Finite Group Databasebeta

← #81375

Subgroups — #81375

Order 384.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5607C211yesmaximal, characteristic
2#5523C211yesmaximal, characteristic, complemented
3C3×#482C211yesmaximal, characteristic, complemented
4#5033C211yesmaximal, characteristic, complemented
5#5003C211yesmaximal, characteristic, complemented
6#4882C211yesmaximal, characteristic, complemented
7#4881C211yesmaximal, characteristic, complemented
8#4866C211yesmaximal, characteristic, complemented
9#4860C211yesmaximal, characteristic, complemented
10#4779C211yesmaximal, characteristic, complemented
11#4777C211yesmaximal, characteristic, complemented
12#4770C211yesmaximal, characteristic, complemented
13D3×#102C211yesmaximal, characteristic, complemented
14#4735C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3096—31nomaximal, Sylow
17C2×D3×D4C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C3×#121C2211yescharacteristic
20C3×#117C2211yescharacteristic, complemented
21#940—21no—
22#938C2211yescharacteristic
23#937—21no—
24#930C2211yescharacteristic
25C2×C4⋊Dic3C2211yescharacteristic, complemented
26C2×ΓL(1,𝔽25)C2211yescharacteristic
27C2×D3×C8C2211yescharacteristic
28#895C2211yescharacteristic
29#892C2211yescharacteristic
30#892C2211yescharacteristic
31#892C2211yescharacteristic, complemented
32D3×C4⋊C4C2211yescharacteristic
33D3×C4⋊C4C2211yescharacteristic, complemented
34#885—21no—
35#882—21no—
36#67⋊C4—21no—
37#67⋊C4—21no—
38C4×D12—21no—
39C4×Dic6—21no—
40C3×#107C2211yescharacteristic, complemented
41C3×#106C2211yescharacteristic, complemented
42C3×#102C2211yescharacteristic
43C3×#102C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic, complemented
45#832C2211yescharacteristic
46#832C2211yescharacteristic
47#830C2211yescharacteristic, complemented
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic, complemented
51(C2×D3)⋊C8C2211yescharacteristic, complemented
52#817C2211yescharacteristic, complemented
53#817C2211yescharacteristic, complemented
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#808C2211yescharacteristic, complemented
57#807C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#535—31no—
61#520—31no—
62#482—31no—
63#482—31no—
64#482—31no—
65#482D311yescharacteristic, complemented
66#481—31no—
67#481—31no—
68#480—31no—
69#480—31no—
70#428—31no—
71#427—31no—
72#418—31no—
73C2×#102—31no—
74#408—31no—
75C23×D3—21no—
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67D411yescharacteristic
83C2×#67D411yescharacteristic
84C22×Dic3—21no—
85D3×D4—22no—
86D3×D4—22no—
87#235—22no—
88#235—22no—
89C2×D12—21no—
90C2×D12D411yescharacteristic
91C2×D12C2311yescharacteristic
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3C2311yescharacteristic
95C2×C4×D3C2311yescharacteristic
96C2×Dic6D411yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C42—21no—
105#217—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117C4×Dic3—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C2×C3⋊C8C2311yescharacteristic
121C2×C3⋊C8C2311yescharacteristic
122ΓL(1,𝔽25)—22no—
123D3×C8—22no—
124C2×#41—31no—
125C22×D4—31no—
126C2×M16—31no—
127C22×C8—31no—
128#127—61no—
129#124—61no—
130#121—31no—
131#121—31no—
132#121—31no—
133#121C2×D311yescharacteristic
134#120—61no—
135#120—61no—
136C4×Q8—61no—
137C4×D4—61no—
138C4×D4—61no—
139C4×D4—61no—
140#117—31no—
141#117—31no—
142#117—31no—
143#117C2×D311yescharacteristic, complemented
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147#107—31no—
148#107—31no—
149#107—31no—
150#107C2×D311yescharacteristic, complemented
151#106—31no—
152#106—31no—
153#106—31no—
154#106C2×D311yescharacteristic, complemented
155#102—31no—
156#102—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161#102C2×D311yescharacteristic
162#102C2×D311yescharacteristic
163C22⋊C8—31no—
164C22⋊C8—31no—
165C22⋊C8—31no—
166C22⋊C8C2×D311yescharacteristic, complemented
167C23×C3—21no—
168C23×C3C2×D411yescharacteristic
169C22×D3—21no—
170C22×D3—42no—
171C22×D3—81no—
172C22×D3—81no—
173C22×D3C2×D411yescharacteristic
174C22×D3C2×D411yescharacteristic
175C3×D4—21no—
176C3×D4—21no—
177C3×D4—81no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185#67—22no—
