Finite Group Databasebeta

← #13159357

Subgroups — #13159357

Order 1728.

948 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 186 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
1C32×#893C211yesmaximal, characteristic, complemented
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C2×#88789C211yesmaximal, characteristic
4#11722177C211yesmaximal, characteristic, complemented
5C3×#63636C211yesmaximal, characteristic, complemented
6#11722032C211yesmaximal, characteristic, complemented
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C4×#3753—31nomaximal
9#10590452—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3—21no—
13C2×Dic3×C3⋊D3—21no—
14C33×C4⋊C4C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×(C2×D3)⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20#88803—21no—
21#88789C414yescomplemented
22C32×C4×D4—31no—
23C2×#3753—31no—
24C4×C4⋊(C3⋊D3)—31no—
25C3×#893D314yescomplemented
26Dic3×D12—34no—
27C2×#3713—34no—
28#63659—31no—
29#63636D311yescharacteristic, complemented
30#63634—31no—
31#63634—34no—
32#63634—64no—
33C4×C3⋊(C3⋊C8)—31no—
34C3×#809—34no—
35#63563—34no—
36Dic3×C3⋊C8—34no—
37C22×C3×C3⋊D3—21no—
38C22×C32×D3—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C3×C4⋊(C3⋊D3)—21no—
43C3×C4⋊(C3⋊D3)C2×C412yescomplemented
44C2×C32×Dic3D411yescharacteristic
45C2×C32×Dic3D411yescharacteristic
46C32×D12—21no—
47C32×D12C2×C412yescomplemented
48C32×C4×D3—42no—
49Dic3×C3⋊D3—42no—
50C3⋊(C3⋊(C3⋊C8))C2×C412yescomplemented
51C3×C3⋊(C3⋊C8)—21no—
52C4×#213—94no—
53#4756—91no—
54C2×C32×D4—31no—
55C22×C32×C4—61no—
56C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
57C2×C4×C3⋊D3—61no—
58C2×C3×D12—34no—
59C2×C3×D12C2×D314yes—
60C2×C3×C4×D3—24no—
61C2×D3×Dic3—64no—
62C32×C4⋊C4—24no—
63C32×C4⋊C4C2×D314yes—
64C32×C4⋊C4C2×D311yescharacteristic
65C32×C22⋊C4—61no—
66C32×C42—31no—
67#3753—34no—
68#3752—61no—
69C4⋊(C3⋊Dic3)—31no—
70C4⋊(C3⋊Dic3)—34no—
71C4⋊(C3⋊Dic3)—64no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
76C3×(C2×D3)⋊C4—24no—
77C3×C4×Dic3C2×D314yes—
78C2×C3×C3⋊C8—34no—
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×C4—41no—
89C33×C4C22×C411yescharacteristic
90C33×C4#4111yescharacteristic
91C3⋊(C3⋊Dic3)—41no—
92C32×Dic3—21no—
93C32×Dic3D812yescomplemented
94C3×C4×D4—34no—
95D4×Dic3—91no—
96C2×#213—94no—
97C2×#213—91no—
98#893C3⋊D311yescharacteristic, complemented
99C4×D12—94no—
100C3×#102—91no—
101#832—94no—
102#821—91no—
103C8×Dic3—91no—
104#809—34no—
105#807—94no—
106#807—91no—
107#807—184no—
108C4×C3⋊C8—94no—
109C23×C32—61no—
110C22×C3⋊D3—21no—
111C22×C3×D3—24no—
112C22×C3×D3—64no—
113C32×D4—32no—
114C32×D4—61no—
115C2×C32×C4—21no—
116C2×C32×C4—24no—
117C2×C32×C4—24no—
118C2×C32×C4—31no—
119C2×C32×C4—31no—
120C2×C32×C4—44no—
121C2×C32×C4—122no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124C2×C3⋊Dic3—61no—
