Finite Group Databasebeta

← #13158932

Subgroups — #13158932

Order 1728.

904 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×#88787C211yesmaximal, characteristic, complemented
3C2×C3⋊D3×C3⋊C8C211yesmaximal, characteristic, complemented
4#11722173C211yesmaximal, characteristic, complemented
5C32×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#11722023C211yesmaximal, characteristic, complemented
7#11722023C211yesmaximal, characteristic, complemented
8#10592535—31nomaximal
9#10590330—34nomaximal
10C2×C3×C4⋊(C3⋊D3)C411yescharacteristic
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C33×C42C2211yescharacteristic
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×#3752C411yescharacteristic
17C3×#3752C411yescharacteristic
18C3×C4⋊(C3⋊Dic3)C411yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88787—22no—
22C3⋊D3×C3⋊C8—22no—
23C2×#3743—31no—
24C2×C8×C3⋊D3—31no—
25C4×C4⋊(C3⋊D3)D311yescharacteristic
26C3×C4×D12—34no—
27C2×#3711—34no—
28C2×D3×C3⋊C8—34no—
29C32×C4⋊C8—31no—
30#63647—31no—
31#63647—31no—
32#63630—31no—
33#63630—34no—
34#63630—64no—
35C3×C4⋊(C3⋊C8)D314yescomplemented
36(C2×D3)⋊(C3⋊C8)—34no—
37(C2×D3)⋊(C3⋊C8)—34no—
38C22×C3×C3⋊D3C2×C411yescharacteristic
39C22×C3×C3⋊D3C2×C411yescharacteristic
40C2×C33×C4C2×C411yescharacteristic
41C2×C33×C4C2×C411yescharacteristic
42C2×C33×C4C2311yescharacteristic
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C2×C3×C3⋊Dic3C2×C411yescharacteristic
45C3×C4⋊(C3⋊D3)—22no—
46C3×C4×C3⋊D3—21no—
47C3×C4×C3⋊D3—21no—
48C3×C4×C3⋊D3—21no—
49C3×C4×C3⋊D3—21no—
50C3⋊(C3⋊(C3⋊C8))—21no—
51C3⋊(C3⋊(C3⋊C8))D412yescomplemented
52C32×C3⋊C8—21no—
53C32×C3⋊C8—21no—
54#5011—91no—
55#4836—94no—
56C2×C4⋊(C3⋊D3)Dic311yescharacteristic
57C2×C4×C3⋊D3C2×D311yescharacteristic
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×C3×D12—34no—
60C2×C3×C4×D3—34no—
61C2×C3×C4×D3—34no—
62C2×C32×C8—31no—
63C2×C32×C8—31no—
64C32×C42—24no—
65C32×C42C2×D314yes—
66C32×C42C2×D311yescharacteristic
67#3752Dic311yescharacteristic
68#3752Dic311yescharacteristic
69C4⋊(C3⋊Dic3)Dic311yescharacteristic
70C2×C3⋊(C3⋊C8)—31no—
71C2×C3⋊(C3⋊C8)—34no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—34no—
74C2×C3⋊(C3⋊C8)—64no—
75C2×C3⋊(C3⋊C8)—64no—
76#3743—62no—
77C8×C3⋊D3—62no—
78C3×(C2×D3)⋊C4—34no—
79C3×(C2×D3)⋊C4—34no—
80C3×C4⋊Dic3—34no—
81C2×C3×C3⋊C8C2×D314yes—
82C2×C3×C3⋊C8C2×D314yes—
83#3711—68no—
84D3×C3⋊C8—68no—
85C22×C33C22×C411yescharacteristic
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C2×C3×C3⋊D3—21no—
90C2×C3×C3⋊D3—21no—
91C2×C3×C3⋊D3M1612yescomplemented
92C33×C4—21no—
93C33×C4—21no—
94C33×C4C2×D411yescharacteristic
95C33×C4#4111yescharacteristic
96C3×C3⋊Dic3—21no—
97C3×C3⋊Dic3—21no—
98C3×C4×D4—91no—
99C2×#208—91no—
100C22×C3⋊C8—91no—
101C2×ΓL(1,𝔽25)—94no—
102C2×D3×C8—94no—
103C4×D12—34no—
104C3×C4⋊C8—34no—
105#830—91no—
106#830—91no—
107(C2×D3)⋊C8—94no—
108(C2×D3)⋊C8—94no—
109C4⋊(C3⋊C8)C3⋊D311yescharacteristic, complemented
110#803—94no—
111#803—91no—
112#803—184no—
113C22×C3⋊D3C2×Dic311yescharacteristic
