Finite Group Databasebeta

← #13159976

Subgroups — #13159976

Order 1728.

920 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 164 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
2#11723516C211yesmaximal, characteristic
3#11723425C211yesmaximal, characteristic, complemented
4C2×#88797C211yesmaximal, characteristic, complemented
5#11722191C211yesmaximal, characteristic, complemented
6C3×#63643C211yesmaximal, characteristic, complemented
7#11722055C211yesmaximal, characteristic, complemented
8#10592833—31nomaximal
9#10590850—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6291—21no—
14C2×C33×C8C2211yescharacteristic
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C32×ΓL(1,𝔽25)—22no—
19#88806—21no—
20#88804—21no—
21(C2×D3)⋊(C3⋊Dic3)—21no—
22#88797—22no—
23C2×C32×M16—31no—
24#64139—31no—
25#64133—31no—
26C2×#3744—31no—
27C2×C3×ΓL(1,𝔽25)D314yescomplemented
28#63909—34no—
29#63856—34no—
30C2×#3717—34no—
31#63648—31no—
32#63648—34no—
33#63648—64no—
34#63645—31no—
35#63643D311yescharacteristic, complemented
36C3×#816—34no—
37#63575—34no—
38C22×C3⋊(C3⋊D3)—21no—
39C22×C32×D3D411yescharacteristic
40C2×C33×C4C2311yescharacteristic
41C2×C3⋊(C3⋊Dic3)—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C2×C3×C3⋊Dic3—21no—
45C2×C3×C3⋊Dic3—21no—
46C3×#634—21no—
47C3×#634—21no—
48C2×C32×Dic3D411yescharacteristic
49C32×C4×D3—21no—
50C32×C4×D3—21no—
51#6291—42no—
52C33×C8—21no—
53C32×C3⋊C8D412yescomplemented
54#5134—94no—
55#4812—91no—
56C22×C32×C4—31no—
57C2×#638—61no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—34no—
60C2×C4⋊(C3⋊D3)—64no—
61C2×#634C2×D311yescharacteristic
62C2×C3×C4×D3C2×D314yes—
63C2×C3×Dic6—34no—
64C2×#626—64no—
65C32×M16—62no—
66C2×C32×C8—24no—
67C2×C32×C8—31no—
68C2×C32×C8C2×D314yes—
69C2×C32×C8C2×D311yescharacteristic
70#3757—61no—
71#3752—61no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—34no—
74C4⋊(C3⋊Dic3)—64no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76#3750—61no—
77#3744—62no—
78C3×C4⋊Dic3—34no—
79C2×C3×C3⋊C8C2×D314yes—
80C3×ΓL(1,𝔽25)—28no—
81#3724—64no—
82#3722—64no—
83(C2×D3)⋊Dic3—64no—
84#3717—68no—
85C22×C33C2×D411yescharacteristic
86C2×C3⋊(C3⋊D3)—41no—
87C2×C3⋊(C3⋊D3)—41no—
88C2×C3⋊(C3⋊D3)—41no—
89C2×C32×D3—21no—
90C2×C32×D3—21no—
91C2×C32×D3—21no—
92C33×C4C2×D411yescharacteristic
93C33×C4#4111yescharacteristic
94C3⋊(C3⋊Dic3)—41no—
95C3×C3⋊Dic3—41no—
96C3×C3⋊Dic3—41no—
97C32×Dic3—21no—
98C2×C3×M16—34no—
99#946—91no—
100C2×#215—91no—
101#930—94no—
102#924—94no—
103C2×#204—94no—
104C2×ΓL(1,𝔽25)C3⋊D311yescharacteristic, complemented
105#895—91no—
106C3×#103—91no—
107#821—94no—
108#821—91no—
109#821—184no—
110#818—94no—
111#816—34no—
112#807—91no—
113C23×C32—31no—
114C22×C3⋊D3—61no—
115C22×C3⋊D3—64no—
116C22×C3⋊D3—124no—
117C22×C3×D3D1214yes—
118C2×C32×C4—24no—
119C2×C32×C4—31no—
120C2×C32×C4—61no—
121C2×C32×C4—61no—
122C2×C32×C4C22×D314yes—
123C2×C32×C4C22×D311yescharacteristic
