Finite Group Databasebeta

← #13159928

Subgroups — #13159928

Order 1728.

952 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 164 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
1#11723501C211yesmaximal, characteristic, complemented
2#11723499C211yesmaximal, characteristic, complemented
3C2×#88789C211yesmaximal, characteristic
4#11722185C211yesmaximal, characteristic, complemented
5C3×#63648C211yesmaximal, characteristic, complemented
6C32×#821C211yesmaximal, characteristic, complemented
7#11722053C211yesmaximal, characteristic, complemented
8#10592484—31nomaximal
9#10590741—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C2×D3×C3⋊Dic3—21no—
14C2×C33×C8C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88803—21no—
20(C2×D3)⋊(C3⋊Dic3)—21no—
21#88789C414yescomplemented
22D4×C3⋊Dic3—31no—
23C2×#3753—31no—
24#64102—31no—
25#63898—34no—
26#63898—34no—
27C2×#3713—34no—
28C32×#102—31no—
29#63648D311yescharacteristic, complemented
30#63642—31no—
31#63642—34no—
32#63642—64no—
33#63635—31no—
34C3×#821—34no—
35C3×#821D314yescomplemented
36#63574—34no—
37C22×C3×C3⋊D3—21no—
38C22×C32×D3—21no—
39C2×C33×C4C2311yescharacteristic
40C2×C3⋊(C3⋊Dic3)D411yescharacteristic
41C2×C3⋊(C3⋊Dic3)D411yescharacteristic
42C2×C3×C3⋊Dic3—21no—
43C3×C4⋊(C3⋊D3)—21no—
44C3×C4⋊(C3⋊D3)C2×C412yescomplemented
45C2×C32×Dic3—21no—
46C32×D12—21no—
47C32×D12C2×C412yescomplemented
48Dic3×C3⋊D3—42no—
49D3×C3⋊Dic3—42no—
50C33×C8—21no—
51C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
52#4785—94no—
53#4785—91no—
54C2×C32×D4—31no—
55C22×C3⋊Dic3—61no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C4×C3⋊D3—61no—
58C2×C3×D12—34no—
59C2×C3×D12C2×D314yes—
60C2×D3×Dic3—64no—
61C2×D3×Dic3—64no—
62C2×C32×C8—24no—
63C2×C32×C8C2×D314yes—
64C2×C32×C8C2×D311yescharacteristic
65C32×C4⋊C4—31no—
66#3757—61no—
67#3753—34no—
68#3752—61no—
69C4⋊(C3⋊Dic3)C2×D311yescharacteristic
70C4×C3⋊Dic3—31no—
71C4×C3⋊Dic3—34no—
72C4×C3⋊Dic3—64no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76C3×C4⋊Dic3—34no—
77C3×C4⋊Dic3C2×D314yes—
78(C2×D3)⋊Dic3—64no—
79(C2×D3)⋊Dic3—64no—
80#3713—316no—
81C22×C33C2×D411yescharacteristic
82C2×C3×C3⋊D3—22no—
83C2×C3×C3⋊D3—41no—
84C2×C3×C3⋊D3—41no—
85C2×C32×D3—22no—
86C2×C32×D3—41no—
87C2×C32×D3—41no—
88C33×C4C22×C411yescharacteristic
89C33×C4#4111yescharacteristic
90C3⋊(C3⋊Dic3)—21no—
91C3⋊(C3⋊Dic3)—21no—
92C3×C3⋊Dic3—41no—
93C32×Dic3—41no—
94D4×Dic3—94no—
95D4×Dic3—91no—
96C2×#213—94no—
97C2×#213—91no—
98#893—94no—
99#893—91no—
100C3×#102—34no—
101C3×#102—91no—
102#821—34no—
103#821C3⋊D311yescharacteristic, complemented
104#815—94no—
105#815—91no—
106#815—184no—
107#808—94no—
108#808—91no—
109C23×C32—61no—
110C22×C3⋊D3—21no—
111C22×C3×D3—24no—
112C22×C3×D3—64no—
113C32×D4—32no—
114C32×D4—61no—
115C2×C32×C4—24no—
116C2×C32×C4—61no—
117C2×C32×C4C22×D314yes—
118C2×C32×C4C22×D311yescharacteristic
119C2×C3⋊Dic3—21no—
