Finite Group Databasebeta

← #13168676

Subgroups — #13168676

Order 1728.

1148 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1148 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
2#11723494C211yesmaximal, characteristic
3#11723440C212yesmaximal
4C3×#63873C211yesmaximal, characteristic, complemented
5C3×#63843C212yesmaximal, complemented
6#11723021C214yesmaximal, complemented
7#11722982C212yesmaximal, complemented
8#11722918C212yesmaximal, complemented
9#10595387—32nomaximal
10#10594350—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C32×Dic6C2212yes—
13C2×#6296C2211yescharacteristic
14C2×#6289C2212yes—
15C3×C4×C3⋊Dic3C2212yes—
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18#88815C2212yes—
19#88815C2211yescharacteristic
20#88804C2212yes—
21#88804C2214yes—
22#88804C2214yes—
23C3×#3724C2212yes—
24C3×#3724C2212yes—
25C3×Dic3⋊Dic3C2214yes—
26C3×Dic32C2214yes—
27#64156—32no—
28#64102—31no—
29C3×Q8×Dic3—32no—
30C3×#944—32no—
31C3×#887—32no—
32C4×#627—31no—
33C4×#627—32no—
34#63895—34no—
35#63894—32no—
36#63894—61no—
37#63874—32no—
38#63873D311yescharacteristic, complemented
39#63868—34no—
40#63868—34no—
41#63865—32no—
42#63864—32no—
43#63864—32no—
44#63864—62no—
45#63848—34no—
46#63843D312yescomplemented
47#63836—34no—
48C22×C3⋊(C3⋊D3)C2312yes—
49C2×C33×C4C2311yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—22no—
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2312yes—
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C2×C32×Dic3C2312yes—
56C2×C32×Dic3C2314yes—
57C32×Dic6—24no—
58#6296—22no—
59#6289—24no—
60#5732—91no—
61#5568—92no—
62C2×C32×Q8—32no—
63C2×C4⋊(C3⋊D3)—31no—
64C2×C4⋊(C3⋊D3)—32no—
65C2×C4⋊(C3⋊D3)—61no—
66C2×C4⋊(C3⋊D3)—62no—
67C2×C4⋊(C3⋊D3)—121no—
68C2×C4×C3⋊D3—32no—
69C2×C3×Dic6—22no—
70C2×C3×Dic6C2×D312yes—
71C2×C3×Dic6C2×D312yes—
72C2×#627—31no—
73C2×#627—32no—
74C2×#624—34no—
75C2×#624—62no—
76C32×C4⋊C4—31no—
77#3752—34no—
78#3752—32no—
79#3752—34no—
80C4×C3⋊Dic3C2×D312yes—
81C4×C3⋊Dic3C2×D311yescharacteristic
82C3×C4⋊Dic3—21no—
83C3×C4⋊Dic3—34no—
84C3×C4⋊Dic3C2×D312yes—
85C3×Dic3⋊C4—34no—
86C3×Dic3⋊C4—34no—
87C3×Dic3⋊C4—34no—
88C3×C4×Dic3—32no—
89C3×C4×Dic3—32no—
90C3×C4×Dic3—32no—
91C3×C4×Dic3—34no—
92C3×C4×Dic3—34no—
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⋊Dic3C2×D314yes—
103#3722—34no—
104#3722—34no—
105#3722—34no—
106#3722—34no—
107#3722—34no—
108#3722—64no—
109#3722—64no—
110#3722—62no—
111Dic32C2×D314yes—
112C22×C33C2411yescharacteristic
113C2×C3⋊(C3⋊D3)—22no—
