Finite Group Databasebeta

← #13180961

Subgroups — #13180961

Order 1728.

920 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 170 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×#89014C211yesmaximal, characteristic
2#11724511C211yesmaximal, characteristic, complemented
3C3×#64283C211yesmaximal, characteristic, complemented
4D3×#3792C214yesmaximal, complemented
5#11724251C214yesmaximal, complemented
6#11724250C214yesmaximal, complemented
7#10596640—31nomaximal
8C2×#6308C411yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×D12C411yescharacteristic
11C2×#6290C412yes—
12C2×D3×C3⋊Dic3C2212yes—
13#89014C412yes—
14#89014C412yes—
15#89014C2212yes—
16#89014C2212yes—
17#88993C2212yes—
18#88989C2212yescomplemented
19#88988C2212yescomplemented
20C3×#3792C2212yes—
21C3×#3788C2212yescomplemented
22C3×#3787C2212yescomplemented
23#88925C2218yescomplemented
24D3×#622C2218yescomplemented
25C2×#3830—31no—
26C2×#3795—62no—
27C2×#3792—34no—
28#64287—34no—
29#64287—34no—
30#64283—31no—
31#64283D311yescharacteristic, complemented
32C22×C3×C3⋊D3C2×C411yescharacteristic
33C22×C32×D3C2×C412yes—
34C2×C33×C4C2×C411yescharacteristic
35C2×C3⋊(C3⋊Dic3)C2×C412yes—
36#6308C2×C412yes—
37#6308C2×C412yes—
38C2×C3×C3⋊Dic3C2311yescharacteristic
39C3×C4×C3⋊D3C2×C411yescharacteristic
40C3×C4×C3⋊D3C2×C411yescharacteristic
41C3×C4×C3⋊D3C2311yescharacteristic
42C3×C4×C3⋊D3C2311yescharacteristic
43C32×D12C2×C412yes—
44C32×D12C2×C412yes—
45#6290C2×C414yes—
46#6290C2×C414yes—
47D3×C3⋊Dic3C2×C414yes—
48D3×C3⋊Dic3C2314yes—
49C3⋊#622C2314yes—
50C3×#622C2314yes—
51#5745—91nomaximal
52C2×C32×D4—31no—
53C2×#638—32no—
54C22×C3⋊Dic3—32no—
55#3830—32no—
56#3830—32no—
57#3830—32no—
58#3830—32no—
59C2×C4×C3⋊D3C2×D311yescharacteristic
60C2×#634—31no—
61C2×#634—62no—
62C2×#634—122no—
63C2×C3×D12—22no—
64C2×C3×C4×D3—62no—
65C2×#625—64no—
66C2×D3×Dic3—64no—
67#3795—64no—
68#3795—64no—
69#3795—64no—
70#3795—64no—
71#3792—32no—
72#3792—38no—
73#3792C2×D312yes—
74C2×#622—38no—
75#3788—32no—
76#3788C2×D312yescomplemented
77#3787—32no—
78#3787C2×D312yescomplemented
79C22×C33C22×C411yescharacteristic
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3C22×C411yescharacteristic
82C2×C3×C3⋊D3C22×C411yescharacteristic
83C2×C32×D3—42no—
84C2×C32×D3C22×C414yes—
85C33×C4C22×C411yescharacteristic
86C33×C4C22×C411yescharacteristic
87C3⋊(C3⋊Dic3)C22×C414yes—
88C3×C3⋊Dic3C22×C411yescharacteristic
89C3×C3⋊Dic3C2411yescharacteristic
90C2×#237—182no—
91C2×#235—91no—
92C2×C3×M16—91no—
93C2×#208—91no—
94#907—94no—
95D3×M16—94no—
96#901—94no—
97C23×C32—32no—
98C22×C3⋊D3C4×D311yescharacteristic
99C22×C3×D3—24no—
100C22×C3×D3—62no—
101C32×D4—32no—
102C32×D4—32no—