125C2×C3⋊Dic3—64no—
126C2×C3⋊Dic3—124no—
127C4⋊(C3⋊D3)—21no—
128C4⋊(C3⋊D3)C4×D312yescomplemented
129C4×C3⋊D3—122no—
130C2×C3×Dic3#6714yes—
131C2×C3×Dic3#6714yes—
132C3×D12—24no—
133C3×D12—38no—
134C3×D12—64no—
135C3×D12C4×D318yescomplemented
136C3×C4×D3—48no—
137D3×Dic3—128no—
138C3⋊(C3⋊C8)—21no—
139C3⋊(C3⋊C8)—32no—
140C3⋊(C3⋊C8)—38no—
141C3⋊(C3⋊C8)—68no—
142C3×C3⋊C8—64no—
143C4×D8—271noSylow
144C2×C33C4×D411yescharacteristic
145C2×C33C2×D811yescharacteristic
146C2×C33#13511yescharacteristic
147C3×C3⋊D3—42no—
148C32×D3—42no—
149C2×C3×D4—34no—
150C2×C3×D4—91no—
151C22×C3×C4—64no—
152C22×Dic3—181no—
153C2×D12—34no—
154C2×D12C2×C3⋊D311yescharacteristic
155C2×C4×D3—21no—
156C2×C4×D3—184no—
157C2×C3×C8—91no—
158C3×C4⋊C4—24no—
159C3×C4⋊C4D3214yes—
160C3×C4⋊C4C2×C3⋊D311yescharacteristic
161C3×C22⋊C4—64no—
162C3×C42—34no—
163#217—181no—
164#213—916no—
165#213—94no—
166(C2×D3)⋊C4—21no—
167(C2×D3)⋊C4—184no—
168C4⋊Dic3—94no—
169C4⋊Dic3—91no—
170C4⋊Dic3—184no—
171C4×Dic3C2×C3⋊D311yescharacteristic
172C2×C3⋊C8—34no—
173C2×C3⋊C8—94no—
174C2×C3⋊C8—91no—
175C2×C3⋊C8—184no—
176C22×C32—24no—
177C22×C32—62no—
178C22×C32—121no—
179C22×C32—121no—
180C22×C32D3×D411yescharacteristic
181C22×C32C2×#6714yes—
182C2×C3⋊D3—22no—
183C2×C3⋊D3—41no—
184C2×C3⋊D3—41no—
185C2×C3×D3—28no—
186C2×C3×D3—44no—
187C2×C3×D3—44no—
188C2×C3×D3—68no—
189C2×C3×D3—124no—
190C2×C3×D3—124no—
191C32×C4—24no—
192C32×C4—24no—
193C32×C4—32no—
194C32×C4—41no—
195C32×C4—44no—
196C32×C4—61no—
197C32×C4—84no—
198C32×C4C2×C4×D314yes—
199C32×C4C2×C4×D311yescharacteristic
200C32×C4#23514yes—
201C32×C4#23711yescharacteristic
202C3⋊Dic3—121no—
203C3⋊Dic3—124no—
204C3⋊Dic3—244no—
205C3×Dic3—24no—
206C3×Dic3#21318yescomplemented
207C2×D8—271no—
208C4×D4—31no—
209C4×D4—271no—
210#107—271no—
211#102—91no—
212#102—271no—
213C4×C8—271no—
214C33C4×D811yescharacteristic, complemented, Sylow
215C23×C3—64no—
216C23×C3—181no—
217C22×D3—21no—
218C22×D3—64no—
219C3×D4—38no—
220C3×D4—64no—
221C3×D4—92no—
222C3×D4—181no—
223C2×C3×C4—24no—
224C2×C3×C4—21no—
225C2×C3×C4—24no—
226C2×C3×C4—34no—
227C2×C3×C4—34no—
228C2×C3×C4—44no—
229C2×C3×C4—128no—
230C2×C3×C4C2×D3214yes—
231C2×C3×C4C22×C3⋊D311yescharacteristic
232C2×Dic3—184no—
233C2×Dic3—181no—
234C2×Dic3—364no—
235C2×Dic3—362no—
236C2×Dic3#63811yescharacteristic
237C2×Dic3#63811yescharacteristic
238D12—21no—
239D12—38no—
240D12—64no—
241D12C4×C3⋊D312yescomplemented
242C4×D3—42no—
243C4×D3—368no—
244C3×C8—181no—
245C3⋊C8—64no—
246C3⋊C8—98no—
247C3⋊C8—92no—
248C3⋊C8—188no—
249C2×C32—24no—
250C2×C32—24no—
251C2×C32—24no—
252C2×C32—122no—
253C2×C32#67⋊C411yescharacteristic
254C2×C32D3×D811yescharacteristic
255C2×C32#91611yescharacteristic
256C2×C32C4×#6714yes—
257C2×C32C2×#21314yes—
258C2×C32#95014yes—
259C3⋊D3—42no—
260C3×D3—48no—
261C3×D3—128no—
262C2×D4—31no—