114C2×C3⋊(C3⋊D3)—22no—
115C2×C3⋊(C3⋊D3)—22no—
116C33×C4#4112yes—
117C3×C3⋊Dic3—22no—
118C3×C3⋊Dic3—22no—
119C3×C3⋊Dic3—21no—
120C3×C3⋊Dic3#4112yes—
121C32×Dic3—22no—
122C32×Dic3—24no—
123C3×C4⋊Q8—91no—
124C3×C4×Q8—92no—
125#947—92no—
126#947—92no—
127#947—182no—
128Q8×Dic3—32no—
129#944—32no—
130#898—94no—
131#898—94no—
132#897—92no—
133#896—94no—
134#893—92no—
135#893—181no—
136#892—92no—
137#887—32no—
138#876—94no—
139#875—92no—
140C4×D12—91no—
141#872—94no—
142C4×Dic6—91no—
143C4×Dic6—92no—
144C22×C3⋊D3—32no—
145C22×C3⋊D3—34no—
146C22×C3⋊D3—62no—
147C22×C3⋊D3—64no—
148C22×C3⋊D3—122no—
149C32×Q8—64no—
150C2×C32×C4—21no—
151C2×C32×C4—22no—
152C2×C32×C4—32no—
153C2×C32×C4—34no—
154C2×C32×C4—41no—
155C2×C32×C4C22×D312yes—
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3C22×D312yes—
158C2×C3⋊Dic3C22×D311yescharacteristic
159C2×C3⋊Dic3C22×D312yes—
160C2×C3⋊Dic3C22×D311yescharacteristic
161C4⋊(C3⋊D3)—62no—
162C4⋊(C3⋊D3)—64no—
163C4⋊(C3⋊D3)—64no—
164C4⋊(C3⋊D3)—124no—
165C4⋊(C3⋊D3)—124no—
166C4×C3⋊D3—64no—
167C2×C3×Dic3—24no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—32no—
170C2×C3×Dic3—32no—
171C2×C3×Dic3—32no—
172C2×C3×Dic3—32no—
173C2×C3×Dic3—32no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—62no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—62no—
179C2×C3×Dic3C22×D314yes—
180C2×C3×Dic3C22×D314yes—
181C2×C3×Dic3C22×D314yes—
182C3×Dic6—24no—
183C3×Dic6—24no—
184C3×Dic6—44no—
185#627—64no—
186#627—62no—
187#624—68no—
188#624—124no—
189#556—271noSylow
190C2×C33C2×#4111yescharacteristic
191C2×C33C2×#4111yescharacteristic
192C2×C332−1+411yescharacteristic
193C3⋊(C3⋊D3)—42no—
194C2×C3×Q8—32no—
195C2×C3×Q8—32no—
196C2×C3×Q8—62no—
197C2×D12—91no—
198C2×D12—92no—
199C2×D12—181no—
200C2×D12—182no—
201C2×D12—361no—
202C2×C4×D3—94no—
203C2×C4×D3—92no—
204C2×C4×D3—182no—
205C2×Dic6—92no—
206C2×Dic6—91no—
207C2×Dic6D3212yes—
208C3×C4⋊C4—32no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4—92no—
211C3×C4⋊C4—94no—
212C3×C4⋊C4—92no—
213C3×C42—94no—
214C3×C42—91no—
215C3×C42—92no—
216(C2×D3)⋊C4—94no—
217(C2×D3)⋊C4—94no—
218(C2×D3)⋊C4—94no—
219(C2×D3)⋊C4—94no—
220(C2×D3)⋊C4—94no—
221(C2×D3)⋊C4—94no—
222(C2×D3)⋊C4—92no—
223(C2×D3)⋊C4—94no—
224(C2×D3)⋊C4—182no—
225(C2×D3)⋊C4—184no—
226(C2×D3)⋊C4—184no—
227C4⋊Dic3—34no—
228C4⋊Dic3—91no—
229C4⋊Dic3—92no—
230C4⋊Dic3D3211yescharacteristic
231Dic3⋊C4—34no—
232Dic3⋊C4—34no—
233Dic3⋊C4—34no—
234Dic3⋊C4—92no—
235Dic3⋊C4—92no—
236Dic3⋊C4—92no—
237C4×Dic3—32no—
