Finite Group Databasebeta

← #13160096

Subgroups — #13160096

Order 1728.

1112 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C2×#89015C211yesmaximal, characteristic
2C2×C32×#204C211yesmaximal, characteristic, complemented
3#11723456C211yesmaximal, characteristic, complemented
4C2×#88798C211yesmaximal, characteristic, complemented
5#11722190C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7#11722031C211yesmaximal, characteristic, complemented
8#10592872—31nomaximal
9#10590961—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6291—21no—
15C2×D3×C3⋊Dic3—21no—
16#89015—22no—
17#89015—22no—
18C2×C33×C8C2211yescharacteristic
19C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C32×#204—22no—
22#88807—21no—
23#88804—21no—
24#88798—22no—
25C2×#3830—31no—
26C2×#3796—34no—
27C2×C32×SD16—31no—
28#64155—31no—
29C2×#3756—31no—
30C2×C3×#204D314yescomplemented
31#63871—34no—
32C2×#3718—34no—
33#63659—31no—
34#63647—31no—
35#63647—34no—
36#63647—64no—
37#63643D311yescharacteristic, complemented
38C3×#816—34no—
39#63562—34no—
40C22×C3⋊(C3⋊D3)D411yescharacteristic
41C22×C32×D3—21no—
42C2×C33×C4C2311yescharacteristic
43C2×C3⋊(C3⋊Dic3)D411yescharacteristic
44C4×C3⋊(C3⋊D3)—21no—
45C4×C3⋊(C3⋊D3)—21no—
46C2×C3×C3⋊Dic3—21no—
47C2×C3×C3⋊Dic3—21no—
48C3×#634—21no—
49C3×#634D412yes—
50C2×C32×Dic3—21no—
51C32×D12—21no—
52C32×D12D412yes—
53C32×Dic6—21no—
54C32×Dic6—21no—
55#6291—24no—
56D3×C3⋊Dic3—24no—
57C33×C8—21no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59#5173—94no—
60#4808—91no—
61C2×C32×Q8—31no—
62C2×C32×D4—31no—
63C2×#638—61no—
64C22×C3⋊Dic3—61no—
65#3830—62no—
66#3830—62no—
67C2×C4×C3⋊D3—31no—
68C2×C4×C3⋊D3—34no—
69C2×C4×C3⋊D3—64no—
70C2×#634C2×D311yescharacteristic
71C2×C3×D12C2×D314yes—
72C2×C3×Dic6—34no—
73C2×C3×Dic6C2×D314yes—
74C2×#626—64no—
75C2×D3×Dic3—64no—
76#3796—68no—
77#3796—68no—
78C32×SD16—62no—
79C2×C32×C8—24no—
80C2×C32×C8C2×D314yes—
81C2×C32×C8C2×D311yescharacteristic
82#3756—62no—
83#3752—61no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85#3750—61no—
86C2×C3⋊(C3⋊C8)—31no—
87C2×C3⋊(C3⋊C8)—34no—
88C2×C3⋊(C3⋊C8)—64no—
89C3×C4⋊Dic3—34no—
90C3×#204—28no—
91Dic3⋊Dic3—64no—
92#3722—64no—
93#3718—68no—
94C22×C33C2×D411yescharacteristic
95C2×C3⋊(C3⋊D3)—21no—
96C2×C3⋊(C3⋊D3)—21no—
97C2×C3⋊(C3⋊D3)—21no—
98C2×C32×D3—22no—
99C2×C32×D3—41no—
100C2×C32×D3—41no—
101C33×C4C2×D411yescharacteristic
102C33×C4C2×D411yescharacteristic
103C3⋊(C3⋊Dic3)—21no—
104C3×C3⋊Dic3—22no—
105C3×C3⋊Dic3—41no—
106C32×Dic3—41no—
107C2×#239—91no—
108C2×#237—94no—
109C2×C3×SD16—34no—
110#946—94no—
111C2×#216—94no—
112C2×#216—91no—
113C2×#204C3⋊D311yescharacteristic, complemented
114#897—91no—
115C3×#103—91no—
116#832—94no—
117(C2×D3)⋊C8—94no—
118(C2×D3)⋊C8—91no—
119(C2×D3)⋊C8—184no—
120#816—34no—
121#809—91no—
122C23×C32—61no—
123C22×C3⋊D3—31no—
124C22×C3⋊D3—34no—
125C22×C3⋊D3—64no—
126C22×C3×D3—24no—
127C32×Q8—61no—
128C32×Q8—61no—
129C32×D4—32no—
130C32×D4—61no—
131C2×C32×C4—24no—
132C2×C32×C4—61no—
133C2×C32×C4C22×D314yes—
