Finite Group Databasebeta

← #80813

Subgroups — #80813

Order 384.

414 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×C22⋊Q8C211yesmaximal, characteristic, complemented
2#5606C211yesmaximal, characteristic
3C3×#468C211yesmaximal, characteristic, complemented
4#5173C211yesmaximal, characteristic, complemented
5#5170C211yesmaximal, characteristic, complemented
6#5137C211yesmaximal, characteristic, complemented
7#5126C211yesmaximal, characteristic, complemented
8#5052C211yesmaximal, characteristic, complemented
9#5042C211yesmaximal, characteristic, complemented
10#4888C211yesmaximal, characteristic, complemented
11#4884C211yesmaximal, characteristic, complemented
12#4815C211yesmaximal, characteristic, complemented
13#4809C211yesmaximal, characteristic, complemented
14#4783C211yesmaximal, characteristic, complemented
15#4777C211yesmaximal, characteristic, complemented
16#3196—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C2×C3×M16C2211yescharacteristic
22C3×C22⋊Q8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic
25C2×#216C2211yescharacteristic, complemented
26#938C2211yescharacteristic
27#937—21no—
28D4×Dic3—21no—
29C2×#214C2211yescharacteristic, complemented
30#924C2211yescharacteristic
31#924C2211yescharacteristic
32C2×#208C2211yescharacteristic
33C2×Dic12C2211yescharacteristic, complemented
34#897C2211yescharacteristic
35#897C2211yescharacteristic
36#896—21no—
37#892C2211yescharacteristic
38D3×C4⋊C4—21no—
39D3×C4⋊C4C2211yescharacteristic
40#883—21no—
41#67⋊C4—21no—
42D3×C22⋊C4—21no—
43#878—21no—
44C3×#107C2211yescharacteristic, complemented
45C3×#103C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic, complemented
47#835C2211yescharacteristic, complemented
48#832C2211yescharacteristic, complemented
49(C2×D3)⋊C8C2211yescharacteristic
50(C2×D3)⋊C8C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53#817C2211yescharacteristic, complemented
54#816C2211yescharacteristic, complemented
55#816C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#525—31no—
61C2×C22⋊Q8—31no—
62#485—31no—
63#485—31no—
64#484—31no—
65#481—31no—
66#470—31no—
67#470—31no—
68#468—31no—
69#468D311yescharacteristic, complemented
70#452—31no—
71#452—31no—
72#451—31no—
73#448—31no—
74#407—31no—
75C23×D3D411yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C22×Dic3—21no—
83C22×Dic3D411yescharacteristic
84D3×Q8—41no—
85D3×Q8—41no—
86#237—41no—
87#237—41no—
88C2×C4×D3—21no—
89C2×C4×D3—21no—
90C2×C4×D3—41no—
91C2×C4×D3—41no—
92C2×C4×D3D411yescharacteristic
93C2×C4×D3D411yescharacteristic
94C2×C4×D3C2311yescharacteristic
95C2×C4×D3C2311yescharacteristic
96C2×Dic6—21no—
97C2×Dic6C2311yescharacteristic
98C2×Dic6C2311yescharacteristic
99C3×SD16—41no—
100C3×M16—41no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C3×C4⋊C4—21no—
104C3×C4⋊C4C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108#217—21no—
109#217—21no—
110#217—21no—
111#216—41no—
112#214—41no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3C2311yescharacteristic
120C4⋊Dic3C2311yescharacteristic
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125C4×Dic3—21no—
126#208—41no—
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129Dic12—41no—
130C2×#41—31no—
131C22×Q8—31no—
132C23×C4—31no—
133C2×Q16—31no—
134C2×Q16—31no—
135C2×SD16—31no—
136C2×SD16C2×D311yescharacteristic, complemented
137C2×M16—31no—
138C2×M16C2×D311yescharacteristic
139#123—61no—
140C22⋊Q8—31no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—31no—
144C22⋊Q8—31no—
145C22⋊Q8—61no—
146C22⋊Q8—61no—
147C22⋊Q8C2×D311yescharacteristic
148#121—31no—
149#121C2×D311yescharacteristic
150#120—61no—
151C4×D4—61no—
152C4×D4—61no—
153#117—31no—
154C2×C4⋊C4—31no—
155C2×C4⋊C4—61no—
156C2×C22⋊C4—61no—
157#107—31no—
158#107C2×D311yescharacteristic, complemented
159#106—31no—
160#106—31no—
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103—31no—
166#103C2×D311yescharacteristic, complemented
167#102—31no—
168#102C2×D311yescharacteristic, complemented
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—22no—
176C22×D3—41no—
177C22×D3—41no—
178C22×D3C2×D411yescharacteristic
179C22×D3C2×D411yescharacteristic
