Finite Group Databasebeta

← #81555

Subgroups — #81555

Order 384.

386 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5632C211yesmaximal, characteristic, complemented
2#5532C211yesmaximal, characteristic
3C3×#485C211yesmaximal, characteristic, complemented
4#5049C211yesmaximal, characteristic, complemented
5#5004C211yesmaximal, characteristic, complemented
6#4886C211yesmaximal, characteristic, complemented
7#4875C211yesmaximal, characteristic, complemented
8#4868C211yesmaximal, characteristic, complemented
9#4852C211yesmaximal, characteristic, complemented
10#4818C211yesmaximal, characteristic, complemented
11#4806C211yesmaximal, characteristic, complemented
12#4802C211yesmaximal, characteristic, complemented
13#4792C211yesmaximal, characteristic, complemented
14#4738C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3329—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C22×Dic6C2211yescharacteristic
19C3×C22⋊Q8C2211yescharacteristic
20C3×#117C2211yescharacteristic
21#944C2211yescharacteristic
22C2×#215C2211yescharacteristic
23#930C2211yescharacteristic
24C4×#67C2211yescharacteristic
25C2×#208C2211yescharacteristic
26C2×#204C2211yescharacteristic
27C2×#204C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic
29#897C2211yescharacteristic
30#896—21no—
31#895C2211yescharacteristic
32#893C2211yescharacteristic
33#892C2211yescharacteristic
34#888—21no—
35#888C2211yescharacteristic
36#887C2211yescharacteristic
37#885—21no—
38#884—21no—
39#67⋊C4—21no—
40#878—21no—
41#875—21no—
42#872—21no—
43C3×#107C2211yescharacteristic
44C3×#106C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#103C2211yescharacteristic
47C3×C22⋊C8C2211yescharacteristic
48#835C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50#816C2211yescharacteristic
51#816C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53C8×Dic3C2211yescharacteristic
54#810C2211yescharacteristic
55#810C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#809C2211yescharacteristic
59#807C2211yescharacteristic
60#555—31no—
61D4×Q8—31no—
62#487—31no—
63#485—31no—
64#485D311yescharacteristic, complemented
65#477—31no—
66#468—31no—
67#468—31no—
68#461—31no—
69#448—31no—
70#448—31no—
71C4×SD16—31no—
72#436—31no—
73#419—31no—
74#419—31no—
75C2×C3×Q8C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C2×#67—21no—
78C2×#67D411yescharacteristic
79C22×Dic3—21no—
80D3×Q8—22no—
81D3×Q8—41no—
82C2×D12—21no—
83C2×D12C2311yescharacteristic
84C2×C4×D3—21no—
85C2×C4×D3—21no—
86C2×C4×D3C2311yescharacteristic
87C2×Dic6—22no—
88C2×Dic6—21no—
89C2×Dic6—21no—
90C2×Dic6—21no—
91C2×Dic6—41no—
92C2×Dic6C2311yescharacteristic
93C2×Dic6C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C3×C4⋊C4—21no—
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102C3×C42—21no—
103#217D411yescharacteristic
104#215—41no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4D411yescharacteristic
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121#208—41no—
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124#204—22no—
125#204—41no—
126ΓL(1,𝔽25)—41no—
127C22×Q8—31no—
128C22×Q8—31no—
129C2×SD16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C2×M16—31no—
134C4⋊Q8—31no—
135C4⋊Q8—31no—
136C4⋊Q8—61no—
137#124—61no—
138#124—61no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141C22⋊Q8—61no—
142C22⋊Q8C2×D311yescharacteristic
143#121—31no—
144#121—31no—
145#121—61no—
146C4×Q8—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—61no—
150#117—31no—
151#117—61no—
152#117C2×D311yescharacteristic
153#107—31no—
154#107C2×D311yescharacteristic
155#106C2×D311yescharacteristic
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103C2×D311yescharacteristic
165#103C2×D311yescharacteristic
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8C2×D311yescharacteristic