134C2×C32×C4C22×D311yescharacteristic
135#638—64no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—31no—
139C2×C3⋊Dic3—34no—
140C2×C3⋊Dic3—64no—
141C2×C3⋊Dic3—64no—
142C4×C3⋊D3—61no—
143C4×C3⋊D3—61no—
144C4×C3⋊D3—64no—
145C4×C3⋊D3—64no—
146C4×C3⋊D3—124no—
147C4×C3⋊D3—124no—
148#634—21no—
149#634D1212yes—
150C2×C3×Dic3—24no—
151C2×C3×Dic3—64no—
152C2×C3×Dic3—64no—
153C3×D12—24no—
154C3×D12#6718yes—
155C3×Dic6—24no—
156C3×Dic6—24no—
157C3×Dic6—38no—
158C3×Dic6—64no—
159#626—616no—
160D3×Dic3—616no—
161C32×C8—21no—
162C32×C8—24no—
163C32×C8—44no—
164C3⋊(C3⋊C8)—61no—
165C3⋊(C3⋊C8)—64no—
166C3⋊(C3⋊C8)—124no—
167#452—271noSylow
168C2×C33#12011yescharacteristic
169C2×C33#13511yescharacteristic
170C2×C33#13711yescharacteristic
171C3⋊(C3⋊D3)—41no—
172C32×D3—42no—
173C2×C3×Q8—34no—
174C2×C3×Q8—91no—
175C2×C3×D4—34no—
176C2×#67—184no—
177C22×Dic3—184no—
178#239—182no—
179#239—182no—
180#237—188no—
181#237—188no—
182C2×D12—181no—
183C2×D12C2×C3⋊D311yescharacteristic
184C2×C4×D3—94no—
185C2×C4×D3—91no—
186C2×C4×D3—184no—
187C2×C4×D3—181no—
188C2×Dic6—34no—
189C2×Dic6C2×C3⋊D311yescharacteristic
190C3×SD16—68no—
191C2×C3×C8—24no—
192C2×C3×C8D3214yes—
193C2×C3×C8C2×C3⋊D311yescharacteristic
194C3×C4⋊C4—91no—
195#216—182no—
196#216—188no—
197(C2×D3)⋊C4—181no—
198(C2×D3)⋊C4—184no—
199C4⋊Dic3—34no—
200C4⋊Dic3—181no—
201Dic3⋊C4—184no—
202C2×C3⋊C8—94no—
203C2×C3⋊C8—91no—
204C2×C3⋊C8—184no—
205#204—22no—
206C22×C32—24no—
207C22×C32—62no—
208C22×C32—121no—
209C22×C32—121no—
210C22×C32D3×D414yes—
211C22×C32D3×D411yescharacteristic
212C2×C3⋊D3—68no—
213C2×C3⋊D3—61no—
214C2×C3⋊D3—64no—
215C2×C3⋊D3—61no—
216C2×C3⋊D3—61no—
217C2×C3⋊D3—64no—
218C2×C3⋊D3—64no—
219C2×C3⋊D3—124no—
220C2×C3⋊D3—124no—
221C2×C3×D3—28no—
222C2×C3×D3—44no—
223C2×C3×D3—44no—
224C32×C4—24no—
225C32×C4—24no—
226C32×C4—121no—
227C32×C4C2×D1211yescharacteristic
228C32×C4D3×D414yes—
229C32×C4D3×D411yescharacteristic
230C32×C4C2×#6714yes—
231C3⋊Dic3—22no—
232C3⋊Dic3—41no—
233C3⋊Dic3—61no—
234C3⋊Dic3—64no—
235C3⋊Dic3—124no—
236C3×Dic3—44no—
237C3×Dic3—68no—
238C3×Dic3—124no—
239C2×#41—271no—
240C2×Q16—271no—
241C2×SD16—31no—
242C22⋊Q8—271no—
243#103—91no—
244#102—271no—
245C22⋊C8—271no—
246C33#45211yescharacteristic, complemented, Sylow
247C23×C3—64no—
248C22×D3—21no—
249C22×D3—94no—
250C22×D3—91no—
251C22×D3—184no—
252C3×Q8—64no—
253C3×Q8—64no—
254C3×Q8—92no—
255C3×Q8—181no—
256C3×D4—38no—
257C3×D4—64no—
258C2×C3×C4—24no—
259C2×C3×C4—64no—
260C2×C3×C4—181no—
261C2×C3×C4—181no—
262C2×C3×C4C2×D3214yes—
263C2×C3×C4C22×C3⋊D311yescharacteristic
264#67—1816no—
265C2×Dic3—21no—
266C2×Dic3—64no—
267C2×Dic3—64no—
268C2×Dic3—94no—
269C2×Dic3—91no—
270C2×Dic3—184no—
271C2×Dic3—1816no—
272D12—21no—
273D12—184no—
274D12#63812yes—
275C4×D3—184no—
276C4×D3—184no—
277C4×D3—184no—
278C4×D3—181no—
279C4×D3—181no—
280C4×D3—364no—
281C4×D3—364no—
282Dic6—21no—
283Dic6—21no—
284Dic6—38no—
285Dic6—64no—
286C3×C8—24no—
287C3×C8—21no—
288C3×C8—44no—
289C3⋊C8—184no—
290C3⋊C8—181no—
291C3⋊C8—364no—