120C2×C3⋊Dic3—31no—
121C2×C3⋊Dic3—31no—
122C2×C3⋊Dic3—34no—
123C2×C3⋊Dic3—34no—
124C2×C3⋊Dic3—64no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—122no—
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)C4×D312yescomplemented
129C4×C3⋊D3—122no—
130C2×C3×Dic3—24no—
131C2×C3×Dic3—64no—
132C3×D12—24no—
133C3×D12—38no—
134C3×D12—64no—
135C3×D12C4×D318yescomplemented
136D3×Dic3—128no—
137D3×Dic3—128no—
138C32×C8—28no—
139C32×C8—24no—
140C32×C8—21no—
141C3⋊(C3⋊C8)—32no—
142C3⋊(C3⋊C8)—38no—
143C3⋊(C3⋊C8)—68no—
144#442—271noSylow
145C2×C33C4×D411yescharacteristic
146C2×C33AGL(1,ℤ/8)11yescharacteristic
147C2×C33AGL(1,ℤ/8)11yescharacteristic
148C3×C3⋊D3—42no—
149C32×D3—42no—
150C2×C3×D4—34no—
151C2×C3×D4—91no—
152C22×Dic3—181no—
153C22×Dic3—184no—
154C2×D12—34no—
155C2×D12C2×C3⋊D311yescharacteristic
156C2×C4×D3—184no—
157C2×C4×D3—181no—
158C2×C3×C8—24no—
159C2×C3×C8D3214yes—
160C2×C3×C8C2×C3⋊D311yescharacteristic
161C3×C4⋊C4—34no—
162C3×C4⋊C4—91no—
163#217—181no—
164#217—184no—
165#213—916no—
166#213—94no—
167(C2×D3)⋊C4—184no—
168(C2×D3)⋊C4—181no—
169C4⋊Dic3—34no—
170C4⋊Dic3C2×C3⋊D311yescharacteristic
171C4×Dic3—94no—
172C4×Dic3—91no—
173C4×Dic3—184no—
174C2×C3⋊C8—94no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—184no—
177C22×C32—24no—
178C22×C32—62no—
179C22×C32—121no—
180C22×C32—121no—
181C22×C32D3×D414yes—
182C22×C32D3×D411yescharacteristic
183C2×C3⋊D3—22no—
184C2×C3⋊D3—41no—
185C2×C3⋊D3—41no—
186C2×C3×D3—28no—
187C2×C3×D3—44no—
188C2×C3×D3—44no—
189C2×C3×D3—68no—
190C2×C3×D3—124no—
191C2×C3×D3—124no—
192C32×C4—24no—
193C32×C4—24no—
194C32×C4—121no—
195C32×C4C2×C4×D314yes—
196C32×C4C2×C4×D311yescharacteristic
197C32×C4#23714yes—
198C32×C4#23711yescharacteristic
199C3⋊Dic3—41no—
200C3⋊Dic3—61no—
201C3⋊Dic3—61no—
202C3⋊Dic3—64no—
203C3⋊Dic3—64no—
204C3⋊Dic3—124no—
205C3⋊Dic3—124no—
206C3×Dic3—44no—
207C3×Dic3—124no—
208C2×D8—271no—
209C4×D4—271no—
210C4×D4—271no—
211#106—271no—
212#102—31no—
213#102—91no—
214#97—271no—
215C33#44211yescharacteristic, complemented, Sylow
216C23×C3—64no—
217C23×C3—181no—
218C22×D3—21no—
219C22×D3—64no—
220C3×D4—38no—
221C3×D4—64no—
222C3×D4—92no—
223C3×D4—181no—
224C2×C3×C4—24no—
225C2×C3×C4—64no—
226C2×C3×C4—181no—
227C2×C3×C4C2×D3214yes—
228C2×C3×C4C22×C3⋊D311yescharacteristic
229C2×Dic3—21no—
230C2×Dic3—64no—
231C2×Dic3—94no—
232C2×Dic3—94no—
233C2×Dic3—91no—
234C2×Dic3—91no—
235C2×Dic3—184no—
236C2×Dic3—184no—
237C2×Dic3—362no—
238C2×Dic3—368no—
239D12—21no—
240D12—38no—
241D12—64no—
242D12C4×C3⋊D312yescomplemented
243C4×D3—368no—
244C4×D3—362no—
245C3×C8—28no—
246C3×C8—24no—
247C3×C8—21no—
248C3⋊C8—98no—
249C3⋊C8—92no—
250C3⋊C8—188no—
251C2×C32—24no—
252C2×C32—24no—
253C2×C32—24no—
254C2×C32—122no—
255C2×C32#67⋊C414yes—
256C2×C32#67⋊C411yescharacteristic
257C2×C32#91114yes—
258C2×C32#91111yescharacteristic
259C2×C32#91414yes—
260C2×C32#91411yescharacteristic
