Finite Group Databasebeta

← #83498

Subgroups — #83498

Order 384.

410 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#546C211yesmaximal, characteristic
2#5543C211yesmaximal, characteristic, complemented
3#5239C211yesmaximal, characteristic, complemented
4#5232C211yesmaximal, characteristic, complemented
5#5218C211yesmaximal, characteristic, complemented
6C2×#832C211yesmaximal, characteristic, complemented
7#5055C211yesmaximal, characteristic, complemented
8#5054C211yesmaximal, characteristic, complemented
9#5051C211yesmaximal, characteristic, complemented
10#5044C211yesmaximal, characteristic, complemented
11#5042C211yesmaximal, characteristic, complemented
12#5033C211yesmaximal, characteristic, complemented
13#5018C211yesmaximal, characteristic, complemented
14#5011C211yesmaximal, characteristic, complemented
15#5007C211yesmaximal, characteristic, complemented
16#3314—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C3×#124C2211yescharacteristic
20C3×#123—21no—
21C3×C22⋊Q8C2211yescharacteristic
22C3×#121—21no—
23C3×#121C2211yescharacteristic
24C3×#120—21no—
25C3×C4×D4C2211yescharacteristic
26C3×C4×D4C2211yescharacteristic
27C2×C3×C22⋊C4—21no—
28C2×#216C2211yescharacteristic, complemented
29D4×Dic3—21no—
30C2×#214C2211yescharacteristic
31C2×#214C2211yescharacteristic
32C2×C4⋊Dic3C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic, complemented
34#924C2211yescharacteristic
35#924C2211yescharacteristic, complemented
36C2×#208C2211yescharacteristic, complemented
37C22×C3⋊C8C2211yescharacteristic
38#890—21no—
39#878—21no—
40#878—21no—
41#869C2211yescharacteristic, complemented
42C4×Dic6C2211yescharacteristic, complemented
43#835C2211yescharacteristic, complemented
44#835C2211yescharacteristic, complemented
45#835C2211yescharacteristic, complemented
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#832C2211yescharacteristic
52#830C2211yescharacteristic
53#830C2211yescharacteristic, complemented
54#810C2211yescharacteristic, complemented
55#807C2211yescharacteristic, complemented
56#807C2211yescharacteristic, complemented
57#807C2211yescharacteristic, complemented
58C4⋊(C3⋊C8)C2211yescharacteristic, complemented
59#803C2211yescharacteristic, complemented
60#554—31no—
61#546D311yescharacteristic
62#489—31no—
63#484—31no—
64#480—31no—
65#474—31no—
66#468—31no—
67#467—31no—
68#464—31no—
69#452—31no—
70#450—31no—
71#440—31no—
72#435—31no—
73#417—31no—
74C2×#102—31no—
75C24×C3—21no—
76C3×#41—22no—
77C3×#41—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—22no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—41no—
84C2×C3×D4D411yescharacteristic
85C2×C3×D4C2311yescharacteristic
86C2×C3×D4C2311yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C22×Dic3—21no—
92C22×Dic3—21no—
93C2×Dic6—21no—
94C2×Dic6C2311yescharacteristic
95C3×C4⋊C4—21no—
96C3×C4⋊C4D411yescharacteristic
97C3×C4⋊C4C2311yescharacteristic
98C3×C22⋊C4—21no—
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4D411yescharacteristic
104C3×C22⋊C4D411yescharacteristic
105C3×C42C2311yescharacteristic
106#217—21no—
107#217—21no—
108#216—41no—
109#214—22no—
110#214—41no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3C2311yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120C4×Dic3—21no—
121#208—41no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×#41C2×D311yescharacteristic
128C22×D4C2×D311yescharacteristic
129C2×Q16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×M16—31no—
133C22×C8—31no—
134C4⋊Q8—31no—
135#125—61no—
136#124C2×D311yescharacteristic
137#123—21no—
138C22⋊Q8—31no—
139C22⋊Q8—31no—
140C22⋊Q8—61no—
141C22⋊Q8—61no—
142C22⋊Q8C2×D311yescharacteristic
143#121—21no—
144#121C2×D311yescharacteristic
145#120—21no—
146C4×Q8—31no—
147C4×D4—61no—
148C4×D4C2×D311yescharacteristic
149C4×D4C2×D311yescharacteristic
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C22⋊C4—21no—
153#107—31no—
154#107—31no—
155#107—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#103—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169#97—31no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—81no—
175C23×C3C2×D411yescharacteristic
176C23×C3C2×D411yescharacteristic
177C3×Q8—21no—
178C3×Q8—21no—
