Finite Group Databasebeta

← #81149

Subgroups — #81149

Order 384.

392 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5691C211yesmaximal, characteristic, complemented
2#5548C211yesmaximal, characteristic, complemented
3C3×#478C211yesmaximal, characteristic, complemented
4#5066C211yesmaximal, characteristic, complemented
5#5016C211yesmaximal, characteristic, complemented
6#4890C211yesmaximal, characteristic, complemented
7#4886C211yesmaximal, characteristic, complemented
8#4865C211yesmaximal, characteristic, complemented
9#4854C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4837C211yesmaximal, characteristic, complemented
12#4786C211yesmaximal, characteristic, complemented
13#4782C211yesmaximal, characteristic, complemented
14#4780C211yesmaximal, characteristic, complemented
15#4762C211yesmaximal, characteristic, complemented
16#3251—31nomaximal, Sylow
17C2×#235—21no—
18C3×#125C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20#940C2211yescharacteristic
21#938C2211yescharacteristic
22C2×#214C2211yescharacteristic
23C2×#213C2211yescharacteristic
24C4×#67—21no—
25C2×D24C2211yescharacteristic
26C2×#204C2211yescharacteristic
27C2×ΓL(1,𝔽25)C2211yescharacteristic
28C2×ΓL(1,𝔽25)C2211yescharacteristic
29#898—21no—
30#897C2211yescharacteristic
31#895C2211yescharacteristic
32#895C2211yescharacteristic
33#894—21no—
34#893—21no—
35#893C2211yescharacteristic
36#892—21no—
37#888C2211yescharacteristic
38#887C2211yescharacteristic
39#885—21no—
40#884—21no—
41#875C2211yescharacteristic
42C4×D12C2211yescharacteristic
43C42×D3C2211yescharacteristic
44C3×#107C2211yescharacteristic
45C3×#106C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#102C2211yescharacteristic
48C3×#102C2211yescharacteristic
49#821C2211yescharacteristic
50#816C2211yescharacteristic
51Dic3⋊C8C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53#810C2211yescharacteristic
54#809C2211yescharacteristic
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#532—31no—
61#529—31no—
62#478—31no—
63#478D311yescharacteristic, complemented
64#475—31no—
65#474—31no—
66#469—31no—
67#469—31no—
68#468—31no—
69#468—31no—
70#463—31no—
71#463—31no—
72#460—31no—
73#459—31no—
74#423—31no—
75C2×C3×D4C2311yescharacteristic
76C22×C3×C4—21no—
77C2×#67—21no—
78C2×#67—21no—
79C2×#67—21no—
80#235—44no—
81C2×D12—21no—
82C2×D12—21no—
83C2×D12C2311yescharacteristic
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3D411yescharacteristic
88C2×C4×D3D411yescharacteristic
89C2×C4×D3C2311yescharacteristic
90C2×Dic6—21no—
91C2×Dic6C2311yescharacteristic
92C2×C3×C8C2311yescharacteristic
93C2×C3×C8C2311yescharacteristic
94C3×C4⋊C4—21no—
95C3×C4⋊C4—21no—
96C3×C4⋊C4C2311yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C42C2311yescharacteristic
101#217—21no—
102#214—41no—
103#213—41no—
104(C2×D3)⋊C4—21no—
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)⋊C4—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115C4×Dic3—21no—
116C4×Dic3D411yescharacteristic
117C4×Dic3D411yescharacteristic
118C4×Dic3C2311yescharacteristic
119C2×C3⋊C8C2311yescharacteristic
120C2×C3⋊C8C2311yescharacteristic
121D24—41no—
122#204—41no—
123ΓL(1,𝔽25)—41no—
124ΓL(1,𝔽25)—41no—
125C2×#41—61no—
126C2×SD16—31no—
127C2×SD16—31no—
128C2×D8—31no—
129C2×D8—31no—
130C2×M16—31no—
131C2×M16—31no—
132C4⋊Q8—31no—
133#127—31no—
134#126—61no—
135#125C2×D311yescharacteristic
136#124—31no—
137#124—61no—
138#123—61no—
139C22⋊Q8—31no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—61no—
144C4×Q8—31no—
145C4×D4—31no—
146C4×D4—31no—
147C4×D4—61no—
148C4×D4—61no—
149C4×D4C2×D311yescharacteristic
150#117—61no—
151C2×C42—31no—
152#107—31no—
153#107C2×D311yescharacteristic
154#106—31no—
155#106C2×D311yescharacteristic
156C4⋊C8—31no—
157C4⋊C8—31no—
158C4⋊C8—31no—
159C4⋊C8C2×D311yescharacteristic
160#103—31no—
161#103—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic
167#102C2×D311yescharacteristic
168C23×C3—21no—
