Finite Group Databasebeta

← #79804

Subgroups — #79804

Order 384.

388 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
1Q8×D12C211yesmaximal, characteristic, complemented
2#5558C211yesmaximal, characteristic
3C3×#440C211yesmaximal, characteristic, complemented
4#5170C211yesmaximal, characteristic, complemented
5#5170C211yesmaximal, characteristic, complemented
6#5022C211yesmaximal, characteristic, complemented
7#5019C211yesmaximal, characteristic, complemented
8#4888C211yesmaximal, characteristic, complemented
9#4884C211yesmaximal, characteristic, complemented
10#4802C211yesmaximal, characteristic, complemented
11D3×#103C211yesmaximal, characteristic, complemented
12#4780C211yesmaximal, characteristic, complemented
13#4778C211yesmaximal, characteristic, complemented
14#4706C211yesmaximal, characteristic, complemented
15#4703C211yesmaximal, characteristic, complemented
16#3320—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22#938C2211yescharacteristic
23#938C2211yescharacteristic
24C2×#214C2211yescharacteristic
25C2×#214C2211yescharacteristic
26#929—21no—
27C4×#67—21no—
28C2×ΓL(1,𝔽25)C2211yescharacteristic
29C2×D3×C8C2211yescharacteristic
30#897—21no—
31#897—21no—
32#897C2211yescharacteristic
33#897C2211yescharacteristic
34#892C2211yescharacteristic
35D3×C4⋊C4C2211yescharacteristic
36#884—21no—
37#879—21no—
38C4×D12—21no—
39C4×D12C2211yescharacteristic
40#870C2211yescharacteristic
41#869—21no—
42#869C2211yescharacteristic
43C3×#107C2211yescharacteristic
44C3×#103C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×#97C2211yescharacteristic
47#835C2211yescharacteristic
48#835C2211yescharacteristic
49(C2×D3)⋊C8C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51#816C2211yescharacteristic
52#816C2211yescharacteristic
53#816C2211yescharacteristic
54#816C2211yescharacteristic
55#810C2211yescharacteristic
56#810C2211yescharacteristic
57#808C2211yescharacteristic
58#807C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#553—31no—
61D4×Q8—31no—
62#491—31no—
63#485—31no—
64#484—31no—
65#479—31no—
66#468—31no—
67#465—31no—
68#461—31no—
69#448—31no—
70#448—31no—
71#440D311yescharacteristic, complemented
72#435—31no—
73#419—31no—
74C2×#103—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C2×#67—21no—
79C2×#67—21no—
80D3×Q8—22no—
81D3×Q8—22no—
82D3×Q8—41no—
83D3×Q8—41no—
84C2×D12D411yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3—21no—
87C2×C4×D3—21no—
88C2×C4×D3C2311yescharacteristic
89C2×C4×D3C2311yescharacteristic
90C2×Dic6—21no—
91C2×Dic6—21no—
92C2×Dic6C2311yescharacteristic
93C2×Dic6C2311yescharacteristic
94C3×SD16—22no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4C2311yescharacteristic
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101C3×C42—21no—
102C3×C42C2311yescharacteristic
103#217—21no—
104#217—21no—
105#214—41no—
106#214—41no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4D411yescharacteristic
111(C2×D3)⋊C4D411yescharacteristic
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3D411yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122C4×Dic3—21no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125ΓL(1,𝔽25)—41no—
126D3×C8—41no—
127C22×Q8—31no—
128C22×Q8—31no—
129C2×SD16—31no—
130C2×SD16—31no—
131C2×SD16C2×D311yescharacteristic
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135C4⋊Q8—61no—
136#126—61no—
137#125—31no—
138#123—61no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143#121—31no—
144#121—31no—
145#121—61no—
146C4×Q8C2×D311yescharacteristic
147C4×D4—31no—
148C4×D4—61no—
149C4×D4—61no—
150C4×D4C2×D311yescharacteristic
151#117—31no—
152C2×C4⋊C4—31no—
153#107—31no—
154#107C2×D311yescharacteristic
155#106—31no—
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#103—31no—
165#103C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97C2×D311yescharacteristic
