Finite Group Databasebeta

← #13160075

Subgroups — #13160075

Order 1728.

1172 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 210 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×#89022C211yesmaximal, characteristic
2C2×C32×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
3D3×C4⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
4#11722191C211yesmaximal, characteristic, complemented
5C3×#63643C211yesmaximal, characteristic, complemented
6#11722040C211yesmaximal, characteristic, complemented
7#11722036C211yesmaximal, characteristic, complemented
8#10592832—31nomaximal
9#10590671—34nomaximal
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
11C2×C3×#634C2211yescharacteristic
12C2×C32×C4×D3C2211yescharacteristic
13C2×#6291—21no—
14C2×#6289—21no—
15C2×D3×C3⋊Dic3—21no—
16#89022—22no—
17#89022C414yescomplemented
18C2×C33×C8C2211yescharacteristic
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
21C2×C32×C3⋊C8C2211yescharacteristic
22C32×ΓL(1,𝔽25)—22no—
23Dic3⋊(C3⋊Dic3)—21no—
24C2×#3828—31no—
25C2×#3799—34no—
26C2×C32×M16—31no—
27C2×C4⋊(C3⋊Dic3)—31no—
28C2×C3×ΓL(1,𝔽25)D314yescomplemented
29D3×C4⋊Dic3—34no—
30#63648—31no—
31#63648—31no—
32#63648—34no—
33#63648—64no—
34#63643—31no—
35#63643D311yescharacteristic, complemented
36C3×#816—34no—
37#63566—34no—
38#63564—34no—
39C22×C3⋊(C3⋊D3)—21no—
40C22×C32×D3D411yescharacteristic
41C2×C33×C4C2311yescharacteristic
42C2×C3⋊(C3⋊Dic3)—21no—
43C4⋊(C3⋊(C3⋊D3))—21no—
44C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
45C2×C3×C3⋊Dic3—21no—
46C2×C3×C3⋊Dic3—21no—
47C3×#634—21no—
48C3×#634C2×C412yescomplemented
49C2×C32×Dic3D411yescharacteristic
50C32×C4×D3C2×C412yes—
51C32×C4×D3D412yes—
52#6291—24no—
53#6289—24no—
54D3×C3⋊Dic3—42no—
55C33×C8—21no—
56C32×C3⋊C8—21no—
57#5133—94no—
58#4803—91no—
59C22×C32×C4—31no—
60C2×#638—61no—
61C22×C3⋊Dic3—61no—
62#3828—34no—
63#3828—62no—
64C2×C4⋊(C3⋊D3)—31no—
65C2×C4⋊(C3⋊D3)—34no—
66C2×C4⋊(C3⋊D3)—64no—
67C2×C4×C3⋊D3—61no—
68C2×#634C2×D311yescharacteristic
69C2×C3×C4×D3C2×D314yes—
70C2×C3×Dic6—34no—
71C2×#626—64no—
72C2×#624—64no—
73C2×D3×Dic3—64no—
74#3799—316no—
75#3799—68no—
76C32×M16—62no—
77C2×C32×C8—24no—
78C2×C32×C8—31no—
79C2×C32×C8C2×D314yes—
80C2×C32×C8C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)—31no—
82C4⋊(C3⋊Dic3)—34no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—64no—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic
86C3×C4⋊Dic3—34no—
87C2×C3×C3⋊C8C2×D314yes—
88C3×ΓL(1,𝔽25)—28no—
89Dic3⋊Dic3—64no—
90C22×C33C2×D411yescharacteristic
91C2×C3⋊(C3⋊D3)—22no—
92C2×C3⋊(C3⋊D3)—41no—
93C2×C3⋊(C3⋊D3)—41no—
94C2×C32×D3—21no—
95C2×C32×D3—21no—
96C2×C32×D3C22⋊C412yes—
97C33×C4C22×C411yescharacteristic
98C33×C4C2×D411yescharacteristic
99C3⋊(C3⋊Dic3)—41no—
100C3×C3⋊Dic3—22no—
101C3×C3⋊Dic3—41no—
102C32×Dic3C22⋊C412yes—
103C2×#239—91no—
104C2×#235—94no—
105C2×C3×M16—34no—
106C2×C4⋊Dic3—94no—
107C2×ΓL(1,𝔽25)C3⋊D311yescharacteristic, complemented
108D3×C4⋊C4—91no—
109C3×#103—91no—
110#835—91no—
111#821—94no—
112#821—94no—
113#821—91no—
114#821—184no—
115#816—34no—
116#816—94no—
117#809—91no—
118C23×C32—31no—
119C22×C3⋊D3—61no—
120C22×C3⋊D3—64no—
121C22×C3⋊D3—124no—
122C22×C3×D3D1214yes—