124#638—122no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—21no—
127C2×C3⋊Dic3—61no—
128C2×C3⋊Dic3—64no—
129C2×C3⋊Dic3—124no—
130C4⋊(C3⋊D3)—68no—
131C4⋊(C3⋊D3)—64no—
132C4⋊(C3⋊D3)—61no—
133C4⋊(C3⋊D3)—61no—
134C4⋊(C3⋊D3)—64no—
135C4⋊(C3⋊D3)—124no—
136#634—21no—
137#634—21no—
138C2×C3×Dic3—64no—
139C2×C3×Dic3—64no—
140C2×C3×Dic3D1214yes—
141C3×C4×D3—24no—
142C3×C4×D3—24no—
143C3×Dic6—64no—
144C3×Dic6—64no—
145#626—128no—
146C32×C8—21no—
147C32×C8—24no—
148C32×C8—32no—
149C32×C8—44no—
150C3×C3⋊C8#6718yescomplemented
151#468—271noSylow
152C2×C33#12111yescharacteristic
153C2×C33AGL(1,ℤ/8)11yescharacteristic
154C2×C33#13711yescharacteristic
155C3⋊(C3⋊D3)—81no—
156C32×D3—41no—
157C2×C3×Q8—91no—
158C22×C3×C4—34no—
159C2×#67—184no—
160C2×D12—94no—
161C2×D12—91no—
162C2×D12—184no—
163C2×D12—181no—
164C2×C4×D3C2×C3⋊D311yescharacteristic
165C2×Dic6—34no—
166C3×M16—68no—
167C2×C3×C8—24no—
168C2×C3×C8—34no—
169C2×C3×C8D3214yes—
170C2×C3×C8C2×C3⋊D311yescharacteristic
171C3×C4⋊C4—91no—
172#217—184no—
173#215—182no—
174(C2×D3)⋊C4—184no—
175(C2×D3)⋊C4—181no—
176(C2×D3)⋊C4—181no—
177C4⋊Dic3—34no—
178C4⋊Dic3—94no—
179C4⋊Dic3—91no—
180C4⋊Dic3—184no—
181Dic3⋊C4—184no—
182Dic3⋊C4—181no—
183C2×C3⋊C8C2×C3⋊D311yescharacteristic
184#204—188no—
185ΓL(1,𝔽25)—22no—
186C22×C32—24no—
187C22×C32—61no—
188C22×C32—61no—
189C22×C32—61no—
190C22×C32C2×D1214yes—
191C22×C32D3×D411yescharacteristic
192C2×C3⋊D3—128no—
193C2×C3⋊D3—121no—
194C2×C3⋊D3—121no—
195C2×C3⋊D3—121no—
196C2×C3⋊D3—124no—
197C2×C3⋊D3—124no—
198C2×C3⋊D3—124no—
199C2×C3⋊D3—244no—
200C2×C3⋊D3—244no—
201C2×C3×D3—24no—
202C2×C3×D3—24no—
203C2×C3×D3—24no—
204C32×C4—24no—
205C32×C4—24no—
206C32×C4—61no—
207C32×C4#23514yes—
208C32×C4#23511yescharacteristic
209C32×C4D3×D411yescharacteristic
210C32×C4C2×#6714yes—
211C3⋊Dic3—41no—
212C3⋊Dic3—41no—
213C3⋊Dic3—121no—
214C3⋊Dic3—124no—
215C3⋊Dic3—244no—
216C3×Dic3—24no—
217C3×Dic3—124no—
218C3×Dic3—124no—
219C2×SD16—271no—
220C2×M16—31no—
221C22⋊Q8—271no—
222#121—271no—
223#107—271no—
224#103—91no—
225#102—271no—
226C33#46811yescharacteristic, complemented, Sylow
227C23×C3—34no—
228C22×D3—184no—
229C22×D3—181no—
230C22×D3—364no—
231C22×D3C4⋊(C3⋊D3)11yescharacteristic
232C3×Q8—181no—
233C3×Q8—181no—
234C2×C3×C4—24no—
235C2×C3×C4—34no—
236C2×C3×C4—64no—
237C2×C3×C4—64no—
238C2×C3×C4—181no—
239C2×C3×C4—181no—
240C2×C3×C4C2×D3214yes—
241C2×C3×C4C22×C3⋊D311yescharacteristic
242#67—368no—
243C2×Dic3—64no—
244C2×Dic3—64no—
245C2×Dic3—184no—
246C2×Dic3—181no—
247C2×Dic3—364no—
248C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
249D12—188no—
250D12—184no—
251D12—184no—
252D12—181no—
253D12—181no—
254D12—364no—
255D12—362no—
256C4×D3—21no—
257C4×D3—21no—
258Dic6—64no—
259Dic6—64no—
260C3×C8—24no—
261C3×C8—21no—
262C3×C8—38no—
263C3×C8—44no—
264C3⋊C8#63812yescomplemented
265C2×C32—24no—
266C2×C32—24no—
267C2×C32—24no—
268C2×C32—121no—
269C2×C32#88411yescharacteristic