114C22×C3⋊D3C2×Dic311yescharacteristic
115C22×C3×D3—34no—
116C22×C3×D3—34no—
117C2×C32×C4—24no—
118C2×C32×C4—24no—
119C2×C32×C4—24no—
120C2×C32×C4C4×D314yes—
121C2×C32×C4C4×D314yes—
122C2×C32×C4C2×Dic311yescharacteristic
123C2×C32×C4C2×Dic311yescharacteristic
124C2×C32×C4C22×D314yes—
125C2×C32×C4C22×D311yescharacteristic
126C2×C3⋊Dic3C2×Dic311yescharacteristic
127C2×C3⋊Dic3C2×Dic311yescharacteristic
128C4⋊(C3⋊D3)—22no—
129C4×C3⋊D3—21no—
130C4×C3⋊D3—21no—
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3—34no—
135C3×D12—68no—
136C3×C4×D3—64no—
137C3×C4×D3—64no—
138C3×C4×D3—64no—
139C3×C4×D3—64no—
140C32×C8—61no—
141C32×C8—61no—
142C3⋊(C3⋊C8)—32no—
143C3⋊(C3⋊C8)—38no—
144C3⋊(C3⋊C8)—68no—
145C3⋊(C3⋊C8)—68no—
146C3⋊(C3⋊C8)—61no—
147C3⋊(C3⋊C8)—64no—
148C3×C3⋊C8—24no—
149C3×C3⋊C8—24no—
150#435—271noSylow
151C2×C33C4×D411yescharacteristic
152C2×C33C2×M1611yescharacteristic
153C2×C33#13111yescharacteristic
154C3×C3⋊D3—22no—
155C3×C3⋊D3—41no—
156C2×C3×D4—91no—
157C22×C3×C4—91no—
158C22×C3×C4—91no—
159C2×D12—34no—
160C2×C4×D3—34no—
161C2×C4×D3—34no—
162C2×C3×C8—34no—
163C2×C3×C8—34no—
164C3×C4⋊C4—91no—
165C3×C22⋊C4—91no—
166C3×C22⋊C4—91no—
167C3×C42—24no—
168C3×C42D3214yes—
169C3×C42C2×C3⋊D311yescharacteristic
170(C2×D3)⋊C4—34no—
171(C2×D3)⋊C4—34no—
172C4⋊Dic3—34no—
173#208—182no—
174C2×C3⋊C8—94no—
175C2×C3⋊C8—91no—
176C2×C3⋊C8—94no—
177C2×C3⋊C8—91no—
178C2×C3⋊C8—184no—
179C2×C3⋊C8—184no—
180C2×C3⋊C8—182no—
181C2×C3⋊C8C2×C3⋊D311yescharacteristic
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183ΓL(1,𝔽25)—188no—
184D3×C8—188no—
185C22×C32—24no—
186C22×C32C2×C4×D314yes—
187C22×C32C22×Dic311yescharacteristic
188C2×C3⋊D3—21no—
189C2×C3⋊D3—21no—
190C2×C3⋊D3—21no—
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3#20812yescomplemented
194C2×C3×D3—38no—
195C2×C3×D3—64no—
196C2×C3×D3—64no—
197C2×C3×D3—64no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C32×C4—21no—
201C32×C4—21no—
202C32×C4—24no—
203C32×C4—24no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—44no—
207C32×C4—44no—
208C32×C4D3×D414yes—
209C32×C4D3×D411yescharacteristic
210C32×C4#23711yescharacteristic
211C32×C4#23914yes—
212C3⋊Dic3—21no—
213C3⋊Dic3—21no—
214C3×Dic3—64no—
215C3×Dic3—64no—
216C2×M16—271no—
217C22×C8—271no—
218C4×D4—91no—
219C4⋊C8—31no—
220C22⋊C8—271no—
221C22⋊C8—271no—
222#97—271no—
223C33#43511yescharacteristic, complemented, Sylow
224C23×C3—91no—
225C23×C3—91no—
226C22×D3—34no—
227C22×D3—34no—
228C3×D4—182no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—24no—
232C2×C3×C4—91no—
233C2×C3×C4—91no—
234C2×C3×C4—181no—
235C2×C3×C4—181no—
236C2×C3×C4—181no—
237C2×C3×C4—181no—
238C2×C3×C4D3×Dic314yes—
239C2×C3×C4D3×Dic314yes—
240C2×C3×C4C4×C3⋊D311yescharacteristic
241C2×C3×C4C4×C3⋊D311yescharacteristic
242C2×C3×C4C2×D3214yes—
243C2×C3×C4C22×C3⋊D311yescharacteristic
244C2×Dic3—34no—
245C2×Dic3—34no—
246D12—68no—
247C4×D3—64no—
248C4×D3—64no—
249C4×D3—64no—
250C4×D3—64no—
251C3×C8—64no—
252C3×C8—64no—
253C3⋊C8—21no—
254C3⋊C8—21no—
255C3⋊C8—98no—
256C3⋊C8—92no—
257C3⋊C8—188no—
