Finite Group Databasebeta

← #13167592

Subgroups — #13167592

Order 1728.

914 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C22×#6295C211yesmaximal, characteristic, complemented
2C2×#88941C212yesmaximal, complemented
3#11723717C211yesmaximal, characteristic, complemented
4C3×C22⋊#622C211yesmaximal, characteristic, complemented
5#11722793C212yesmaximal, complemented
6#10593519—31nomaximal
7C23×C3×C3⋊D3—21no—
8C2×C3×#638—21no—
9C22×C3×C3⋊Dic3C2211yescharacteristic
10C2×#6295—21no—
11C2×#6295—24no—
12C2×#6295C2212yes—
13C2×#6295C2211yescharacteristic, complemented
14C2×C3×#622C2212yescomplemented
15#88941—24no—
16#88815C2211yescharacteristic, complemented
17#88814—21no—
18C22×#626—61no—
19C22×#625—31no—
20C2×#3774—32no—
21#63980—31no—
22#63978—61no—
23C22⋊#622D311yescharacteristic, complemented
24#63739—32no—
25C23×C33D411yescharacteristic
26C22×C3×C3⋊D3—24no—
27C22×C3×C3⋊D3—22no—
28C22×C3×C3⋊D3—21no—
29C22×C3×C3⋊D3—42no—
30C22×C3×C3⋊D3—42no—
31C3×#638—42no—
32C2×C3×C3⋊Dic3—22no—
33C2×C3×C3⋊Dic3—21no—
34C2×C3×C3⋊Dic3D411yescharacteristic
35C2×C3×C3⋊Dic3C2311yescharacteristic
36#6295—24no—
37#6295—22no—
38#6295—22no—
39#6295D414yescomplemented
40C3×#622—22no—
41#4732—91nomaximal
42C23×C3⋊D3—21no—
43C23×C3×D3—61no—
44C23×C3×D3—61no—
45C23×C3×D3—121no—
46C2×#638—21no—
47C22×C3⋊Dic3C2×D311yescharacteristic
48C2×C3×#67—61no—
49C2×C3×#67—61no—
50C2×C3×#67—121no—
51C22×C3×Dic3—61no—
52C22×C3×Dic3—61no—
53C2×#626—62no—
54C2×#626—68no—
55C2×#626—62no—
56C2×#626—61no—
57C2×#625—32no—
58C2×#625—31no—
59C2×#625—64no—
60C2×#625—61no—
61C2×#622C2×D312yescomplemented
62#3774—64no—
63#3724—31no—
64Dic3⋊Dic3—61no—
65#3722—61no—
66(C2×D3)⋊Dic3—61no—
67(C2×D3)⋊Dic3—61no—
68C22×C33—21no—
69C22×C33—21no—
70C22×C33D812yescomplemented
71C22×C33C2×D411yescharacteristic
72C2×C3×C3⋊D3—24no—
73C2×C3×C3⋊D3—48no—
74C2×C3×C3⋊D3—42no—
75C2×C3×C3⋊D3—42no—
76C2×C3×C3⋊D3—41no—
77C2×C3×C3⋊D3—41no—
78C2×C3×C3⋊D3—41no—
79C3×C3⋊Dic3—21no—
80C3×C3⋊Dic3—41no—
81C3×C3⋊Dic3C2×D412yes—
82C22×#67—181no—
83C22×D12—91no—
84#939—181no—
85#930—91no—
86C2×D24—92no—
87C3×C22⋊C8—91no—
88#821—92no—
89C23×C32—21no—
90C23×C32—21no—
91C23×C32—41no—
92C23×C32—41no—
93C23×C32D1211yescharacteristic
94C22×C3⋊D3—24no—
95C22×C3⋊D3—22no—
96C22×C3⋊D3—21no—
97C22×C3⋊D3—42no—
98C22×C3⋊D3—42no—
99C22×C3×D3—64no—
100C22×C3×D3—64no—
101C22×C3×D3—62no—
102C22×C3×D3—62no—
103C22×C3×D3—61no—
104C22×C3×D3—61no—
105C22×C3×D3—122no—
106C22×C3×D3—124no—
107C22×C3×D3—121no—
108C22×C3×D3—122no—
109C22×C3×D3—122no—
110C22×C3×D3—122no—
111C22×C3×D3—124no—
112C22×C3×D3—124no—
113C22×C3×D3—122no—
114#638—42no—
115C2×C3⋊Dic3—22no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3D1211yescharacteristic
118C2×C3⋊Dic3C22×D311yescharacteristic
119C3×#67—122no—
120C3×#67—124no—
121C3×#67—122no—
122C2×C3×Dic3—61no—
123C2×C3×Dic3—61no—
124C2×C3×Dic3—61no—
125C2×C3×Dic3—61no—
126C2×C3×Dic3—64no—
127C2×C3×Dic3—61no—
128C2×C3×Dic3—61no—
129C2×C3×Dic3—122no—
130C2×C3×Dic3—121no—
131#626—64no—
132#626—64no—
133#626—68no—
134#626—122no—
135#625—34no—
136#625—64no—
137#625—62no—
138#625—62no—
139#622—22no—
140#447—271noSylow
141C2×C33—22no—