238C4×Dic3—32no—
239C4×Dic3—32no—
240C4×Dic3—34no—
241C4×Dic3—34no—
242C4×Dic3—61no—
243C4×Dic3—62no—
244C22×C32—21no—
245C22×C32—22no—
246C22×C32—41no—
247C22×C32C23×D312yes—
248C22×C32C23×D311yescharacteristic
249C2×C3⋊D3—62no—
250C2×C3⋊D3—64no—
251C2×C3⋊D3—64no—
252C2×C3⋊D3—62no—
253C2×C3⋊D3—62no—
254C2×C3⋊D3—64no—
255C2×C3⋊D3—64no—
256C2×C3⋊D3—124no—
257C2×C3⋊D3—124no—
258C2×C3⋊D3—122no—
259C2×C3⋊D3—122no—
260C2×C3⋊D3—124no—
261C2×C3⋊D3—124no—
262C2×C3⋊D3—242no—
263C2×C3⋊D3—242no—
264C32×C4—22no—
265C32×C4—24no—
266C32×C4—42no—
267C32×C4—64no—
268C32×C4—62no—
269C32×C4#23514yes—
270C32×C4#23712yes—
271C3⋊Dic3—21no—
272C3⋊Dic3—22no—
273C3⋊Dic3—22no—
274C3⋊Dic3#23912yes—
275C3×Dic3—24no—
276C3×Dic3—24no—
277C3×Dic3—24no—
278C3×Dic3—34no—
279C3×Dic3—42no—
280C3×Dic3—44no—
281C3×Dic3—62no—
282C3×Dic3—62no—
283C3×Dic3—62no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C3×Dic3—121no—
288C3×Dic3—122no—
289C3×Dic3—122no—
290C4⋊Q8—91no—
291#126—274no—
292#124—272no—
293C22⋊Q8—272no—
294C4×Q8—92no—
295C4×Q8—271no—
296C4×D4—271no—
297#117—272no—
298C33#55611yescharacteristic, complemented, Sylow
299C22×D3—92no—
300C22×D3—94no—
301C22×D3—182no—
302C22×D3—184no—
303C22×D3—362no—
304C3×Q8—64no—
305C3×Q8—64no—
306C3×Q8—124no—
307C2×C3×C4—21no—
308C2×C3×C4—22no—
309C2×C3×C4—34no—
310C2×C3×C4—34no—
311C2×C3×C4—34no—
312C2×C3×C4—41no—
313C2×C3×C4—64no—
314C2×C3×C4—62no—
315C2×C3×C4—91no—
316C2×C3×C4—92no—
317C2×C3×C4—91no—
318C2×C3×C4—92no—
319C2×C3×C4C2×D3211yescharacteristic
320C2×C3×C4C2×D3212yes—
321C2×Dic3—32no—
322C2×Dic3—32no—
323C2×Dic3—32no—
324C2×Dic3—32no—
325C2×Dic3—32no—
326C2×Dic3—32no—
327C2×Dic3—61no—
328C2×Dic3—62no—
329C2×Dic3—61no—
330C2×Dic3—62no—
331C2×Dic3C2×D3212yes—
332C2×Dic3C2×D3214yes—
333D12—184no—
334D12—182no—
335D12—184no—
336D12—364no—
337D12—364no—
338C4×D3—184no—
339C4×D3—188no—
340C4×D3—364no—
341Dic6—24no—
342Dic6—184no—
343Dic6—182no—
344C2×C32—21no—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—22no—
348C2×C32—22no—
349C2×C32—22no—
350C2×C32—41no—
351C2×C32—41no—
352C2×C32—41no—
353C2×C32C2×#23512yes—
354C2×C32C2×#23711yescharacteristic
355C2×C32C2×#23911yescharacteristic
356C2×C32#100711yescharacteristic
357C2×C32#100712yes—
358C2×C32D3×#4112yes—
359C3⋊D3—122no—
360C3⋊D3—124no—
361C3⋊D3—124no—
362C3⋊D3—244no—
363C3⋊D3—244no—
364C2×Q8—32no—
365C2×Q8—271no—
366C2×D4—271no—
367C22×C4—272no—
368C4⋊C4—31no—
369C4⋊C4—92no—
370C4⋊C4—94no—
371C4⋊C4—92no—
372C4⋊C4—271no—
373C4⋊C4—272no—
374C22⋊C4—274no—
