Finite Group Databasebeta

← #13159363

Subgroups — #13159363

Order 1728.

1004 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1004 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
2#11723484C211yesmaximal, characteristic, complemented
3C2×#88790C211yesmaximal, characteristic
4#11722179C211yesmaximal, characteristic, complemented
5C3×#63634C211yesmaximal, characteristic, complemented
6#11722031C211yesmaximal, characteristic, complemented
7Dic3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8C4×#3753—31nomaximal
9#10590456—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C32×D12C2211yescharacteristic
12C2×C32×C4×D3—21no—
13C2×#6289—21no—
14C33×C4⋊C4C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×(C2×D3)⋊C4—21no—
19C32×C4×Dic3C2211yescharacteristic
20#88804—21no—
21#88790C414yescomplemented
22C32×C4×D4—31no—
23C2×#3753—31no—
24C4×C4⋊(C3⋊D3)—31no—
25C3×#893D314yescomplemented
26#63894—34no—
27C2×#3714—34no—
28#63659—31no—
29#63636—31no—
30#63636—34no—
31#63636—64no—
32#63634D311yescharacteristic, complemented
33C4×C3⋊(C3⋊C8)—31no—
34C3×#807—34no—
35#63562—34no—
36Dic3×C3⋊C8—34no—
37C22×C3⋊(C3⋊D3)—21no—
38C22×C32×D3—21no—
39C2×C33×C4—21no—
40C2×C33×C4C2311yescharacteristic
41C4⋊(C3⋊(C3⋊D3))—21no—
42C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
43C2×C3×C3⋊Dic3—21no—
44C2×C32×Dic3D411yescharacteristic
45C2×C32×Dic3D411yescharacteristic
46C32×D12—21no—
47C32×D12C2×C412yescomplemented
48C32×C4×D3—42no—
49#6289—42no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51C3×C3⋊(C3⋊C8)C2×C412yescomplemented
52C4×#213—94no—
53#4872—91no—
54C2×C32×D4—31no—
55C22×C32×C4—61no—
56C2×C4⋊(C3⋊D3)—31no—
57C2×C4⋊(C3⋊D3)—34no—
58C2×C4⋊(C3⋊D3)—64no—
59C2×C4×C3⋊D3—61no—
60C2×C3×D12C2×D314yes—
61C2×C3×C4×D3—24no—
62C2×#624—64no—
63C32×C4⋊C4—24no—
64C32×C4⋊C4C2×D314yes—
65C32×C4⋊C4C2×D311yescharacteristic
66C32×C22⋊C4—61no—
67C32×C42—31no—
68#3753—34no—
69#3752—61no—
70C4⋊(C3⋊Dic3)C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—34no—
73C2×C3⋊(C3⋊C8)—64no—
74C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
75C3×(C2×D3)⋊C4—24no—
76C3×C4⋊Dic3—34no—
77C3×C4×Dic3C2×D314yes—
78C2×C3×C3⋊C8—34no—
79#3722—64no—
80#3714—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—
95C2×#213—94no—
96C2×D24—91no—
97#893—91no—
98#893C3⋊D311yescharacteristic, complemented
99C4×D12—94no—
100C3×#107—91no—
101#832—94no—
102C8×Dic3—91no—
103#809—91no—
104#809—94no—
105#809—91no—
106#809—184no—
107#807—34no—
108C4×C3⋊C8—94no—
109C23×C32—61no—
110C22×C3⋊D3—61no—
111C22×C3⋊D3—64no—
112C22×C3⋊D3—124no—
113C22×C3×D3—24no—
114C32×D4—32no—
115C32×D4—61no—
116C2×C32×C4—21no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—31no—
120C2×C32×C4—31no—
121C2×C32×C4—44no—
122C2×C32×C4—122no—
123C2×C32×C4C22×D314yes—
124C2×C32×C4C22×D311yescharacteristic
125C2×C3⋊Dic3—21no—
126C4⋊(C3⋊D3)—32no—
127C4⋊(C3⋊D3)—38no—
128C4⋊(C3⋊D3)—68no—