103C2×C32×C4—22no—
104C2×C32×C4—42no—
105C2×C32×C4C4×D311yescharacteristic
106#638—34no—
107#638—34no—
108C2×C3⋊Dic3—32no—
109C2×C3⋊Dic3—34no—
110C2×C3⋊Dic3—34no—
111C2×C3⋊Dic3—64no—
112C2×C3⋊Dic3—124no—
113C2×C3⋊Dic3C22×D311yescharacteristic
114C4×C3⋊D3C4×D311yescharacteristic
115C4×C3⋊D3C4×D311yescharacteristic
116C4×C3⋊D3C22×D311yescharacteristic
117C4×C3⋊D3C22×D311yescharacteristic
118#634—32no—
119#634—32no—
120#634—64no—
121#634—64no—
122#634—124no—
123#634—124no—
124C2×C3×Dic3—62no—
125C3×D12—24no—
126C3×D12—24no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129C3×C4×D3—62no—
130C3×C4×D3—62no—
131#625—68no—
132#625—68no—
133D3×Dic3—68no—
134D3×Dic3—68no—
135#622—34no—
136#622C22×D314yes—
137#568—271noSylow
138C2×C33—21no—
139C2×C33C23×C411yescharacteristic
140C3×C3⋊D3—21no—
141C3×C3⋊D3—21no—
142C32×D3—24no—
143C2×C3×D4—62no—
144C22×C3×C4—91no—
145C2×#67—92no—
146C2×#67—184no—
147C22×Dic3—184no—
148#237—184no—
149#237—184no—
150#237—184no—
151#237—184no—
152#235—92no—
153#235—92no—
154#235—92no—
155#235—92no—
156C2×D12#15711yescharacteristic
157C2×C4×D3—62no—
158C2×C4×D3—92no—
159C2×Dic6—91no—
160C2×Dic6—182no—
161C2×Dic6—362no—
162C3×M16—92no—
163C3×M16—92no—
164C2×C3×C8—92no—
165#208—92no—
166#208—92no—
167C2×C3⋊C8—92no—
168ΓL(1,𝔽25)—98no—
169D3×C8—98no—
170C22×C32—22no—
171C22×C32—34no—
172C22×C32—42no—
173C22×C32—122no—
174C22×C32C2×C4×D311yescharacteristic
175C2×C3⋊D3—41no—
176C2×C3⋊D3C2×C4×D311yescharacteristic
177C2×C3⋊D3C2×C4×D311yescharacteristic
178C2×C3×D3—28no—
179C2×C3×D3—48no—
180C2×C3×D3—62no—
181C2×C3×D3—62no—
182C2×C3×D3—124no—
183C32×C4—22no—
184C32×C4—22no—
185C32×C4—42no—
186C32×C4—42no—
187C32×C4C2×C4×D311yescharacteristic
188C32×C4C2×C4×D311yescharacteristic
189C3⋊Dic3—34no—
190C3⋊Dic3—68no—
191C3⋊Dic3—128no—
192C3⋊Dic3C2×C4×D311yescharacteristic
193C3⋊Dic3C23×D311yescharacteristic
194C3×Dic3—62no—
195C3×Dic3—62no—
196C2×#41—271no—
197#131—274no—
198#131—274no—
199C2×M16—91no—
200C2×M16—274no—
201C2×M16—271no—
202C33#56811yescharacteristic, complemented, Sylow
203C23×C3—64no—
204C23×C3—91no—
205C22×D3—62no—
206C22×D3C2×#15712yes—
207C3×D4—64no—
208C3×D4—64no—
209C2×C3×C4—22no—
210C2×C3×C4—42no—
211C2×C3×C4—91no—
212C2×C3×C4—91no—
213C2×C3×C4—91no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4C2×#15711yescharacteristic
217#67—94no—
218#67—94no—
219#67—188no—
220#67—188no—
221C2×Dic3—62no—
222C2×Dic3—92no—
223C2×Dic3—188no—
224C2×Dic3—188no—
225C2×Dic3—184no—
226C2×Dic3—364no—
227D12C2×#15712yes—
228D12C2×#15712yes—
229C4×D3—62no—
230C4×D3—62no—
231C4×D3—62no—
232C4×D3—62no—
233C4×D3—94no—
234C4×D3—94no—
235Dic6—92no—
236Dic6—92no—
237Dic6—184no—
238Dic6—184no—
239Dic6—364no—
