Finite Group Databasebeta

← #13159641

Subgroups — #13159641

Order 1728.

908 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 908 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C4×#6290C211yesmaximal, characteristic
2C2×#88787C211yesmaximal, characteristic, complemented
3C2×#88786C211yesmaximal, characteristic, complemented
4C8×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6C32×(C2×D3)⋊C8C211yesmaximal, characteristic, complemented
7#11722050C211yesmaximal, characteristic, complemented
8#10592428—31nomaximal
9#10590734—34nomaximal
10C2×C3×C4×C3⋊D3C2211yescharacteristic
11C2×C32×C4×D3C2211yescharacteristic
12C2×#6290C411yescharacteristic
13C2×C33×C8C2211yescharacteristic
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88806C411yescharacteristic
19#88803C411yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
21#88787—22no—
22#88786—22no—
23C4×#638—31no—
24C2×#3748—31no—
25C2×#3743—31no—
26C4×#625—34no—
27C2×#3711—34no—
28C2×#3711—34no—
29C32×C22⋊C8—31no—
30#63647D311yescharacteristic, complemented
31#63641—31no—
32C8×C3⋊Dic3—31no—
33C8×C3⋊Dic3—34no—
34C8×C3⋊Dic3—64no—
35C3×(C2×D3)⋊C8—34no—
36C3×(C2×D3)⋊C8D314yescomplemented
37#63572—34no—
38C22×C3×C3⋊D3C2×C411yescharacteristic
39C22×C32×D3C2×C411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
42C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
43C2×C3×C3⋊Dic3C2×C411yescharacteristic
44C3×C4×C3⋊D3—21no—
45C3×C4×C3⋊D3—21no—
46C2×C32×Dic3C2×C411yescharacteristic
47C32×C4×D3—21no—
48C32×C4×D3—21no—
49#6290—22no—
50C33×C8—21no—
51C3⋊(C3⋊(C3⋊C8))D412yescomplemented
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8—21no—
54#4729—94no—
55#4729—91no—
56C22×C32×C4—31no—
57C2×#638—31no—
58C2×C4×C3⋊D3C2×D311yescharacteristic
59C2×C3×C4×D3—34no—
60C2×C3×C4×D3C2×D314yes—
61C2×#625—34no—
62C2×C32×C8—24no—
63C2×C32×C8—31no—
64C2×C32×C8C2×D314yes—
65C2×C32×C8C2×D311yescharacteristic
66#3757—31no—
67#3752—31no—
68#3750—31no—
69C4×C3⋊Dic3—31no—
70C4×C3⋊Dic3—34no—
71C4×C3⋊Dic3—64no—
72#3748—62no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—34no—
75C2×C3⋊(C3⋊C8)—64no—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77#3743—62no—
78C2×C3×C3⋊C8—34no—
79C2×C3×C3⋊C8C2×D314yes—
80#3724—34no—
81(C2×D3)⋊Dic3—34no—
82(C2×D3)⋊Dic3—34no—
83#3711—68no—
84#3711—68no—
85C22×C33C22×C411yescharacteristic
86C2×C3×C3⋊D3—21no—
87C2×C3×C3⋊D3—21no—
88C2×C3×C3⋊D3—21no—
89C2×C32×D3—21no—
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—21no—
95C3⋊(C3⋊Dic3)M1612yescomplemented
96C3×C3⋊Dic3—21no—
97C32×Dic3—21no—
98C4×#67—94no—
99C4×#67—91no—
100C2×#208—94no—
101C2×#208—91no—
102C2×ΓL(1,𝔽25)—94no—
103C2×ΓL(1,𝔽25)—91no—
104C3×C22⋊C8—34no—
105C3×C22⋊C8—91no—
106(C2×D3)⋊C8—34no—
107(C2×D3)⋊C8C3⋊D311yescharacteristic, complemented
108Dic3⋊C8—94no—
109Dic3⋊C8—91no—
110C8×Dic3—94no—
111C8×Dic3—91no—
112C8×Dic3—184no—
113C23×C32—31no—
114C22×C3⋊D3C4×D311yescharacteristic
115C22×C3×D3—34no—
116C22×C3×D3C4×D314yes—