129C4⋊(C3⋊D3)—61no—
130C4⋊(C3⋊D3)—64no—
131C4⋊(C3⋊D3)—124no—
132C4×C3⋊D3—122no—
133C2×C3×Dic3—64no—
134C2×C3×Dic3#6714yes—
135C2×C3×Dic3#6714yes—
136C3×D12—24no—
137C3×D12C4×D318yescomplemented
138C3×C4×D3—48no—
139#624—128no—
140C3⋊(C3⋊C8)—61no—
141C3⋊(C3⋊C8)—64no—
142C3⋊(C3⋊C8)—124no—
143C3⋊(C3⋊C8)C4×D312yescomplemented
144C3×C3⋊C8—38no—
145C4×D8—271noSylow
146C2×C33C4×D411yescharacteristic
147C2×C33C2×D811yescharacteristic
148C2×C33#13511yescharacteristic
149C3⋊(C3⋊D3)—42no—
150C32×D3—42no—
151C2×C3×D4—34no—
152C22×C3×C4—64no—
153C2×D12—94no—
154C2×D12—91no—
155C2×D12—184no—
156C2×D12C2×C3⋊D311yescharacteristic
157C2×C4×D3—21no—
158C2×C4×D3—184no—
159C2×C4×D3—181no—
160C2×C3×C8—91no—
161C3×C4⋊C4—24no—
162C3×C4⋊C4—91no—
163C3×C4⋊C4D3214yes—
164C3×C4⋊C4C2×C3⋊D311yescharacteristic
165C3×C22⋊C4—64no—
166C3×C42—34no—
167#213—916no—
168(C2×D3)⋊C4—21no—
169(C2×D3)⋊C4—184no—
170(C2×D3)⋊C4—181no—
171C4⋊Dic3—34no—
172C4×Dic3C2×C3⋊D311yescharacteristic
173C2×C3⋊C8—34no—
174C2×C3⋊C8—94no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—184no—
177D24—94no—
178C22×C32—24no—
179C22×C32—62no—
180C22×C32—121no—
181C22×C32—121no—
182C22×C32D3×D411yescharacteristic
183C22×C32C2×#6714yes—
184C2×C3⋊D3—62no—
185C2×C3⋊D3—68no—
186C2×C3⋊D3—128no—
187C2×C3⋊D3—121no—
188C2×C3⋊D3—121no—
189C2×C3⋊D3—124no—
190C2×C3⋊D3—124no—
191C2×C3⋊D3—244no—
192C2×C3⋊D3—244no—
193C2×C3×D3—28no—
194C2×C3×D3—44no—
195C2×C3×D3—44no—
196C32×C4—24no—
197C32×C4—24no—
198C32×C4—32no—
199C32×C4—41no—
200C32×C4—44no—
201C32×C4—61no—
202C32×C4—84no—
203C32×C4C2×C4×D314yes—
204C32×C4C2×C4×D311yescharacteristic
205C32×C4#23514yes—
206C32×C4#23911yescharacteristic
207C3⋊Dic3—41no—
208C3×Dic3—24no—
209C3×Dic3—124no—
210C3×Dic3#21318yescomplemented
211C2×D8—271no—
212C4×D4—31no—
213C4×D4—271no—
214#107—91no—
215#102—271no—
216#102—271no—
217C4×C8—271no—
218C33C4×D811yescharacteristic, complemented, Sylow
219C23×C3—64no—
220C22×D3—21no—
221C22×D3—184no—
222C22×D3—181no—
223C22×D3—364no—
224C3×D4—38no—
225C3×D4—64no—
226C2×C3×C4—24no—
227C2×C3×C4—21no—
228C2×C3×C4—24no—
229C2×C3×C4—34no—
230C2×C3×C4—34no—
231C2×C3×C4—44no—
232C2×C3×C4—128no—
233C2×C3×C4—181no—
234C2×C3×C4C2×D3214yes—
235C2×C3×C4C22×C3⋊D311yescharacteristic
236C2×Dic3—64no—
237C2×Dic3#63811yescharacteristic
238C2×Dic3#63811yescharacteristic
239D12—21no—
240D12—98no—
241D12—92no—
242D12—188no—
243D12—184no—
244D12—181no—
245D12—364no—
246D12C4×C3⋊D312yescomplemented
247C4×D3—42no—
248C4×D3—368no—
249C4×D3—362no—
250C3×C8—92no—
251C3⋊C8—38no—
252C3⋊C8—184no—
253C3⋊C8—181no—
254C3⋊C8—364no—
255C2×C32—24no—
256C2×C32—24no—
257C2×C32—24no—
258C2×C32—122no—
259C2×C32#89311yescharacteristic
260C2×C32D3×D811yescharacteristic
261C2×C32#91911yescharacteristic
262C2×C32C4×#6714yes—
263C2×C32C2×#21314yes—
264C2×C32#95014yes—
265C3⋊D3—122no—
266C3⋊D3—128no—
267C3⋊D3—248no—
268C3×D3—48no—
269C2×D4—31no—
270C2×D4—271no—