169C4×C8—31no—
170C23×C3C2×D411yescharacteristic
171C22×D3—21no—
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2411yescharacteristic
183#67—22no—
184#67—81no—
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—42no—
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192D12—21no—
193D12—21no—
194D12—81no—
195C4×D3—21no—
196C4×D3—21no—
197C4×D3—42no—
198C4×D3—42no—
199Dic6—21no—
200Dic6—21no—
201Dic6—21no—
202Dic6—42no—
203Dic6—42no—
204Dic6—81no—
205Dic6#4112yes—
206C3×C8—21no—
207C3×C8—21no—
208C3⋊C8—21no—
209C3⋊C8—21no—
210C2×Q8—31no—
211C2×Q8—31no—
212C2×Q8—61no—
213C2×Q8—61no—
214C2×Q8—62no—
215C2×Q8—61no—
216C2×Q8—62no—
217C2×Q8—121no—
218C2×Q8—121no—
219C2×Q8C22×D311yescharacteristic
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—61no—
223C2×D4—61no—
224C22×C4—31no—
225C22×C4—61no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4C22×D311yescharacteristic
229SD16—62no—
230SD16—121no—
231SD16—121no—
232M16—121no—
233M16—121no—
234C2×C8—31no—
235C2×C8—31no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—61no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4C22×D311yescharacteristic
247C4⋊C4C22×D311yescharacteristic
248C4⋊C4C22×D311yescharacteristic
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—31no—
252C22⋊C4—31no—
253C22⋊C4—61no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C42—21no—
259C42—31no—
260C42—61no—
261C42—61no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—21no—
265C22×C3C22×D411yescharacteristic
266C2×D3—21no—
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—41no—
270C2×D3—41no—
271C2×D3—41no—
272C3×C4—21no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4C2×#4111yescharacteristic
278C3×C4H(5,𝔽2)11yescharacteristic
279Dic3—22no—
280Dic3—21no—
281Dic3—21no—
282Dic3—41no—
283Dic3—41no—
284Dic3#13712yes—
285C23—31no—
286C23—61no—
287C23D3×D411yescharacteristic
288Q8—21no—
289Q8—21no—
290Q8—32no—
291Q8—61no—
292Q8—61no—
293Q8—61no—
294Q8—122no—
295Q8—122no—
296Q8—241no—
297D4—62no—
298D4—61no—
299D4—61no—
300D4—241no—
301D4—241no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—61no—
315C2×C4—61no—
316C2×C4—122no—
317C2×C4—122no—
318C2×C4—122no—
319C2×C4D3×D411yescharacteristic
320C2×C4C23×D311yescharacteristic
321C8—21no—
322C8—21no—
323C8—61no—
324C8—61no—
325C2×C3—41no—
326C2×C3#54611yescharacteristic
327C2×C3C2×#13711yescharacteristic
328C2×C3#57711yescharacteristic
329D3—41no—
330D3—81no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—61no—
335C22—61no—
336C22—61no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22C2×D3×D411yescharacteristic
341C4—21no—
342C4—32no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—61no—
348C4—62no—
349C4—61no—
350C4—121no—
351C4—121no—
352C4D3×#4111yescharacteristic
353C4#100911yescharacteristic
354C3#332911yescharacteristic, complemented, Sylow
355C2—41no—
356C2—121no—
357C2—241no—
358C2#565411yescharacteristic
359C2#577011yescharacteristic
360C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#485C2
socleC22×C3C22×D4
prime radical#5532C2
hypercenter#485D3
hypocenterC3#3329
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #485

Complements

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

NH ≅ G/Ntype
1. #5632C2semidirect
3. C3×#485C2semidirect
4. #5049C2semidirect
5. #5004C2semidirect
6. #4886C2semidirect
7. #4875C2semidirect
8. #4868C2semidirect
9. #4852C2semidirect
10. #4818C2semidirect
11. #4806C2semidirect
12. #4802C2semidirect
13. #4792C2semidirect
14. #4738C2semidirect
15. #4729C2semidirect
64. #485D3semidirect
354. C3#3329semidirect

Maximal subgroups

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

subgroupindex
#56322
#55322
C3×#4852
#50492
#50042
#48862
#48752
#48682
#48522
#48182
#48062
#48022
#47922
#47382
#47292
#33293