Finite Group Databasebeta

← #80235

Subgroups — #80235

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
1#5828C211yesmaximal, characteristic
2D3×C22⋊Q8C211yesmaximal, characteristic, complemented
3C3×#452C211yesmaximal, characteristic, complemented
4#5239C211yesmaximal, characteristic, complemented
5#5189C211yesmaximal, characteristic, complemented
6#5186C211yesmaximal, characteristic, complemented
7#5174C211yesmaximal, characteristic, complemented
8#5170C211yesmaximal, characteristic, complemented
9#5045C211yesmaximal, characteristic, complemented
10#4816C211yesmaximal, characteristic, complemented
11#4811C211yesmaximal, characteristic, complemented
12#4780C211yesmaximal, characteristic, complemented
13#4777C211yesmaximal, characteristic, complemented
14#4726C211yesmaximal, characteristic, complemented
15#4723C211yesmaximal, characteristic, complemented
16#3196—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C2×C3×Q16C2211yescharacteristic, complemented
21C2×C3×SD16C2211yescharacteristic, complemented
22C3×C22⋊Q8C2211yescharacteristic
23#946C2211yescharacteristic
24#946C2211yescharacteristic
25#946C2211yescharacteristic
26C2×#216C2211yescharacteristic, complemented
27#938C2211yescharacteristic
28#938C2211yescharacteristic
29#937—21no—
30#935—21no—
31C2×#214C2211yescharacteristic, complemented
32C4×#67—21no—
33C4×#67—21no—
34#926C2211yescharacteristic
35#924C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37C2×ΓL(1,𝔽25)C2211yescharacteristic
38#897C2211yescharacteristic
39#896—21no—
40D3×C4⋊C4—21no—
41D3×C4⋊C4C2211yescharacteristic
42D3×C22⋊C4—21no—
43#878—21no—
44C3×#103C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46C3×C22⋊C8C2211yescharacteristic
47#835C2211yescharacteristic, complemented
48#835C2211yescharacteristic, complemented
49#835C2211yescharacteristic, complemented
50#832C2211yescharacteristic, complemented
51#830C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#818C2211yescharacteristic, complemented
55#817C2211yescharacteristic, complemented
56#816C2211yescharacteristic, complemented
57#816C2211yescharacteristic, 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#468—31no—
70#452—31no—
71#452D311yescharacteristic, complemented
72#451—31no—
73#448—31no—
74#407—31no—
75C23×D3D411yescharacteristic
76C3×#41—41no—
77C3×#41—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C22×Dic3D411yescharacteristic
87D3×Q8—41no—
88D3×Q8—41no—
89C2×C4×D3—21no—
90C2×C4×D3—21no—
91C2×C4×D3—41no—
92C2×C4×D3—41no—
93C2×C4×D3D411yescharacteristic
94C2×C4×D3D411yescharacteristic
95C2×C4×D3C2311yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C2×Dic6—21no—
98C2×Dic6C2311yescharacteristic
99C3×Q16—41no—
100C3×SD16—41no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C3×C4⋊C4—21no—
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106#217—21no—
107#217—21no—
108#217—21no—
109#217—21no—
110#216—41no—
111#214—41no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116C4⋊Dic3—21no—
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125C4×Dic3—21no—
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128ΓL(1,𝔽25)—41no—
129ΓL(1,𝔽25)—41no—
130C2×#41C2×D311yescharacteristic
131C22×Q8—31no—
132C23×C4—31no—
133C2×Q16—31no—
134C2×Q16C2×D311yescharacteristic, complemented
135C2×SD16—31no—
136C2×SD16C2×D311yescharacteristic, complemented
137C2×M16—31no—
138C2×M16—31no—
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#121—31no—
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#107—31no—
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⋊C8C2×D311yescharacteristic
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×Q8—21no—
183C3×Q8—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—42no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—42no—
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2411yescharacteristic
195#67—42no—
196#67—42no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—41no—
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204C4×D3—21no—
205C4×D3—21no—
206C4×D3—21no—
207C4×D3—21no—
208C4×D3—41no—
209C4×D3—41no—
210C4×D3—42no—
211C4×D3—42no—
212Dic6—21no—
213Dic6—21no—
214Dic6—42no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—31no—
220#41—41no—
221#41—41no—
222C2×Q8—31no—
223C2×Q8—61no—
224C2×Q8—121no—
225C2×Q8—121no—
226C2×Q8C22×D311yescharacteristic
227C2×Q8C22×D311yescharacteristic
228C2×D4—21no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4C22×D311yescharacteristic
232C22×C4—21no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4—121no—
241C22×C4—121no—
242C22×C4C22×D311yescharacteristic
243Q16—41no—
244Q16—121no—
245SD16—41no—
246SD16—121no—
247M16—121no—
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—31no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4—61no—
262C4⋊C4—61no—
263C4⋊C4C22×D311yescharacteristic
264C22⋊C4—21no—
265C22⋊C4—61no—
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—21no—
310Q8—21no—
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—42no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—121no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—121no—
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#100411yescharacteristic
379C3#319611yescharacteristic, complemented, Sylow
380C2—41no—
381C2—62no—
382C2—81no—
383C2—121no—
384C2#558911yescharacteristic
385C2#576811yescharacteristic
386C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
prime radical#5828C2
hypercenter#452D3
hypocenterC3#3196
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. D3×C22⋊Q8C2semidirect
3. C3×#452C2semidirect
4. #5239C2semidirect
5. #5189C2semidirect
6. #5186C2semidirect
7. #5174C2semidirect
8. #5170C2semidirect
9. #5045C2semidirect
10. #4816C2semidirect
11. #4811C2semidirect
12. #4780C2semidirect
13. #4777C2semidirect
14. #4726C2semidirect
15. #4723C2semidirect
20. C2×C3×Q16C22semidirect
21. C2×C3×SD16C22semidirect
26. C2×#216C22semidirect
31. C2×#214C22semidirect
44. C3×#103C22semidirect
45. C3×#102C22semidirect
47. #835C22semidirect
48. #835C22semidirect
49. #835C22semidirect
50. #832C22semidirect
54. #818C22semidirect
55. #817C22semidirect
56. #816C22semidirect
57. #816C22semidirect
58. #808C22semidirect
59. #807C22semidirect
71. #452D3semidirect
134. C2×Q16C2×D3semidirect
136. C2×SD16C2×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
#58282
D3×C22⋊Q82
C3×#4522
#52392
#51892
#51862
#51742
#51702
#50452
#48162
#48112
#47802
#47772
#47262
#47232
#31963