271C22×C4—61no—
272C22×C4—541no—
273D8—274no—
274C2×C8—91no—
275C2×C8—271no—
276C4⋊C4—91no—
277C4⋊C4D3×C3⋊D311yescharacteristic
278C22⋊C4—61no—
279C22⋊C4—541no—
280C42—31no—
281C22×C3—24no—
282C22×C3—68no—
283C22×C3—124no—
284C22×C3—124no—
285C22×C3D3×#6714yes—
286C22×C3C2×#63811yescharacteristic
287C2×D3—22no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—182no—
291C2×D3—188no—
292C2×D3—368no—
293C2×D3—361no—
294C2×D3—361no—
295C2×D3—364no—
296C2×D3—364no—
297C2×D3—724no—
298C2×D3—724no—
299C3×C4—24no—
300C3×C4—24no—
301C3×C4—38no—
302C3×C4—41no—
303C3×C4—44no—
304C3×C4—64no—
305C3×C4—84no—
306C3×C4—361no—
307C3×C4#379914yes—
308C3×C4C4×D3214yes—
309C3×C4C2×C4×C3⋊D311yescharacteristic
310C3×C4#382811yescharacteristic
311Dic3—21no—
312Dic3—124no—
313Dic3#375312yescomplemented
314C32—24no—
315C32#487211yescharacteristic, complemented
316C32C4×#21314yescomplemented
317C23—61no—
318C23—541no—
319D4—32no—
320D4—61no—
321D4—272no—
322D4—541no—
323C2×C4—21no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—122no—
327C2×C4—181no—
328C2×C4—1082no—
329C2×C4C2×D3×C3⋊D311yescharacteristic
330C8—92no—
331C8—541no—
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—128no—
336C2×C3#6390414yes—
337C2×C3D3×#21314yes—
338C2×C3#6394914yes—
339C2×C3C4×#63811yescharacteristic
340C2×C3C2×#375311yescharacteristic
341C2×C3#6415911yescharacteristic
342D3—42no—
343D3—362no—
344D3—368no—
345D3—728no—
346C22—62no—
347C22—121no—
348C22—121no—
349C22—542no—
350C22—1081no—
351C22—1081no—
352C22D3×#63811yescharacteristic
353C4—32no—
354C4—41no—
355C4—61no—
356C4—361no—
357C4#8902211yescharacteristic
358C4C4×D3×C3⋊D311yescharacteristic
359C3—24no—
360C3#1059045614yescomplemented
361C3C4×#375311yescharacteristic, complemented
362C2—122no—
363C2—1082no—
364C2#1172351011yescharacteristic
365C2D3×#375311yescharacteristic
366C2#1172358211yescharacteristic
subgroupG/H
centerC22D3×#638
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC33×C4⋊C4C22
socleC22×C33C2×D4
prime radicalC2×#88790C2
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. #11723484C2semidirect
4. #11722179C2semidirect
5. C3×#63634C2semidirect
6. #11722031C2semidirect
7. Dic3×C3⋊(C3⋊C8)C2semidirect
21. #88790C4semidirect
25. C3×#893D3semidirect
32. #63634D3semidirect
42. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
47. C32×D12C2×C4semidirect
51. C3×C3⋊(C3⋊C8)C2×C4semidirect
93. C32×Dic3D8semidirect
98. #893C3⋊D3semidirect
137. C3×D12C4×D3semidirect
143. C3⋊(C3⋊C8)C4×D3semidirect
210. C3×Dic3#213semidirect
218. C33C4×D8semidirect
246. D12C4×C3⋊D3semidirect
313. Dic3#3753semidirect
315. C32#4872semidirect
316. C32C4×#213semidirect
360. C3#10590456semidirect
361. C3C4×#3753semidirect

Maximal subgroups

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

subgroupindex
C32×#8932
#117234842
C2×#887902
#117221792
C3×#636342
#117220312
Dic3×C3⋊(C3⋊C8)2
C4×#37533
#105904563