240Dic6—364no—
241C3×C8—94no—
242C3⋊C8—94no—
243C2×C32—21no—
244C2×C32—24no—
245C2×C32—22no—
246C2×C32—44no—
247C2×C32—42no—
248C2×C32—64no—
249C2×C32C22×C4×D311yescharacteristic
250C3⋊D3—21no—
251C3⋊D3—21no—
252C3×D3—48no—
253C3×D3—122no—
254C3×D3—122no—
255#41—272no—
256#41—272no—
257#41—272no—
258#41—272no—
259C2×Q8—271no—
260C2×D4—31no—
261C2×D4—272no—
262C22×C4—91no—
263C22×C4—272no—
264M16—92no—
265M16—92no—
266M16—278no—
267M16—272no—
268M16—272no—
269C2×C8—92no—
270C2×C8—278no—
271C2×C8—272no—
272C22×C3—22no—
273C22×C3—42no—
274C22×C3—68no—
275C22×C3—91no—
276C22×C3—91no—
277C22×C3—128no—
278C22×C3—361no—
279C22×C3C22×#15711yescharacteristic
280C2×D3—42no—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—124no—
284C2×D3C22×#15714yes—
285C3×C4—22no—
286C3×C4—22no—
287C3×C4—42no—
288C3×C4—42no—
289C3×C4—91no—
290C3×C4—91no—
291C3×C4C22×#15711yescharacteristic
292C3×C4C22×#15711yescharacteristic
293Dic3—62no—
294Dic3—62no—
295Dic3—94no—
296Dic3—188no—
297Dic3—368no—
298C32—22no—
299C32—42no—
300C32#574511yescharacteristic, complemented
301C23—32no—
302C23—91no—
303Q8—272no—
304Q8—272no—
305D4—32no—
306D4—32no—
307D4—274no—
308D4—274no—
309C2×C4—91no—
310C2×C4—91no—
311C2×C4—91no—
312C2×C4—91no—
313C2×C4—91no—
314C2×C4—274no—
315C2×C4—274no—
316C2×C4—272no—
317C2×C4D3×#15711yescharacteristic
318C8—94no—
319C8—274no—
320C2×C3—24no—
321C2×C3—21no—
322C2×C3—22no—
323C2×C3—44no—
324C2×C3—42no—
325C2×C3—128no—
326C2×C3—181no—
327C2×C3—181no—
328C2×C3C23×#15711yescharacteristic
329D3—24no—
330D3—122no—
331D3—122no—
332C22—34no—
333C22—91no—
334C22—91no—
335C22—122no—
336C22—361no—
337C22C2×D3×#15711yescharacteristic
338C4—91no—
339C4—91no—
340C4—274no—
341C4C2×D3×#15711yescharacteristic
342C4C2×D3×#15711yescharacteristic
343C3—22no—
344C3—42no—
345C3#1059664011yescharacteristic, complemented
346C2—21no—
347C2—64no—
348C2—181no—
349C2—181no—
350C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC2×C33×C4C2×C4
prime radicalC2×#89014C2
cosocleC3×C3⋊Dic3C24
hypercenterC2×C4D3×#157
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724511C2semidirect
3. C3×#64283C2semidirect
4. D3×#3792C2semidirect
5. #11724251C2semidirect
6. #11724250C2semidirect
18. #88989C22semidirect
19. #88988C22semidirect
21. C3×#3788C22semidirect
22. C3×#3787C22semidirect
23. #88925C22semidirect
24. D3×#622C22semidirect
31. #64283D3semidirect
76. #3788C2×D3semidirect
78. #3787C2×D3semidirect
202. C33#568semidirect
300. C32#5745semidirect
345. C3#10596640semidirect

Maximal subgroups

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

subgroupindex
C2×#890142
#117245112
C3×#642832
D3×#37922
#117242512
#117242502
#105966403
#57459