261C3⋊D3—42no—
262C3×D3—48no—
263C3×D3—128no—
264C2×D4—31no—
265C2×D4—91no—
266C22×C4—541no—
267C22×C4—541no—
268D8—274no—
269C2×C8—271no—
270C2×C8D3×C3⋊D311yescharacteristic
271C4⋊C4—31no—
272C4⋊C4—91no—
273C22⋊C4—541no—
274C22⋊C4—541no—
275C42—271no—
276C22×C3—24no—
277C22×C3—68no—
278C22×C3—124no—
279C22×C3—124no—
280C22×C3—182no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3#380214yes—
284C22×C3D4×C3⋊D311yescharacteristic
285C2×D3—22no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—68no—
289C2×D3—124no—
290C2×D3—124no—
291C3×C4—24no—
292C3×C4—24no—
293C3×C4—124no—
294C3×C4—361no—
295C3×C4C4×D3214yes—
296C3×C4#380514yes—
297C3×C4C2×C4×C3⋊D311yescharacteristic
298C3×C4#383011yescharacteristic
299Dic3—41no—
300Dic3—124no—
301Dic3—184no—
302Dic3—184no—
303Dic3—181no—
304Dic3—181no—
305Dic3—364no—
306Dic3—364no—
307C32—24no—
308C32#478514yescomplemented
309C32#478511yescharacteristic, complemented
310C23—61no—
311C23—181no—
312D4—32no—
313D4—61no—
314D4—92no—
315D4—181no—
316C2×C4—61no—
317C2×C4—181no—
318C2×C4—271no—
319C2×C4—271no—
320C2×C4—1082no—
321C2×C4—1082no—
322C2×C4C2×D3×C3⋊D311yescharacteristic
323C8—21no—
324C8—272no—
325C2×C3—24no—
326C2×C3—24no—
327C2×C3—24no—
328C2×C3—128no—
329C2×C3—362no—
330C2×C3#6379814yes—
331C2×C3#6379814yes—
332C2×C3#6390214yes—
333C2×C3#6409011yescharacteristic
334C2×C3#6412011yescharacteristic
335C2×C3#6412311yescharacteristic
336D3—42no—
337D3—128no—
338C22—62no—
339C22—121no—
340C22—121no—
341C22—182no—
342C22—361no—
343C22—361no—
344C22#8902611yescharacteristic
345C4—121no—
346C4—361no—
347C4—541no—
348C4—541no—
349C4C4×D3×C3⋊D311yescharacteristic
350C4#8902911yescharacteristic
351C3—24no—
352C3#1059074114yescomplemented
353C3#1059248411yescharacteristic, complemented
354C2—122no—
355C2—362no—
356C2#1172332311yescharacteristic
357C2#1172332411yescharacteristic
358C2#1172350811yescharacteristic
subgroupG/H
centerC22#89026
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#88789C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#442
LCSG ▷ 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
1. #11723501C2semidirect
2. #11723499C2semidirect
4. #11722185C2semidirect
5. C3×#63648C2semidirect
6. C32×#821C2semidirect
7. #11722053C2semidirect
21. #88789C4semidirect
29. #63648D3semidirect
35. C3×#821D3semidirect
44. C3×C4⋊(C3⋊D3)C2×C4semidirect
47. C32×D12C2×C4semidirect
51. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
103. #821C3⋊D3semidirect
128. C4⋊(C3⋊D3)C4×D3semidirect
135. C3×D12C4×D3semidirect
215. C33#442semidirect
242. D12C4×C3⋊D3semidirect
308. C32#4785semidirect
309. C32#4785semidirect
352. C3#10590741semidirect
353. C3#10592484semidirect

Maximal subgroups

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

subgroupindex
#117235012
#117234992
C2×#887892
#117221852
C3×#636482
C32×#8212
#117220532
#105924843
#105907413