Finite Group Databasebeta

← #13159640

Subgroups — #13159640

Order 1728.

884 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 176 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
1C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
2C4×#6290C211yesmaximal, characteristic
3C2×#88787C211yesmaximal, characteristic, complemented
4#11722184C211yesmaximal, characteristic, complemented
5C3×#63647C211yesmaximal, characteristic, complemented
6(C2×D3)⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
7#11722018C211yesmaximal, characteristic, complemented
8#10592827—31nomaximal
9#10590735—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
18C32×ΓL(1,𝔽25)—22no—
19#88806C411yescharacteristic
20#88803C411yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
22#88787—22no—
23C2×C32×M16—31no—
24C4×#638—31no—
25C2×#3743—31no—
26C2×C3×ΓL(1,𝔽25)D314yescomplemented
27C4×#625—34no—
28C2×#3711—34no—
29#63657—31no—
30#63647D311yescharacteristic, complemented
31#63641—31no—
32#63641—34no—
33#63641—64no—
34C8×C3⋊Dic3—31no—
35C3×(C2×D3)⋊C8—34no—
36(C2×D3)⋊(C3⋊C8)—34no—
37#63553—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))—21no—
52C3×C3⋊(C3⋊C8)—21no—
53C32×C3⋊C8D412yescomplemented
54#5128—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—
62C32×M16—62no—
63C2×C32×C8—24no—
64C2×C32×C8—31no—
65C2×C32×C8C2×D314yes—
66C2×C32×C8C2×D311yescharacteristic
67#3757—31no—
68#3752—31no—
69#3750—31no—
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—
76C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
77#3743—62no—
78C2×C3×C3⋊C8—34no—
79C2×C3×C3⋊C8C2×D314yes—
80C3×ΓL(1,𝔽25)—28no—
81#3724—34no—
82(C2×D3)⋊Dic3—34no—
83(C2×D3)⋊Dic3—34no—
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—
98C2×C3×M16—34no—
99C4×#67—94no—
100C4×#67—91no—
101C2×#208—91no—
102C2×ΓL(1,𝔽25)—94no—
103C2×ΓL(1,𝔽25)C3⋊D311yescharacteristic, complemented
104C3×C22⋊C8—91no—
105#830—94no—
106(C2×D3)⋊C8—34no—
107(C2×D3)⋊C8—91no—
108Dic3⋊C8—94no—
109Dic3⋊C8—91no—
110Dic3⋊C8—184no—
111C8×Dic3—94no—
112C8×Dic3—91no—
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—21no—
141C32×C8—24no—
142C32×C8—32no—
143C32×C8—44no—
144C3⋊(C3⋊C8)—21no—
145C3⋊(C3⋊C8)—61no—
146C3⋊(C3⋊C8)—64no—
147C3⋊(C3⋊C8)—124no—
148C3×C3⋊C8—64no—
149C3×C3⋊C8#6718yescomplemented
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
162C3×M16—68no—
163C2×C3×C8—24no—
164C2×C3×C8—34no—
165C2×C3×C8—91no—
166C2×C3×C8D3214yes—
167C2×C3×C8C2×C3⋊D311yescharacteristic
168#217—94no—
169#217—91no—
170(C2×D3)⋊C4—94no—
171(C2×D3)⋊C4—91no—
172Dic3⋊C4—94no—
173Dic3⋊C4—91no—
174C4×Dic3—94no—
175C4×Dic3—91no—
176C4×Dic3—184no—
177#208—182no—
178C2×C3⋊C8—34no—
179C2×C3⋊C8—94no—
180C2×C3⋊C8—91no—
181C2×C3⋊C8—184no—
182C2×C3⋊C8C2×C3⋊D311yescharacteristic
183ΓL(1,𝔽25)—22no—
184ΓL(1,𝔽25)—188no—
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×C4#23514yes—
204C32×C4D3×D411yescharacteristic
205C32×C4#23711yescharacteristic
206C32×C4C2×#6714yes—
207C3⋊Dic3—21no—
208C3⋊Dic3—32no—
209C3⋊Dic3—38no—
210C3⋊Dic3—68no—
211C3⋊Dic3—61no—
212C3⋊Dic3—64no—
213C3⋊Dic3—124no—
214C3×Dic3—24no—
215C3×Dic3—64no—
216C2×M16—31no—
217C2×M16—271no—
218C4×D4—271no—
219C4⋊C8—271no—
220C22⋊C8—91no—
221C22⋊C8—271no—
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—24no—
252C3×C8—21no—
253C3×C8—38no—
254C3×C8—44no—
255C3×C8—181no—
256C3⋊C8—64no—
257C3⋊C8—184no—
258C3⋊C8—181no—
259C3⋊C8—364no—
260C3⋊C8#63812yescomplemented
261C2×C32—24no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—121no—
265C2×C32#67⋊C411yescharacteristic
266C2×C32#90111yescharacteristic
267C2×C32D3×M1614yes—
268C2×C32D3×M1611yescharacteristic
269C2×C32#90714yes—
270C2×C32C4×#6714yes—
271C3⋊D3—41no—
272C3×D3—44no—
273C3×D3—124no—
274C2×D4—271no—
275C22×C4—31no—
276C22×C4—91no—
277M16—62no—
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#379514yes—
307C3×C4D3×#6714yes—
308C3×C4#382811yescharacteristic
309C3×C4C2×#63811yescharacteristic
310Dic3—21no—
311Dic3—64no—
312Dic3—98no—
313Dic3—92no—
314Dic3—188no—
315Dic3—184no—
316Dic3—181no—
317Dic3—364no—
318C32—24no—
319C32#472911yescharacteristic, complemented
320C32#512814yescomplemented
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—32no—
335C8—181no—
336C8—541no—
337C2×C3—24no—
338C2×C3—24no—
339C2×C3—24no—
340C2×C3—124no—
341C2×C3—361no—
342C2×C3#6379114yes—
343C2×C3#6380414yes—
344C2×C3#6387914yes—
345C2×C3M16×C3⋊D311yescharacteristic
346C2×C3#6411611yescharacteristic
347C2×C3C4×#63811yescharacteristic
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#8902111yescharacteristic
362C4D3×#63811yescharacteristic
363C3—24no—
364C3#1059073514yescomplemented
365C3#1059282711yescharacteristic, complemented
366C2—121no—
367C2—361no—
368C2#1172331211yescharacteristic
369C2#1172333311yescharacteristic
370C2#1172350611yescharacteristic
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
1. C2×C32×ΓL(1,𝔽25)C2semidirect
3. C2×#88787C2semidirect
4. #11722184C2semidirect
5. C3×#63647C2semidirect
6. (C2×D3)⋊(C3⋊(C3⋊C8))C2semidirect
7. #11722018C2semidirect
26. C2×C3×ΓL(1,𝔽25)D3semidirect
30. #63647D3semidirect
53. C32×C3⋊C8D4semidirect
95. C3⋊(C3⋊Dic3)M16semidirect
103. C2×ΓL(1,𝔽25)C3⋊D3semidirect
149. C3×C3⋊C8#67semidirect
223. C33#436semidirect
260. C3⋊C8#638semidirect
319. C32#4729semidirect
320. C32#5128semidirect
364. C3#10590735semidirect
365. C3#10592827semidirect

Maximal subgroups

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

subgroupindex
C2×C32×ΓL(1,𝔽25)2
C4×#62902
C2×#887872
#117221842
C3×#636472
(C2×D3)⋊(C3⋊(C3⋊C8))2
#117220182
#105928273
#105907353