169C22×D3—21no—
170C22×D3—21no—
171C22×D3C2×D411yescharacteristic
172C3×D4—21no—
173C3×D4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—42no—
178C2×C3×C4C2×D411yescharacteristic
179C2×C3×C4C2×D411yescharacteristic
180C2×C3×C4C2411yescharacteristic
181#67—42no—
182#67—42no—
183#67—42no—
184C2×Dic3—21no—
185C2×Dic3—21no—
186C2×Dic3C2×D411yescharacteristic
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189D12—21no—
190D12—21no—
191D12—21no—
192D12—21no—
193D12—42no—
194D12—42no—
195C4×D3—21no—
196C4×D3—21no—
197C4×D3—22no—
198C4×D3—41no—
199C4×D3—42no—
200C4×D3—42no—
201Dic6—21no—
202Dic6—21no—
203Dic6—42no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—124no—
209C2×Q8—31no—
210C2×Q8—61no—
211C2×D4—31no—
212C2×D4—31no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4—61no—
218C2×D4C22×D311yescharacteristic
219C22×C4—21no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—61no—
224C22×C4—61no—
225SD16—121no—
226SD16—121no—
227D8—121no—
228D8—121no—
229M16—121no—
230M16—121no—
231C2×C8—31no—
232C2×C8—31no—
233C2×C8C22×D311yescharacteristic
234C2×C8C22×D311yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4C22×D311yescharacteristic
242C4⋊C4C22×D311yescharacteristic
243C4⋊C4C22×D311yescharacteristic
244C22⋊C4—21no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C42—31no—
254C42—31no—
255C42—31no—
256C42—61no—
257C42C22×D311yescharacteristic
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3C22×D411yescharacteristic
262C2×D3—21no—
263C2×D3—21no—
264C2×D3—21no—
265C2×D3—41no—
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—21no—
272C3×C4—21no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4H(5,𝔽2)11yescharacteristic
277C3×C42−1+411yescharacteristic
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282Dic3AGL(1,ℤ/8)12yes—
283C23—21no—
284C23—31no—
285C23—61no—
286C23—61no—
287Q8—61no—
288Q8—61no—
289Q8—122no—
290D4—21no—
291D4—21no—
292D4—61no—
293D4—61no—
294D4—61no—
295D4—61no—
296D4—122no—
297D4—122no—
298D4—122no—
299D4—122no—
300D4—122no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—42no—
308C2×C4—62no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—121no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4D3×D411yescharacteristic
317C2×C4D3×D411yescharacteristic
318C2×C4C23×D311yescharacteristic
319C8—21no—
320C8—21no—
321C8—61no—
322C8—61no—
323C2×C3—81no—
324C2×C3#53711yescharacteristic
325C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
326C2×C3#57511yescharacteristic
327D3—41no—
328D3—81no—
329D3—81no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—61no—
334C22—61no—
335C22—61no—
336C22—121no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22—121no—
341C22—121no—
342C22C2×D3×D411yescharacteristic
343C4—21no—
344C4—21no—
345C4—32no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—61no—
350C4—61no—
351C4—121no—
352C4—121no—
353C4#100711yescharacteristic
354C4#100911yescharacteristic
355C3#325111yescharacteristic, complemented, Sylow
356C2—81no—
357C2—121no—
358C2—241no—
359C2—241no—
360C2#562911yescharacteristic
361C2#576811yescharacteristic
362C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#478C2
socleC22×C3C22×D4
hypercenter#478D3
hypocenterC3#3251
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #478

Complements

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

NH ≅ G/Ntype
1. #5691C2semidirect
2. #5548C2semidirect
3. C3×#478C2semidirect
4. #5066C2semidirect
5. #5016C2semidirect
6. #4890C2semidirect
7. #4886C2semidirect
8. #4865C2semidirect
9. #4854C2semidirect
10. #4844C2semidirect
11. #4837C2semidirect
12. #4786C2semidirect
13. #4782C2semidirect
14. #4780C2semidirect
15. #4762C2semidirect
63. #478D3semidirect
355. C3#3251semidirect

Maximal subgroups

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

subgroupindex
#56912
#55482
C3×#4782
#50662
#50162
#48902
#48862
#48652
#48542
#48442
#48372
#47862
#47822
#47802
#47622
#32513