—92no—
424C2×C4—92no—
425C2×C4—91no—
426C2×C4—272no—
427C2×C4—544no—
428C2×C4D3311yescharacteristic
429C2×C3—21no—
430C2×C3—22no—
431C2×C3—22no—
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—21no—
435C2×C3—41no—
436C2×C3—41no—
437C2×C3—41no—
438C2×C3—124no—
439C2×C3—124no—
440C2×C3#6430911yescharacteristic
441C2×C3#6431211yescharacteristic
442C2×C3#6431512yes—
443C2×C3D3×#23912yes—
444C2×C3C2×#380511yescharacteristic
445C2×C3C2×#380512yes—
446D3—42no—
447C22—62no—
448C22—62no—
449C22—62no—
450C22C2×D3311yescharacteristic
451C4—64no—
452C4—92no—
453C4—182no—
454C4—182no—
455C4—181no—
456C4—542no—
457C4#8896312yes—
458C3—21no—
459C3—22no—
460C3—41no—
461C3#1059511911yescharacteristic, complemented
462C3#1059538412yescomplemented
463C2—122no—
464C2#1172437711yescharacteristic
465C2#1172439411yescharacteristic
466C2C2×#8896311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×C32×D12C22
hypercenterC2×C4D33
hypocenterC33#556
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C4×#6296C2semidirect
3. Dic6×C3⋊Dic3C2semidirect
4. C3×#63873C2semidirect
6. #11723022C2semidirect
7. #11722981C2semidirect
8. #11722917C2semidirect
41. #63873D3semidirect
292. C33#556semidirect
415. C32#5671semidirect
416. C32#5732semidirect
461. C3#10595119semidirect
462. C3#10595384semidirect

Maximal subgroups

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

subgroupindex
C4×#62962
D12×C3⋊Dic32
Dic6×C3⋊Dic32
C3×#638732
C3×#638632
#117230222
#117229812
#117229172
#105953843
#105951193