142C2×C33#12011yescharacteristic
143C2×C33C2×D811yescharacteristic
144C2×C33AGL(1,ℤ/8)11yescharacteristic
145C3×C3⋊D3—44no—
146C3×C3⋊D3—81no—
147C24×C3—181no—
148C23×D3—61no—
149C23×D3—61no—
150C23×D3—121no—
151C2×C3×D4—181no—
152C22×C3×C4—91no—
153C2×#67—61no—
154C2×#67—61no—
155C2×#67—121no—
156C2×#67—182no—
157C2×#67—181no—
158C2×#67—182no—
159C2×#67—188no—
160C22×Dic3—61no—
161C22×Dic3—61no—
162C2×D12—92no—
163C2×D12—91no—
164C2×D12—184no—
165C2×D12—181no—
166C2×C3×C8—92no—
167#217—181no—
168(C2×D3)⋊C4—181no—
169(C2×D3)⋊C4—181no—
170C4⋊Dic3—91no—
171Dic3⋊C4—181no—
172D24—184no—
173C22×C32—22no—
174C22×C32—22no—
175C22×C32—21no—
176C22×C32—21no—
177C22×C32—21no—
178C22×C32—21no—
179C22×C32—22no—
180C22×C32—22no—
181C22×C32—42no—
182C22×C32—41no—
183C22×C32—41no—
184C22×C32—42no—
185C22×C32—41no—
186C22×C32—41no—
187C22×C32—42no—
188C22×C32—42no—
189C22×C32—81no—
190C22×C32—81no—
191C22×C32D2412yescomplemented
192C22×C32C2×D1211yescharacteristic
193C2×C3⋊D3—24no—
194C2×C3⋊D3—48no—
195C2×C3⋊D3—42no—
196C2×C3⋊D3—42no—
197C2×C3⋊D3—41no—
198C2×C3⋊D3—41no—
199C2×C3⋊D3—41no—
200C2×C3×D3—64no—
201C2×C3×D3—64no—
202C2×C3×D3—124no—
203C2×C3×D3—128no—
204C2×C3×D3—128no—
205C2×C3×D3—122no—
206C2×C3×D3—122no—
207C2×C3×D3—124no—
208C2×C3×D3—124no—
209C2×C3×D3—1216no—
210C2×C3×D3—122no—
211C2×C3×D3—122no—
212C2×C3×D3—121no—
213C2×C3×D3—121no—
214C2×C3×D3—121no—
215C2×C3×D3—122no—
216C2×C3×D3—121no—
217C2×C3×D3—121no—
218C2×C3×D3—121no—
219C2×C3×D3—241no—
220C2×C3×D3—241no—
221C3⋊Dic3—21no—
222C3⋊Dic3—41no—
223C3⋊Dic3D3×D412yes—
224C3×Dic3—62no—
225C3×Dic3—62no—
226C3×Dic3—62no—
227C3×Dic3—121no—
228C3×Dic3—121no—
229C3×Dic3—122no—
230C3×Dic3—121no—
231C22×D4—271no—
232C2×D8—272no—
233#121—271no—
234#102—272no—
235C22⋊C8—91no—
236C33#44711yescharacteristic, complemented, Sylow
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—184no—
242C23×C3—182no—
243C23×C3—181no—
244C23×C3—362no—
245C23×C3—362no—
246C23×C3GO+(4,𝔽2)11yescharacteristic
247C22×D3—64no—
248C22×D3—64no—
249C22×D3—62no—
250C22×D3—62no—
251C22×D3—61no—
252C22×D3—61no—
253C22×D3—124no—
254C22×D3—122no—
255C22×D3—121no—
256C22×D3—122no—
257C22×D3—122no—
258C22×D3—122no—
259C22×D3—124no—
260C22×D3—124no—
261C22×D3—122no—
262C3×D4—362no—
263C2×C3×C4—91no—
264C2×C3×C4—91no—
265C2×C3×C4—182no—
266C2×C3×C4—181no—
267#67—122no—
268#67—124no—
269#67—122no—
270#67—184no—
271#67—184no—
272#67—188no—
273#67—362no—
274C2×Dic3—61no—
275C2×Dic3—61no—
276C2×Dic3—61no—
277C2×Dic3—61no—
278C2×Dic3—64no—
279C2×Dic3—61no—
280C2×Dic3—61no—
281C2×Dic3—122no—
282C2×Dic3—121no—
283D12—94no—
284D12—182no—
285D12—182no—
286D12—184no—
287C3×C8—182no—
288C2×C32—24no—
289C2×C32—22no—
290C2×C32—21no—
291C2×C32—21no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—41no—
297C2×C32—41no—
298C2×C32—41no—
299C2×C32—41no—
300C2×C32—41no—
301C2×C32—41no—
302C2×C32—42no—
303C2×C32—44no—
304C2×C32—82no—
305C2×C32#88211yescharacteristic
306C2×C32C2×D2411yescharacteristic
307C2×C32#90811yescharacteristic
308C3⋊D3—44no—
309C3⋊D3—81no—
310C3×D3—124no—
311C3×D3—128no—
312C3×D3—124no—
313C3×D3—241no—
314C3×D3—242no—
315C3×D3—241no—
316C24—181no—