375C22⋊C4—272no—
376C22⋊C4—274no—
377C42—94no—
378C42—91no—
379C42—92no—
380C22×C3—21no—
381C22×C3—22no—
382C22×C3—41no—
383C22×C3C22×D3211yescharacteristic
384C22×C3C22×D3212yes—
385C2×D3—184no—
386C2×D3—182no—
387C2×D3—182no—
388C2×D3—182no—
389C2×D3—184no—
390C2×D3—184no—
391C2×D3—184no—
392C2×D3—362no—
393C2×D3—362no—
394C2×D3—364no—
395C2×D3—364no—
396C2×D3—364no—
397C2×D3—364no—
398C2×D3—722no—
399C2×D3—722no—
400C3×C4—22no—
401C3×C4—24no—
402C3×C4—42no—
403C3×C4—64no—
404C3×C4—64no—
405C3×C4—64no—
406C3×C4—92no—
407C3×C4—122no—
408C3×C4—124no—
409C3×C4—181no—
410C3×C4—182no—
411C3×C4—182no—
412C3×C4#379812yes—
413C3×C4#380414yes—
414Dic3—22no—
415Dic3—24no—
416Dic3—34no—
417Dic3—62no—
418Dic3—62no—
419Dic3—62no—
420Dic3—62no—
421Dic3—62no—
422Dic3—62no—
423Dic3—122no—
424Dic3—122no—
425Dic3—121no—
426C32—21no—
427C32—22no—
428C32—41no—
429C32#556812yescomplemented
430C32#573211yescharacteristic, complemented
431C23—272no—
432Q8—64no—
433Q8—542no—
434D4—542no—
435C2×C4—32no—
436C2×C4—34no—
437C2×C4—91no—
438C2×C4—92no—
439C2×C4—91no—
440C2×C4—92no—
441C2×C4—544no—
442C2×C4D3311yescharacteristic
443C2×C3—21no—
444C2×C3—22no—
445C2×C3—21no—
446C2×C3—22no—
447C2×C3—22no—
448C2×C3—21no—
449C2×C3—41no—
450C2×C3—41no—
451C2×C3—41no—
452C2×C3C2×#379811yescharacteristic
453C2×C3#6431211yescharacteristic
454C2×C3#6431611yescharacteristic
455C2×C3#6431612yes—
456C2×C3D3×#23912yes—
457C2×C3C2×#380412yes—
458D3—364no—
459D3—362no—
460D3—364no—
461D3—724no—
462D3—724no—
463C22—542no—
464C22—542no—
465C22—542no—
466C22C2×D3311yescharacteristic
467C4—62no—
468C4—64no—
469C4—92no—
470C4—182no—
471C4—182no—
472C4—181no—
473C4#8895912yes—
474C3—21no—
475C3—22no—
476C3—41no—
477C3#1059435011yescharacteristic, complemented
478C3#1059538712yescomplemented
479C2—1082no—
480C2#1172437711yescharacteristic
481C2#1172439911yescharacteristic
482C2C2×#8895911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
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
4. C3×#63873C2semidirect
5. C3×#63843C2semidirect
6. #11723021C2semidirect
7. #11722982C2semidirect
8. #11722918C2semidirect
38. #63873D3semidirect
46. #63843D3semidirect
298. C33#556semidirect
429. C32#5568semidirect
430. C32#5732semidirect
477. C3#10594350semidirect
478. C3#10595387semidirect

Maximal subgroups

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

subgroupindex
C4×#62962
#117234942
#117234402
C3×#638732
C3×#638432
#117230212
#117229822
#117229182
#105953873
#105943503