170C23×C3—21no—
171C22×D3C2×D411yescharacteristic
172C22×D3C2×D411yescharacteristic
173C3×Q8—21no—
174C3×Q8#4112yes—
175C3×D4—21no—
176C3×D4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—81no—
181C2×C3×C4C2×D411yescharacteristic
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184#67—42no—
185#67—81no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192D12—22no—
193C4×D3—21no—
194C4×D3—21no—
195C4×D3—21no—
196C4×D3—21no—
197C4×D3—42no—
198C4×D3—42no—
199C4×D3—81no—
200Dic6—21no—
201Dic6—21no—
202Dic6—21no—
203Dic6—21no—
204Dic6—42no—
205Dic6—42no—
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—62no—
216C2×Q8—121no—
217C2×Q8—121no—
218C2×Q8C22×D311yescharacteristic
219C2×D4—31no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4C22×D311yescharacteristic
223C22×C4—21no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—61no—
227C22×C4—61no—
228C22×C4—61no—
229SD16—22no—
230SD16—121no—
231SD16—121no—
232M16—121no—
233C2×C8—31no—
234C2×C8—31no—
235C2×C8—121no—
236C2×C8C22×D311yescharacteristic
237C2×C8C22×D311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—31no—
240C4⋊C4—31no—
241C4⋊C4—31no—
242C4⋊C4—61no—
243C4⋊C4—61no—
244C4⋊C4—61no—
245C4⋊C4—61no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4C22×D311yescharacteristic
250C4⋊C4C22×D311yescharacteristic
251C22⋊C4—21no—
252C22⋊C4—31no—
253C22⋊C4—31no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C42—21no—
260C42—61no—
261C42C22×D311yescharacteristic
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3C22×D411yescharacteristic
266C2×D3—21no—
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—21no—
270C2×D3—21no—
271C2×D3#13712yes—
272C3×C4—22no—
273C3×C4—21no—
274C3×C4—21no—
275C3×C4—41no—
276C3×C4—41no—
277C3×C4C2×#4111yescharacteristic
278C3×C4H(5,𝔽2)11yescharacteristic
279Dic3—21no—
280Dic3—21no—
281Dic3—41no—
282Dic3—41no—
283Dic3—41no—
284Dic3—41no—
285C23—21no—
286C23—31no—
287C23—31no—
288Q8—21no—
289Q8—61no—
290Q8—61no—
291Q8—61no—
292Q8—61no—
293Q8—122no—
294Q8—122no—
295Q8#23512yes—
296D4—21no—
297D4—21no—
298D4—62no—
299D4—122no—
300D4—241no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—61no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—61no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—81no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4—241no—
318C2×C4D3×D411yescharacteristic
319C2×C4D3×D411yescharacteristic
320C2×C4C23×D311yescharacteristic
321C8—21no—
322C8—21no—
323C8—61no—
324C8—61no—
325C2×C3—81no—
326C2×C3#54611yescharacteristic
327C2×C3C2×#13711yescharacteristic
328C2×C3#57711yescharacteristic
329D3—22no—
330D3—41no—
331C22—32no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—61no—
336C22—61no—
337C22—61no—
338C22—61no—
339C22—61no—
340C22C2×D3×D411yescharacteristic
341C4—21no—
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—61no—
347C4—61no—
348C4—121no—
349C4—121no—
350C4—121no—
351C4—121no—
352C4C2×#23511yescharacteristic
353C4#100911yescharacteristic
354C3#332011yescharacteristic, complemented, Sylow
355C2—62no—
356C2—81no—
357C2—121no—
358C2#555011yescharacteristic
359C2#577511yescharacteristic
360C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
prime radical#5558C2
hypercenter#440D3
hypocenterC3#3320
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. Q8×D12C2semidirect
3. C3×#440C2semidirect
4. #5170C2semidirect
5. #5170C2semidirect
6. #5022C2semidirect
7. #5019C2semidirect
8. #4888C2semidirect
9. #4884C2semidirect
10. #4802C2semidirect
11. D3×#103C2semidirect
12. #4780C2semidirect
13. #4778C2semidirect
14. #4706C2semidirect
15. #4703C2semidirect
71. #440D3semidirect
354. C3#3320semidirect

Maximal subgroups

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

subgroupindex
Q8×D122
#55582
C3×#4402
#51702
#50222
#50192
#48882
#48842
#48022
D3×#1032
#47802
#47782
#47062
#47032
#33203