117C2×C32×C4—24no—
118C2×C32×C4—31no—
119C2×C32×C4—61no—
120C2×C32×C4—61no—
121C2×C32×C4C22×D314yes—
122C2×C32×C4C22×D311yescharacteristic
123#638—62no—
124C2×C3⋊Dic3—31no—
125C2×C3⋊Dic3—31no—
126C2×C3⋊Dic3—34no—
127C2×C3⋊Dic3—34no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—64no—
130C2×C3⋊Dic3C4×D311yescharacteristic
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133C2×C3×Dic3—34no—
134C2×C3×Dic3C4×D314yes—
135C3×C4×D3—24no—
136C3×C4×D3—24no—
137C3×C4×D3—64no—
138C3×C4×D3—64no—
139#625—68no—
140C32×C8—28no—
141C32×C8—24no—
142C32×C8—21no—
143C32×C8—61no—
144C3⋊(C3⋊C8)—21no—
145C3⋊(C3⋊C8)—32no—
146C3⋊(C3⋊C8)—38no—
147C3⋊(C3⋊C8)—68no—
148C3×C3⋊C8—24no—
149C3×C3⋊C8—64no—
150#436—271noSylow
151C2×C33C4×D411yescharacteristic
152C2×C33C2×M1611yescharacteristic
153C2×C33#13111yescharacteristic
154C3×C3⋊D3—41no—
155C32×D3—41no—
156C22×C3×C4—34no—
157C22×C3×C4—91no—
158C2×#67—94no—
159C2×#67—91no—
160C2×C4×D3—34no—
161C2×C4×D3C2×C3⋊D311yescharacteristic
162C2×C3×C8—24no—
163C2×C3×C8—34no—
164C2×C3×C8—91no—
165C2×C3×C8D3214yes—
166C2×C3×C8C2×C3⋊D311yescharacteristic
167#217—94no—
168#217—91no—
169(C2×D3)⋊C4—94no—
170(C2×D3)⋊C4—91no—
171Dic3⋊C4—94no—
172Dic3⋊C4—91no—
173C4×Dic3—94no—
174C4×Dic3—91no—
175C4×Dic3—184no—
176#208—182no—
177#208—188no—
178C2×C3⋊C8—34no—
179C2×C3⋊C8—94no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—184no—
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183ΓL(1,𝔽25)—188no—
184ΓL(1,𝔽25)—182no—
185C22×C32—24no—
186C22×C32—61no—
187C22×C32—61no—
188C22×C32—61no—
189C22×C32C2×C4×D314yes—
190C22×C32C2×C4×D311yescharacteristic
191C2×C3⋊D3—21no—
192C2×C3⋊D3—21no—
193C2×C3⋊D3—21no—
194C2×C3×D3—24no—
195C2×C3×D3—24no—
196C2×C3×D3—24no—
197C2×C3×D3—64no—
198C2×C3×D3—64no—
199C2×C3×D3—64no—
200C32×C4—24no—
201C32×C4—24no—
202C32×C4—61no—
203C32×C4D3×D414yes—
204C32×C4D3×D411yescharacteristic
205C32×C4#23714yes—
206C32×C4#23711yescharacteristic
207C3⋊Dic3—21no—
208C3⋊Dic3—32no—
209C3⋊Dic3—38no—
210C3⋊Dic3—68no—
211C3⋊Dic3—68no—
212C3⋊Dic3—64no—
213C3⋊Dic3—61no—
214C3×Dic3—24no—
215C3×Dic3—64no—
216C2×M16—271no—
217C2×M16—271no—
218C4×D4—271no—
219C4⋊C8—271no—
220C22⋊C8—31no—
221C22⋊C8—91no—
222C4×C8—271no—
223C33#43611yescharacteristic, complemented, Sylow
224C23×C3—34no—
225C23×C3—91no—
226C22×D3—34no—
227C22×D3C4×C3⋊D311yescharacteristic
228C2×C3×C4—24no—
229C2×C3×C4—34no—
230C2×C3×C4—64no—
231C2×C3×C4—64no—
232C2×C3×C4—91no—
233C2×C3×C4—181no—
234C2×C3×C4—181no—
235C2×C3×C4C2×D3214yes—
236C2×C3×C4C22×C3⋊D311yescharacteristic
237#67—182no—
238#67—188no—
239C2×Dic3—34no—
240C2×Dic3—94no—
241C2×Dic3—91no—
242C2×Dic3—94no—
243C2×Dic3—91no—
244C2×Dic3—184no—
245C2×Dic3—184no—
246C2×Dic3C4×C3⋊D311yescharacteristic
247C4×D3—21no—
248C4×D3—21no—
249C4×D3—64no—
250C4×D3—64no—
251C3×C8—28no—
252C3×C8—24no—
253C3×C8—21no—
254C3×C8—64no—
255C3×C8—181no—
256C3⋊C8—21no—
257C3⋊C8—64no—
258C3⋊C8—98no—
259C3⋊C8—92no—
260C3⋊C8—188no—