292C2×C32—24no—
293C2×C32—24no—
294C2×C32—24no—
295C2×C32—122no—
296C2×C32#88211yescharacteristic
297C2×C32#91514yes—
298C2×C32#91614yes—
299C2×C32#91611yescharacteristic
300C2×C32#91811yescharacteristic
301C2×C32#93714yes—
302C3⋊D3—128no—
303C3⋊D3—124no—
304C3⋊D3—121no—
305C3×D3—48no—
306#41—542no—
307#41—542no—
308C2×Q8—31no—
309C2×Q8—91no—
310C2×D4—31no—
311C2×D4—541no—
312C22×C4—271no—
313C22×C4—541no—
314Q16—542no—
315SD16—62no—
316C2×C8—271no—
317C2×C8D3×C3⋊D311yescharacteristic
318C4⋊C4—91no—
319C4⋊C4—541no—
320C22⋊C4—541no—
321C22×C3—24no—
322C22×C3—68no—
323C22×C3—124no—
324C22×C3—124no—
325C22×C3#380214yes—
326C22×C3D4×C3⋊D311yescharacteristic
327C2×D3—22no—
328C2×D3—41no—
329C2×D3—41no—
330C2×D3—188no—
331C2×D3—184no—
332C2×D3—184no—
333C2×D3—181no—
334C2×D3—181no—
335C2×D3—181no—
336C2×D3—184no—
337C2×D3—364no—
338C2×D3—364no—
339C3×C4—24no—
340C3×C4—24no—
341C3×C4—124no—
342C3×C4—182no—
343C3×C4—361no—
344C3×C4D3×D1214yes—
345C3×C4D3×#6714yes—
346C3×C4D4×C3⋊D311yescharacteristic
347C3×C4C2×#63811yescharacteristic
348Dic3—41no—
349Dic3—68no—
350Dic3—124no—
351Dic3—181no—
352Dic3—184no—
353Dic3—364no—
354C32—24no—
355C32#480811yescharacteristic, complemented
356C32#517314yescomplemented
357C23—61no—
358C23—271no—
359Q8—61no—
360Q8—61no—
361Q8—92no—
362Q8—181no—
363D4—32no—
364D4—61no—
365D4—544no—
366C2×C4—61no—
367C2×C4—181no—
368C2×C4—181no—
369C2×C4—271no—
370C2×C4—544no—
371C2×C4—541no—
372C2×C4—541no—
373C2×C4C2×D3×C3⋊D311yescharacteristic
374C8—21no—
375C8—541no—
376C2×C3—24no—
377C2×C3—24no—
378C2×C3—24no—
379C2×C3—128no—
380C2×C3#6380914yes—
381C2×C3#6381114yes—
382C2×C3#6392214yes—
383C2×C3#6412411yescharacteristic
384C2×C3#6412511yescharacteristic
385C2×C3#6414611yescharacteristic
386D3—42no—
387D3—368no—
388D3—361no—
389D3—364no—
390C22—62no—
391C22—121no—
392C22—121no—
393C22—541no—
394C22—541no—
395C22—541no—
396C22#8902611yescharacteristic
397C4—121no—
398C4—182no—
399C4—361no—
400C4—541no—
401C4C3⋊D3×D1211yescharacteristic
402C4D3×#63811yescharacteristic
403C3—24no—
404C3#1059096114yescomplemented
405C3#1059287211yescharacteristic, complemented
406C2—122no—
407C2—1081no—
408C2#1172334211yescharacteristic
409C2#1172334411yescharacteristic
410C2#1172354011yescharacteristic
subgroupG/H
centerC22#89026
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89015C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#452
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×C32×#204C2semidirect
3. #11723456C2semidirect
4. C2×#88798C2semidirect
5. #11722190C2semidirect
6. C3×#63643C2semidirect
7. #11722031C2semidirect
30. C2×C3×#204D3semidirect
37. #63643D3semidirect
113. C2×#204C3⋊D3semidirect
246. C33#452semidirect
355. C32#4808semidirect
356. C32#5173semidirect
404. C3#10590961semidirect
405. C3#10592872semidirect

Maximal subgroups

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

subgroupindex
C2×#890152
C2×C32×#2042
#117234562
C2×#887982
#117221902
C3×#636432
#117220312
#105928723
#105909613