270C2×C32#90814yes—
271C2×C32#90914yes—
272C2×C32#91411yescharacteristic
273C2×C32#91811yescharacteristic
274C2×C32#93014yes—
275C3⋊D3—248no—
276C3⋊D3—244no—
277C3⋊D3—241no—
278C3×D3—44no—
279C2×Q8—91no—
280C2×D4—271no—
281C2×D4—541no—
282C22×C4—31no—
283SD16—542no—
284M16—62no—
285C2×C8—31no—
286C2×C8D3×C3⋊D311yescharacteristic
287C4⋊C4—91no—
288C4⋊C4—271no—
289C4⋊C4—541no—
290C22⋊C4—541no—
291C22⋊C4—541no—
292C22×C3—24no—
293C22×C3—64no—
294C22×C3—64no—
295C22×C3—64no—
296C22×C3D3×D1214yes—
297C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
298C2×D3—21no—
299C2×D3—21no—
300C2×D3—21no—
301C2×D3—368no—
302C2×D3—364no—
303C2×D3—364no—
304C2×D3—361no—
305C2×D3—361no—
306C2×D3—361no—
307C2×D3—364no—
308C2×D3—724no—
309C2×D3—724no—
310C3×C4—24no—
311C3×C4—24no—
312C3×C4—64no—
313C3×C4—361no—
314C3×C4—361no—
315C3×C4#379814yes—
316C3×C4D3×#6714yes—
317C3×C4#382811yescharacteristic
318C3×C4C2×#63811yescharacteristic
319Dic3—21no—
320Dic3—124no—
321Dic3—124no—
322Dic3—364no—
323Dic3—361no—
324Dic3—724no—
325C32—24no—
326C32#481211yescharacteristic, complemented
327C32#513414yescomplemented
328C23—31no—
329C23—541no—
330Q8—181no—
331Q8—181no—
332D4—541no—
333D4—541no—
334D4—1082no—
335C2×C4—31no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—181no—
339C2×C4—181no—
340C2×C4—541no—
341C2×C4C2×D3×C3⋊D311yescharacteristic
342C8—21no—
343C8—32no—
344C2×C3—24no—
345C2×C3—24no—
346C2×C3—24no—
347C2×C3—124no—
348C2×C3#6379414yes—
349C2×C3#6380114yes—
350C2×C3#6390614yes—
351C2×C3#6411711yescharacteristic
352C2×C3#6411811yescharacteristic
353C2×C3#6413911yescharacteristic
354D3—41no—
355D3—728no—
356D3—721no—
357D3—724no—
358C22—61no—
359C22—61no—
360C22—61no—
361C22—1081no—
362C22—1081no—
363C22—1081no—
364C22D3×C4⋊(C3⋊D3)11yescharacteristic
365C4—61no—
366C4—361no—
367C4—361no—
368C4—1081no—
369C4#8902011yescharacteristic
370C4D3×#63811yescharacteristic
371C3—24no—
372C3#1059085014yescomplemented
373C3#1059283311yescharacteristic, complemented
374C2—121no—
375C2—2161no—
376C2#1172331511yescharacteristic
377C2#1172332711yescharacteristic
378C2#1172351411yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radical#11723516C2
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#468
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ 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. #11723425C2semidirect
4. C2×#88797C2semidirect
5. #11722191C2semidirect
6. C3×#63643C2semidirect
7. #11722055C2semidirect
27. C2×C3×ΓL(1,𝔽25)D3semidirect
35. #63643D3semidirect
53. C32×C3⋊C8D4semidirect
104. C2×ΓL(1,𝔽25)C3⋊D3semidirect
150. C3×C3⋊C8#67semidirect
226. C33#468semidirect
264. C3⋊C8#638semidirect
326. C32#4812semidirect
327. C32#5134semidirect
372. C3#10590850semidirect
373. C3#10592833semidirect

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
#117235162
#117234252
C2×#887972
#117221912
C3×#636432
#117220552
#105928333
#105908503