263C2×D4—91no—
264C22×C4—61no—
265C22×C4—541no—
266D8—274no—
267C2×C8—91no—
268C2×C8—271no—
269C4⋊C4—271no—
270C4⋊C4D3×C3⋊D311yescharacteristic
271C22⋊C4—61no—
272C22⋊C4—541no—
273C42—31no—
274C22×C3—24no—
275C22×C3—68no—
276C22×C3—124no—
277C22×C3—124no—
278C22×C3—182no—
279C22×C3—361no—
280C22×C3—361no—
281C22×C3D3×#6714yes—
282C22×C3C2×#63811yescharacteristic
283C2×D3—22no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—68no—
287C2×D3—124no—
288C2×D3—124no—
289C3×C4—24no—
290C3×C4—24no—
291C3×C4—38no—
292C3×C4—41no—
293C3×C4—44no—
294C3×C4—64no—
295C3×C4—84no—
296C3×C4#379514yes—
297C3×C4C4×D3214yes—
298C3×C4C2×C4×C3⋊D311yescharacteristic
299C3×C4#382811yescharacteristic
300Dic3—21no—
301Dic3—361no—
302Dic3—364no—
303Dic3—724no—
304Dic3#375312yescomplemented
305C32—24no—
306C32#475611yescharacteristic, complemented
307C32C4×#21314yescomplemented
308C23—61no—
309C23—181no—
310D4—32no—
311D4—61no—
312D4—92no—
313D4—181no—
314C2×C4—21no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—122no—
318C2×C4—541no—
319C2×C4—1082no—
320C2×C4C2×D3×C3⋊D311yescharacteristic
321C8—181no—
322C8—272no—
323C2×C3—24no—
324C2×C3—24no—
325C2×C3—24no—
326C2×C3—128no—
327C2×C3—362no—
328C2×C3#6387914yes—
329C2×C3D3×#21314yes—
330C2×C3#6394714yes—
331C2×C3C4×#63811yescharacteristic
332C2×C3C2×#375311yescharacteristic
333C2×C3#6415911yescharacteristic
334D3—42no—
335D3—128no—
336C22—62no—
337C22—121no—
338C22—121no—
339C22—182no—
340C22—361no—
341C22—361no—
342C22D3×#63811yescharacteristic
343C4—32no—
344C4—41no—
345C4—61no—
346C4—1081no—
347C4#8902111yescharacteristic
348C4C4×D3×C3⋊D311yescharacteristic
349C3—24no—
350C3#1059045214yescomplemented
351C3C4×#375311yescharacteristic, complemented
352C2—122no—
353C2—362no—
354C2#1172350611yescharacteristic
355C2D3×#375311yescharacteristic
356C2#1172358011yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#88789C2
cosocleC2×C33×C4C23
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33C4×D8
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C32×#893C2semidirect
2. Dic3×C4⋊(C3⋊D3)C2semidirect
4. #11722177C2semidirect
5. C3×#63636C2semidirect
6. #11722032C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
21. #88789C4semidirect
25. C3×#893D3semidirect
29. #63636D3semidirect
43. C3×C4⋊(C3⋊D3)C2×C4semidirect
47. C32×D12C2×C4semidirect
50. C3⋊(C3⋊(C3⋊C8))C2×C4semidirect
93. C32×Dic3D8semidirect
98. #893C3⋊D3semidirect
128. C4⋊(C3⋊D3)C4×D3semidirect
135. C3×D12C4×D3semidirect
206. C3×Dic3#213semidirect
214. C33C4×D8semidirect
241. D12C4×C3⋊D3semidirect
304. Dic3#3753semidirect
306. C32#4756semidirect
307. C32C4×#213semidirect
350. C3#10590452semidirect
351. C3C4×#3753semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
Dic3×C4⋊(C3⋊D3)2
C2×#887892
#117221772
C3×#636362
#117220322
Dic3×C3⋊(C3⋊C8)2
C4×#37533
#105904523