258C3⋊C8—188no—
259C3⋊C8—184no—
260C3⋊C8—181no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32#89314yes—
265C2×C32#90114yes—
266C2×C32D3×M1614yes—
267C2×C32C2×#20811yescharacteristic
268C2×C32D4×Dic311yescharacteristic
269C2×C32#94811yescharacteristic
270C3⋊D3—22no—
271C3⋊D3—41no—
272C3×D3—68no—
273C3×D3—124no—
274C2×D4—91no—
275C22×C4—91no—
276C22×C4—91no—
277M16—542no—
278C2×C8—31no—
279C2×C8—31no—
280C2×C8—271no—
281C2×C8—271no—
282C2×C8—542no—
283C4⋊C4—91no—
284C22⋊C4—91no—
285C22⋊C4—91no—
286C42D3×C3⋊D311yescharacteristic
287C22×C3—24no—
288C22×C3—92no—
289C22×C3—181no—
290C22×C3—181no—
291C22×C3—181no—
292C22×C3—181no—
293C22×C3—181no—
294C22×C3C2×D3×Dic314yes—
295C22×C3C2×C4×C3⋊D311yescharacteristic
296C2×D3—38no—
297C2×D3—64no—
298C2×D3—64no—
299C2×D3—64no—
300C2×D3—64no—
301C2×D3—64no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—21no—
305C3×C4—21no—
306C3×C4—24no—
307C3×C4—24no—
308C3×C4—44no—
309C3×C4—44no—
310C3×C4—181no—
311C3×C4—181no—
312C3×C4#379614yes—
313C3×C4#380214yes—
314C3×C4D4×C3⋊D311yescharacteristic
315C3×C4#383211yescharacteristic
316Dic3—64no—
317Dic3—64no—
318C32—24no—
319C32#483614yescomplemented
320C32#501111yescharacteristic, complemented
321C23—91no—
322C23—91no—
323D4—182no—
324C2×C4—91no—
325C2×C4—91no—
326C2×C4—181no—
327C2×C4—181no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4Dic3×C3⋊D311yescharacteristic
331C2×C4Dic3×C3⋊D311yescharacteristic
332C2×C4C2×D3×C3⋊D311yescharacteristic
333C8—61no—
334C8—61no—
335C8—272no—
336C8—541no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—182no—
341C2×C3—361no—
342C2×C3D3×#20814yes—
343C2×C3#6381414yes—
344C2×C3#6389814yes—
345C2×C3#6410211yescharacteristic
346C2×C3#6411011yescharacteristic
347C2×C3M16×C3⋊D311yescharacteristic
348D3—68no—
349D3—124no—
350C22—92no—
351C22—181no—
352C22—181no—
353C22—181no—
354C22—181no—
355C22—181no—
356C22C2×Dic3×C3⋊D311yescharacteristic
357C4—21no—
358C4—21no—
359C4—181no—
360C4—181no—
361C4#8901711yescharacteristic
362C4#8902611yescharacteristic
363C3—24no—
364C3#1059033014yescomplemented
365C3#1059253511yescharacteristic, complemented
366C2—182no—
367C2—361no—
368C2#1172335211yescharacteristic
369C2C3⋊D3×#20811yescharacteristic
370C2#1172350111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC33×C42C22
prime radicalC2×C3×C4⋊(C3⋊D3)C4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#435
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C2×#88787C2semidirect
3. C2×C3⋊D3×C3⋊C8C2semidirect
4. #11722173C2semidirect
5. C32×C4⋊(C3⋊C8)C2semidirect
6. #11722023C2semidirect
7. #11722023C2semidirect
35. C3×C4⋊(C3⋊C8)D3semidirect
51. C3⋊(C3⋊(C3⋊C8))D4semidirect
91. C2×C3×C3⋊D3M16semidirect
109. C4⋊(C3⋊C8)C3⋊D3semidirect
193. C2×C3⋊D3#208semidirect
223. C33#435semidirect
319. C32#4836semidirect
320. C32#5011semidirect
364. C3#10590330semidirect
365. C3#10592535semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊(C3⋊D3)2
C2×#887872
C2×C3⋊D3×C3⋊C82
#117221732
C32×C4⋊(C3⋊C8)2
#117220232
#105925353
#105903303