180C3×Q8—21no—
181C3×Q8—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—42no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2411yescharacteristic
192#67—42no—
193#67—42no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—41no—
199C2×Dic3—42no—
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C4×D3—21no—
203C4×D3—21no—
204C4×D3—21no—
205C4×D3—21no—
206C4×D3—41no—
207C4×D3—41no—
208C4×D3—42no—
209C4×D3—42no—
210Dic6—21no—
211Dic6—21no—
212Dic6—21no—
213Dic6—21no—
214Dic6—42no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—31no—
220#41—121no—
221#41—121no—
222C2×Q8—31no—
223C2×Q8—31no—
224C2×Q8—61no—
225C2×Q8—121no—
226C2×Q8—121no—
227C2×Q8C22×D311yescharacteristic
228C2×D4—21no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4C22×D311yescharacteristic
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—121no—
241C22×C4—121no—
242C22×C4C22×D311yescharacteristic
243Q16—121no—
244Q16—121no—
245SD16—41no—
246SD16—121no—
247M16—41no—
248M16—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8C22×D311yescharacteristic
252C2×C8C22×D311yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4C22×D311yescharacteristic
263C4⋊C4C22×D311yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—21no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C42—61no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3C22×D411yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—44no—
287C2×D3#13712yes—
288C3×C4—21no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4H(5,𝔽2)11yescharacteristic
293C3×C4H(5,𝔽2)11yescharacteristic
294Dic3—21no—
295Dic3—21no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300C23—21no—
301C23—31no—
302C23—31no—
303C23—62no—
304C23—121no—
305C23—121no—
306C23D3×D411yescharacteristic
307Q8—21no—
308Q8—21no—
309Q8—61no—
310Q8—61no—
311Q8—61no—
312Q8—61no—
313Q8—122no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—122no—
318D4—122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—121no—
335C2×C4—121no—
336C2×C4—122no—
337C2×C4—122no—
338C2×C4—121no—
339C2×C4—122no—
340C2×C4D3×D411yescharacteristic
341C2×C4C23×D311yescharacteristic
342C8—21no—
343C8—21no—
344C8—61no—
345C8—61no—
346C2×C3—41no—
347C2×C3—81no—
348C2×C3#53411yescharacteristic
349C2×C3C2×#13711yescharacteristic
350C2×C3#57511yescharacteristic
351D3—22no—
352D3—41no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—32no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—61no—
361C22—61no—
362C22—61no—
363C22—61no—
364C22—61no—
365C22—124no—
366C22C2×D3×D411yescharacteristic
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—61no—
372C4—61no—
373C4—121no—
374C4—121no—
375C4—121no—
376C4—121no—
377C4#100411yescharacteristic
378C4#100911yescharacteristic
379C3#319611yescharacteristic, complemented, Sylow
380C2—41no—
381C2—62no—
382C2—81no—
383C2—121no—
384C2#560511yescharacteristic
385C2#577311yescharacteristic
386C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
prime radical#5606C2
hypercenter#468D3
hypocenterC3#3196
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. D3×C22⋊Q8C2semidirect
3. C3×#468C2semidirect
4. #5173C2semidirect
5. #5170C2semidirect
6. #5137C2semidirect
7. #5126C2semidirect
8. #5052C2semidirect
9. #5042C2semidirect
10. #4888C2semidirect
11. #4884C2semidirect
12. #4815C2semidirect
13. #4809C2semidirect
14. #4783C2semidirect
15. #4777C2semidirect
20. C2×C3×SD16C22semidirect
25. C2×#216C22semidirect
29. C2×#214C22semidirect
33. C2×Dic12C22semidirect
44. C3×#107C22semidirect
45. C3×#103C22semidirect
46. C3×#102C22semidirect
47. #835C22semidirect
48. #832C22semidirect
53. #817C22semidirect
54. #816C22semidirect
55. #816C22semidirect
56. #810C22semidirect
57. #810C22semidirect
58. #808C22semidirect
59. #807C22semidirect
69. #468D3semidirect
136. C2×SD16C2×D3semidirect
158. #107C2×D3semidirect
166. #103C2×D3semidirect
168. #102C2×D3semidirect
379. C3#3196semidirect

Maximal subgroups

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

subgroupindex
D3×C22⋊Q82
#56062
C3×#4682
#51732
#51702
#51372
#51262
#50522
#50422
#48882
#48842
#48152
#48092
#47832
#47772
#31963