186#67—22no—
187#67—42no—
188#67—81no—
189#67—81no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—81no—
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196D12—22no—
197D12—21no—
198D12—21no—
199D12—81no—
200C4×D3—21no—
201C4×D3—21no—
202C4×D3—81no—
203C4×D3—81no—
204C4×D3#4112yes—
205C4×D3#4112yes—
206Dic6—22no—
207C3×C8—21no—
208C3×C8—21no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211C24—61no—
212#41—62no—
213#41—62no—
214C2×Q8—31no—
215C2×D4—21no—
216C2×D4—31no—
217C2×D4—31no—
218C2×D4—31no—
219C2×D4—31no—
220C2×D4—61no—
221C2×D4—62no—
222C2×D4—62no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4C22×D311yescharacteristic
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4—61no—
232C22×C4C22×D311yescharacteristic
233M16—62no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8—62no—
237C2×C8C22×D311yescharacteristic
238C2×C8C22×D311yescharacteristic
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4C22×D311yescharacteristic
246C4⋊C4C22×D311yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C42—21no—
257C42—61no—
258C42—61no—
259C42—61no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3C22×D411yescharacteristic
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—21no—
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—84no—
276C2×D3#12312yes—
277C3×C4—21no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4C2×#4111yescharacteristic
282C3×C4C2×#4111yescharacteristic
283Dic3—21no—
284Dic3—41no—
285Dic3—41no—
286Dic3—41no—
287Dic3#12312yes—
288C23—21no—
289C23—31no—
290C23—31no—
291C23—61no—
292C23—122no—
293C23—241no—
294C23—241no—
295C23D3×D411yescharacteristic
296Q8—62no—
297D4—21no—
298D4—21no—
299D4—62no—
300D4—62no—
301D4—62no—
302D4—61no—
303D4—61no—
304D4—81no—
305D4—122no—
306D4—241no—
307D4—241no—
308D4—241no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—31no—
315C2×C4—32no—
316C2×C4—32no—
317C2×C4—31no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—61no—
323C2×C4—241no—
324C2×C4—241no—
325C2×C4—241no—
326C2×C4D3×D411yescharacteristic
327C2×C4C23×D311yescharacteristic
328C8—21no—
329C8—21no—
330C8—61no—
331C8—61no—
332C2×C3—41no—
333C2×C3—81no—
334C2×C3C2×#12311yescharacteristic
335C2×C3#57611yescharacteristic
336C2×C3#57711yescharacteristic
337D3—22no—
338D3—41no—
339D3—81no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—32no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—61no—
348C22—61no—
349C22—61no—
350C22—61no—
351C22—61no—
352C22—121no—
353C22—121no—
354C22—121no—
355C22—244no—
356C22C2×D3×D411yescharacteristic
357C4—21no—
358C4—32no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—61no—
363C4—121no—
364C4—121no—
365C4—121no—
366C4D3×#4111yescharacteristic
367C4D3×#4111yescharacteristic
368C3#309611yescharacteristic, complemented, Sylow
369C2—41no—
370C2—62no—
371C2—81no—
372C2—121no—
373C2—241no—
374C2D3×#12311yescharacteristic
375C2#577011yescharacteristic
376C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#482C2
socleC22×C3C22×D4
prime radical#5607C2
hypercenter#482D3
hypocenterC3#3096
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
2. #5523C2semidirect
3. C3×#482C2semidirect
4. #5033C2semidirect
5. #5003C2semidirect
6. #4882C2semidirect
7. #4881C2semidirect
8. #4866C2semidirect
9. #4860C2semidirect
10. #4779C2semidirect
11. #4777C2semidirect
12. #4770C2semidirect
13. D3×#102C2semidirect
14. #4735C2semidirect
15. #4727C2semidirect
20. C3×#117C22semidirect
25. C2×C4⋊Dic3C22semidirect
31. #892C22semidirect
33. D3×C4⋊C4C22semidirect
40. C3×#107C22semidirect
41. C3×#106C22semidirect
44. C3×C22⋊C8C22semidirect
47. #830C22semidirect
50. (C2×D3)⋊C8C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. #817C22semidirect
53. #817C22semidirect
56. #808C22semidirect
57. #807C22semidirect
58. #807C22semidirect
59. #807C22semidirect
65. #482D3semidirect
143. #117C2×D3semidirect
150. #107C2×D3semidirect
154. #106C2×D3semidirect
166. C22⋊C8C2×D3semidirect
368. C3#3096semidirect

Maximal subgroups

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

subgroupindex
#56072
#55232
C3×#4822
#50332
#50032
#48822
#48812
#48662
#48602
#47792
#47772
#47702
D3×#1022
#47352
#47272
#30963