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—121no—
265C2×C32#67⋊C414yes—
266C2×C32#67⋊C411yescharacteristic
267C2×C32#90114yes—
268C2×C32#90111yescharacteristic
269C2×C32D3×M1614yes—
270C2×C32D3×M1611yescharacteristic
271C3⋊D3—41no—
272C3×D3—44no—
273C3×D3—124no—
274C2×D4—271no—
275C22×C4—31no—
276C22×C4—91no—
277M16—542no—
278M16—542no—
279C2×C8—31no—
280C2×C8—91no—
281C2×C8—271no—
282C2×C8D3×C3⋊D311yescharacteristic
283C4⋊C4—271no—
284C22⋊C4—271no—
285C22⋊C4—271no—
286C42—271no—
287C22×C3—24no—
288C22×C3—64no—
289C22×C3—64no—
290C22×C3—64no—
291C22×C3—181no—
292C22×C3—181no—
293C22×C3—181no—
294C22×C3C4×D3214yes—
295C22×C3C2×C4×C3⋊D311yescharacteristic
296C2×D3—21no—
297C2×D3—21no—
298C2×D3—21no—
299C2×D3—64no—
300C2×D3—64no—
301C2×D3—64no—
302C3×C4—24no—
303C3×C4—24no—
304C3×C4—64no—
305C3×C4—181no—
306C3×C4#380214yes—
307C3×C4#380514yes—
308C3×C4D4×C3⋊D311yescharacteristic
309C3×C4#383011yescharacteristic
310Dic3—21no—
311Dic3—64no—
312Dic3—98no—
313Dic3—92no—
314Dic3—188no—
315Dic3—188no—
316Dic3—184no—
317Dic3—181no—
318C32—24no—
319C32#472914yescomplemented
320C32#472911yescharacteristic, complemented
321C23—31no—
322C23—91no—
323D4—542no—
324C2×C4—31no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—91no—
328C2×C4—181no—
329C2×C4—181no—
330C2×C4—271no—
331C2×C4—271no—
332C2×C4C2×D3×C3⋊D311yescharacteristic
333C8—21no—
334C8—61no—
335C8—181no—
336C8—272no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—124no—
341C2×C3—361no—
342C2×C3#6379114yes—
343C2×C3#6380314yes—
344C2×C3#6390214yes—
345C2×C3#6409011yescharacteristic
346C2×C3#6411011yescharacteristic
347C2×C3M16×C3⋊D311yescharacteristic
348D3—41no—
349D3—124no—
350C22—61no—
351C22—61no—
352C22—61no—
353C22—181no—
354C22—181no—
355C22—181no—
356C22C4×D3×C3⋊D311yescharacteristic
357C4—61no—
358C4—181no—
359C4—272no—
360C4—541no—
361C4#8902611yescharacteristic
362C4#8902911yescharacteristic
363C3—24no—
364C3#1059073414yescomplemented
365C3#1059242811yescharacteristic, complemented
366C2—121no—
367C2—361no—
368C2#1172331211yescharacteristic
369C2#1172333111yescharacteristic
370C2#1172350811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutator, socleC22×C33C22×C4
FittingC2×C33×C8C22
prime radicalC2×#6290C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#436
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88787C2semidirect
3. C2×#88786C2semidirect
4. C8×C3⋊(C3⋊Dic3)C2semidirect
5. C3×#63647C2semidirect
6. C32×(C2×D3)⋊C8C2semidirect
7. #11722050C2semidirect
30. #63647D3semidirect
36. C3×(C2×D3)⋊C8D3semidirect
51. C3⋊(C3⋊(C3⋊C8))D4semidirect
95. C3⋊(C3⋊Dic3)M16semidirect
107. (C2×D3)⋊C8C3⋊D3semidirect
223. C33#436semidirect
319. C32#4729semidirect
320. C32#4729semidirect
364. C3#10590734semidirect
365. C3#10592428semidirect

Maximal subgroups

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

subgroupindex
C4×#62902
C2×#887872
C2×#887862
C8×C3⋊(C3⋊Dic3)2
C3×#636472
C32×(C2×D3)⋊C82
#117220502
#105924283
#105907343