179C3×D4—22no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—42no—
184C3×D4—42no—
185C3×D4—81no—
186C3×D4#4112yes—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—42no—
194C2×C3×C4—81no—
195C2×C3×C4C2×D411yescharacteristic
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2411yescharacteristic
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—81no—
205C2×Dic3—81no—
206Dic6—21no—
207Dic6—21no—
208Dic6—81no—
209C3⋊C8—21no—
210C3⋊C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213C24—21no—
214#41—22no—
215#41—41no—
216C2×Q8—31no—
217C2×Q8—61no—
218C2×Q8C22×D311yescharacteristic
219C2×D4—22no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—41no—
224C2×D4#6711yescharacteristic
225C2×D4C22×D311yescharacteristic
226C2×D4C22×D311yescharacteristic
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—61no—
230C22×C4—61no—
231C22×C4C22×D311yescharacteristic
232C22×C4C22×D311yescharacteristic
233Q16—121no—
234SD16—62no—
235SD16—121no—
236M16—121no—
237C2×C8—31no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8—121no—
242C4⋊C4—21no—
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—61no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4#6711yescharacteristic
253C4⋊C4C22×D311yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4#6711yescharacteristic
262C22⋊C4#6711yescharacteristic
263C42—61no—
264C42C22×D311yescharacteristic
265C22×C3—21no—
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—22no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—84no—
277C22×C3#13712yes—
278C22×C3C22×D411yescharacteristic
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4C2×#4111yescharacteristic
286C3×C4H(5,𝔽2)11yescharacteristic
287Dic3—41no—
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291C23—21no—
292C23—21no—
293C23—42no—
294C23—42no—
295C23—81no—
296C23C2×#6711yescharacteristic
297C23C2×#6711yescharacteristic
298Q8—21no—
299Q8—21no—
300Q8—61no—
301Q8—61no—
302Q8—241no—
303D4—22no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—42no—
308D4—42no—
309D4—81no—
310D4#23712yes—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—42no—
318C2×C4—61no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—81no—
323C2×C4—241no—
324C2×C4—241no—
325C2×C4C2×#6711yescharacteristic
326C2×C4C2×#6711yescharacteristic
327C2×C4C2×#6711yescharacteristic
328C2×C4C2×#6711yescharacteristic
329C2×C4C23×D311yescharacteristic
330C8—61no—
331C8—61no—
332C8—61no—
333C8—61no—
334C2×C3—22no—
335C2×C3—41no—
336C2×C3—42no—
337C2×C3—81no—
338C2×C3#54611yescharacteristic
339C2×C3C2×#13711yescharacteristic
340C2×C3#57611yescharacteristic
341C22—21no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—21no—
346C22—21no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—84no—
353C22#95112yes—
354C22C22×#6711yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—121no—
362C4—121no—
363C4—121no—
364C4—121no—
365C4C2×#23711yescharacteristic
366C4#100411yescharacteristic
367C3#331411yescharacteristic, complemented, Sylow
368C2—22no—
369C2—41no—
370C2—42no—
371C2—81no—
372C2#580611yescharacteristic
373C2C2×#95111yescharacteristic
374C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#546C2
socleC22×C3C22×D4
hypercenter#546D3
hypocenterC3#3314
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
2. #5543C2semidirect
3. #5239C2semidirect
4. #5232C2semidirect
5. #5218C2semidirect
6. C2×#832C2semidirect
7. #5055C2semidirect
8. #5054C2semidirect
9. #5051C2semidirect
10. #5044C2semidirect
11. #5042C2semidirect
12. #5033C2semidirect
13. #5018C2semidirect
14. #5011C2semidirect
15. #5007C2semidirect
28. C2×#216C22semidirect
33. C2×C4⋊Dic3C22semidirect
35. #924C22semidirect
36. C2×#208C22semidirect
41. #869C22semidirect
42. C4×Dic6C22semidirect
43. #835C22semidirect
44. #835C22semidirect
45. #835C22semidirect
53. #830C22semidirect
54. #810C22semidirect
55. #807C22semidirect
56. #807C22semidirect
57. #807C22semidirect
58. C4⋊(C3⋊C8)C22semidirect
59. #803C22semidirect
367. C3#3314semidirect

Maximal subgroups

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

subgroupindex
C3×#5462
#55432
#52392
#52322
#52182
C2×#8322
#50552
#50542
#50512
#50442
#50422
#50332
#50182
#50112
#50072
#33143