123C2×C32×C4—24no—
124C2×C32×C4—31no—
125C2×C32×C4—32no—
126C2×C32×C4—32no—
127C2×C32×C4C22×D314yes—
128C2×C32×C4C22×D311yescharacteristic
129#638—64no—
130C2×C3⋊Dic3—21no—
131C2×C3⋊Dic3—21no—
132C2×C3⋊Dic3—61no—
133C2×C3⋊Dic3—64no—
134C2×C3⋊Dic3—124no—
135C2×C3⋊Dic3—122no—
136C4⋊(C3⋊D3)—32no—
137C4⋊(C3⋊D3)—38no—
138C4⋊(C3⋊D3)—68no—
139C4⋊(C3⋊D3)—61no—
140C4⋊(C3⋊D3)—64no—
141C4⋊(C3⋊D3)—124no—
142C4×C3⋊D3—64no—
143#634—21no—
144#634C4×D312yescomplemented
145C2×C3×Dic3—64no—
146C2×C3×Dic3—64no—
147C2×C3×Dic3D1214yes—
148C3×C4×D3C4×D318yes—
149C3×C4×D3#6718yes—
150C3×Dic6—38no—
151C3×Dic6—64no—
152#626—616no—
153#624—616no—
154D3×Dic3—128no—
155C32×C8—21no—
156C32×C8—24no—
157C32×C8—44no—
158C32×C8—61no—
159C3×C3⋊C8—24no—
160#417—271noSylow
161C2×C33C2×C22⋊C411yescharacteristic
162C2×C33AGL(1,ℤ/8)11yescharacteristic
163C2×C33#13711yescharacteristic
164C3⋊(C3⋊D3)—42no—
165C32×D3—22no—
166C2×C3×Q8—91no—
167C22×C3×C4—34no—
168C2×#67—184no—
169C22×Dic3—184no—
170#239—94no—
171#239—182no—
172#235—916no—
173#235—188no—
174C2×D12—94no—
175C2×D12—91no—
176C2×D12—184no—
177C2×D12—181no—
178C2×C4×D3—184no—
179C2×C4×D3—181no—
180C2×C4×D3—181no—
181C2×C4×D3C2×C3⋊D311yescharacteristic
182C2×Dic6—34no—
183C3×M16—68no—
184C2×C3×C8—24no—
185C2×C3×C8—34no—
186C2×C3×C8D3214yes—
187C2×C3×C8C2×C3⋊D311yescharacteristic
188C3×C4⋊C4—91no—
189C4⋊Dic3—34no—
190C4⋊Dic3—94no—
191C4⋊Dic3—91no—
192C4⋊Dic3—184no—
193C4⋊Dic3—184no—
194Dic3⋊C4—181no—
195C2×C3⋊C8C2×C3⋊D311yescharacteristic
196ΓL(1,𝔽25)—22no—
197C22×C32—24no—
198C22×C32—32no—
199C22×C32—61no—
200C22×C32—61no—
201C22×C32C2×D1214yes—
202C22×C32D3×D411yescharacteristic
203C2×C3⋊D3—62no—
204C2×C3⋊D3—68no—
205C2×C3⋊D3—128no—
206C2×C3⋊D3—121no—
207C2×C3⋊D3—121no—
208C2×C3⋊D3—124no—
209C2×C3⋊D3—124no—
210C2×C3⋊D3—244no—
211C2×C3⋊D3—244no—
212C2×C3×D3—24no—
213C2×C3×D3—24no—
214C2×C3×D3(C2×D3)⋊C418yes—
215C32×C4—24no—
216C32×C4—24no—
217C32×C4—32no—
218C32×C4C2×C4×D314yes—
219C32×C4C2×C4×D311yescharacteristic
220C32×C4D3×D411yescharacteristic
221C32×C4C2×#6714yes—
222C3⋊Dic3—22no—
223C3⋊Dic3—41no—
224C3⋊Dic3—121no—
225C3⋊Dic3—124no—
226C3⋊Dic3—244no—
227C3×Dic3—68no—
228C3×Dic3—124no—
229C3×Dic3(C2×D3)⋊C418yes—
230C2×#41—271no—
231C2×M16—31no—
232C2×C4⋊C4—271no—
233#103—91no—
234#103—271no—
235#102—271no—
236#102—271no—
237C33#41711yescharacteristic, complemented, Sylow
238C23×C3—34no—
239C22×D3—184no—
240C22×D3—181no—
241C22×D3—364no—
242C22×D3C4⋊(C3⋊D3)11yescharacteristic
243C3×Q8—92no—
244C3×Q8—181no—
245C2×C3×C4—24no—
246C2×C3×C4—34no—
247C2×C3×C4—38no—
248C2×C3×C4—38no—
249C2×C3×C4—181no—
250C2×C3×C4—181no—
251C2×C3×C4C2×D3214yes—
252C2×C3×C4C22×C3⋊D311yescharacteristic
253#67—1816no—
254C2×Dic3—64no—
255C2×Dic3—64no—
256C2×Dic3—184no—
257C2×Dic3—181no—
258C2×Dic3—364no—
259C2×Dic3—368no—
260C2×Dic3C4⋊(C3⋊D3)11yescharacteristic
261D12—98no—
262D12—92no—
263D12—188no—
264D12—184no—
265D12—184no—
266D12—181no—
267D12—364no—
268C4×D3—1816no—
269C4×D3—184no—
270C4×D3—362no—
271C4×D3C4×C3⋊D312yes—
272C4×D3#63812yes—
273Dic6—38no—
274Dic6—64no—
275C3×C8—24no—
276C3×C8—21no—
277C3×C8—44no—
278C3×C8—64no—
279C3⋊C8—21no—
280C2×C32—24no—
281C2×C32—24no—
282C2×C32—24no—
283C2×C32—62no—