317C2×D4—181no—
318C2×D4—272no—
319C2×D4—271no—
320C2×D4—541no—
321C2×D4—544no—
322C22×C4—91no—
323D8—544no—
324C2×C8—92no—
325C4⋊C4—271no—
326C22⋊C4—541no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—22no—
334C22×C3—22no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—42no—
342C22×C3—42no—
343C22×C3—81no—
344C22×C3—81no—
345C22×C3—184no—
346C22×C3—368no—
347C22×C3—362no—
348C22×C3—362no—
349C22×C3—361no—
350C22×C3—361no—
351C22×C3—361no—
352C22×C3#377412yescomplemented
353C22×C3C2×GO+(4,𝔽2)11yescharacteristic
354C2×D3—64no—
355C2×D3—64no—
356C2×D3—124no—
357C2×D3—128no—
358C2×D3—128no—
359C2×D3—122no—
360C2×D3—122no—
361C2×D3—1216no—
362C2×D3—124no—
363C2×D3—124no—
364C2×D3—122no—
365C2×D3—122no—
366C2×D3—121no—
367C2×D3—121no—
368C2×D3—121no—
369C2×D3—122no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—121no—
373C2×D3—241no—
374C2×D3—241no—
375C3×C4—92no—
376C3×C4—181no—
377C3×C4—361no—
378Dic3—62no—
379Dic3—62no—
380Dic3—62no—
381Dic3—121no—
382Dic3—121no—
383Dic3—122no—
384Dic3—121no—
385C32—21no—
386C32—21no—
387C32—41no—
388C32—41no—
389C32#473211yescharacteristic, complemented
390C23—184no—
391C23—182no—
392C23—181no—
393C23—362no—
394C23—362no—
395C23#157⋊D311yescharacteristic
396D4—274no—
397D4—362no—
398D4—542no—
399D4—542no—
400D4—544no—
401C2×C4—91no—
402C2×C4—91no—
403C2×C4—182no—
404C2×C4—181no—
405C8—182no—
406C2×C3—24no—
407C2×C3—22no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—42no—
415C2×C3—44no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—41no—
419C2×C3—41no—
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—82no—
423C2×C3—364no—
424C2×C3—721no—
425C2×C3C2×#377411yescharacteristic
426C2×C3#6423611yescharacteristic
427C2×C3#6424111yescharacteristic
428D3—124no—
429D3—128no—
430D3—124no—
431D3—241no—
432D3—242no—
433D3—241no—
434C22—21no—
435C22—21no—
436C22—184no—
437C22—368no—
438C22—362no—
439C22—362no—
440C22—361no—
441C22—361no—
442C22—361no—
443C22#8894112yescomplemented
444C22C2×#157⋊D311yescharacteristic
445C4—92no—
446C4—181no—
447C4—361no—
448C3—21no—
449C3—21no—
450C3—41no—
451C3—41no—
452C3#1059351911yescharacteristic, complemented
453C2—22no—
454C2—364no—
455C2—721no—
456C2C2×#8894111yescharacteristic
457C2#1172435011yescharacteristic
458C2#1172435411yescharacteristic
subgroupG/H
center, FrattiniC22C2×#157⋊D3
commutator, cosocleC2×C3×C3⋊Dic3C23
FittingC23×C33D4
socleC22×C33C2×D4
hypercenterC23#157⋊D3
hypocenterC33#447
LCSG ▷ C2×C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C22×#6295C2semidirect
2. C2×#88941C2semidirect
3. #11723717C2semidirect
4. C3×C22⋊#622C2semidirect
5. #11722793C2semidirect
13. C2×#6295C22semidirect
14. C2×C3×#622C22semidirect
16. #88815C22semidirect
23. C22⋊#622D3semidirect
39. #6295D4semidirect
61. C2×#622C2×D3semidirect
70. C22×C33D8semidirect
191. C22×C32D24semidirect
236. C33#447semidirect
352. C22×C3#3774semidirect
389. C32#4732semidirect
443. C22#88941semidirect
452. C3#10593519semidirect

Maximal subgroups

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

subgroupindex
C22×#62952
C2×#889412
#117237172
C3×C22⋊#6222
#117227932
#105935193
#47329