284C2×C32D3×C22⋊C411yescharacteristic
285C2×C32#90814yes—
286C2×C32#90914yes—
287C2×C32#91411yescharacteristic
288C2×C32#91811yescharacteristic
289C2×C32C2×(C2×D3)⋊C414yes—
290C3⋊D3—122no—
291C3⋊D3—128no—
292C3⋊D3—248no—
293C3×D3—28no—
294#41—274no—
295#41—542no—
296C2×Q8—91no—
297C2×D4—271no—
298C2×D4—541no—
299C22×C4—31no—
300C22×C4—541no—
301C22×C4—541no—
302M16—62no—
303C2×C8—31no—
304C2×C8D3×C3⋊D311yescharacteristic
305C4⋊C4—91no—
306C4⋊C4—271no—
307C4⋊C4—541no—
308C22×C3—24no—
309C22×C3—38no—
310C22×C3—64no—
311C22×C3—64no—
312C22×C3D3×D1214yes—
313C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
314C2×D3—21no—
315C2×D3—21no—
316C2×D3—182no—
317C2×D3—188no—
318C2×D3—368no—
319C2×D3—364no—
320C2×D3—364no—
321C2×D3—361no—
322C2×D3—361no—
323C2×D3—724no—
324C2×D3—724no—
325C2×D3#375212yes—
326C3×C4—24no—
327C3×C4—24no—
328C3×C4—38no—
329C3×C4—182no—
330C3×C4—361no—
331C3×C4C4×D3214yes—
332C3×C4D3×#6714yes—
333C3×C4C2×C4×C3⋊D311yescharacteristic
334C3×C4C2×#63811yescharacteristic
335Dic3—68no—
336Dic3—124no—
337Dic3—364no—
338Dic3—361no—
339Dic3—724no—
340Dic3#375212yes—
341C32—24no—
342C32#480311yescharacteristic, complemented
343C32#513314yescomplemented
344C23—31no—
345C23—541no—
346Q8—92no—
347Q8—181no—
348D4—272no—
349D4—544no—
350D4—541no—
351C2×C4—31no—
352C2×C4—32no—
353C2×C4—32no—
354C2×C4—181no—
355C2×C4—181no—
356C2×C4—544no—
357C2×C4—541no—
358C2×C4—1082no—
359C2×C4C2×D3×C3⋊D311yescharacteristic
360C8—21no—
361C8—61no—
362C2×C3—24no—
363C2×C3—24no—
364C2×C3—24no—
365C2×C3—68no—
366C2×C3#6379414yes—
367C2×C3#6380114yes—
368C2×C3D3×(C2×D3)⋊C414yes—
369C2×C3#6411711yescharacteristic
370C2×C3#6411811yescharacteristic
371C2×C3C2×#375211yescharacteristic
372D3—22no—
373D3—362no—
374D3—368no—
375D3—728no—
376C22—32no—
377C22—61no—
378C22—61no—
379C22—542no—
380C22—1081no—
381C22—1081no—
382C22D3×C4⋊(C3⋊D3)11yescharacteristic
383C4—32no—
384C4—182no—
385C4—361no—
386C4—1081no—
387C4C4×D3×C3⋊D311yescharacteristic
388C4D3×#63811yescharacteristic
389C3—24no—
390C3#1059067114yescomplemented
391C3#1059283211yescharacteristic, complemented
392C2—62no—
393C2—1082no—
394C2#1172331511yescharacteristic
395C2#1172332711yescharacteristic
396C2D3×#375211yescharacteristic
subgroupG/H
centerC22D3×C4⋊(C3⋊D3)
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×C8C22
socleC22×C33C2×D4
prime radicalC2×#89022C2
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#417
LCSG ▷ 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
2. C2×C32×ΓL(1,𝔽25)C2semidirect
3. D3×C4⋊(C3⋊Dic3)C2semidirect
4. #11722191C2semidirect
5. C3×#63643C2semidirect
6. #11722040C2semidirect
7. #11722036C2semidirect
17. #89022C4semidirect
28. C2×C3×ΓL(1,𝔽25)D3semidirect
35. #63643D3semidirect
44. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
48. C3×#634C2×C4semidirect
107. C2×ΓL(1,𝔽25)C3⋊D3semidirect
144. #634C4×D3semidirect
237. C33#417semidirect
342. C32#4803semidirect
343. C32#5133semidirect
390. C3#10590671semidirect
391. C3#10592832semidirect

Maximal subgroups

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

subgroupindex
C2×#890222
C2×C32×ΓL(1,𝔽25)2
D3×C4⋊(C3⋊Dic3)2
#117221912
C3×#636432
#117220402
#117220362
#105928323
#105906713