Finite Group Databasebeta

← #80874

Subgroups — #80874

Order 384.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5612C211yesmaximal, characteristic, complemented
2#5611C211yesmaximal, characteristic, complemented
3C3×#469C211yesmaximal, characteristic, complemented
4#5156C211yesmaximal, characteristic, complemented
5#5153C211yesmaximal, characteristic, complemented
6#5133C211yesmaximal, characteristic, complemented
7#5127C211yesmaximal, characteristic, complemented
8#5037C211yesmaximal, characteristic, complemented
9#5034C211yesmaximal, characteristic, complemented
10#4870C211yesmaximal, characteristic, complemented
11#4862C211yesmaximal, characteristic, complemented
12#4785C211yesmaximal, characteristic, complemented
13#4778C211yesmaximal, characteristic, complemented
14D3×#102C211yesmaximal, characteristic, complemented
15#4765C211yesmaximal, characteristic, complemented
16#3306—31nomaximal, Sylow
17C2×#237—21no—
18C2×D3×D4C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic, complemented
22C3×#121C2211yescharacteristic
23C3×#121C2211yescharacteristic, complemented
24#938—21no—
25#938C2211yescharacteristic
26#937—21no—
27#936C2211yescharacteristic
28#935—21no—
29D4×Dic3C2211yescharacteristic, complemented
30C2×#214C2211yescharacteristic, complemented
31C2×#213C2211yescharacteristic
32C4×#67C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic, complemented
34C2×D3×C8C2211yescharacteristic
35#897C2211yescharacteristic
36#893C2211yescharacteristic
37D3×C4⋊C4C2211yescharacteristic
38#888C2211yescharacteristic, complemented
39#883—21no—
40#883—21no—
41D3×C22⋊C4—21no—
42#878—21no—
43C3×#106C2211yescharacteristic, complemented
44C3×#102C2211yescharacteristic
45C3×#102C2211yescharacteristic, complemented
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#830C2211yescharacteristic, complemented
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52#816C2211yescharacteristic, complemented
53#816C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#808C2211yescharacteristic, complemented
60#547—31no—
61#546—31no—
62#489—31no—
63#483—31no—
64#481—31no—
65#476—31no—
66#469D311yescharacteristic, complemented
67#465—31no—
68#463—31no—
69#449—31no—
70#447—31no—
71#442—31no—
72#435—31no—
73#417—31no—
74C2×#102—31no—
75C23×D3—21no—
76C2×C3×D4—21no—
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67D411yescharacteristic
85C22×Dic3—21no—
86C22×Dic3—21no—
87#237—44no—
88D3×D4—22no—
89D3×D4—41no—
90#235—22no—
91#235—41no—
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3—21no—
96C2×C4×D3C2311yescharacteristic
97C2×Dic6—21no—
98C2×Dic6C2311yescharacteristic
99C3×D8—41no—
100C3×M16—41no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106C3×C22⋊C4—21no—
107#217—21no—
108#217—21no—
109#217—21no—
110#217—21no—
111#217D411yescharacteristic
112#214—41no—
113#213—22no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4D411yescharacteristic
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3C2311yescharacteristic
120C4⋊Dic3C2311yescharacteristic
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4D411yescharacteristic
124C4×Dic3C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127D3×C8—41no—
128C2×#41—31no—
129C2×#41—61no—
130C22×D4—31no—
131C2×SD16—31no—
132C2×D8—31no—
133C2×D8C2×D311yescharacteristic
134C2×M16C2×D311yescharacteristic, complemented
135C22×C8—31no—
136C4⋊Q8—31no—
137#124—31no—
138#123—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143#121—31no—
144#121—61no—
145#121C2×D311yescharacteristic
146#121C2×D311yescharacteristic, complemented
147#120—61no—
148C4×D4—31no—
149C4×D4—31no—
150C4×D4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C22⋊C4—61no—
154#106—31no—
155#106—31no—
156#106C2×D311yescharacteristic, complemented
157C4⋊C8—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#102C2×D311yescharacteristic
167#102C2×D311yescharacteristic, complemented
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97—31no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3C2×D411yescharacteristic
174C22×D3—21no—
175C22×D3—42no—
176C22×D3—42no—
177C22×D3—81no—
178C22×D3C2×D411yescharacteristic
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—42no—
184C3×D4—81no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2411yescharacteristic
191#67—22no—
192#67—42no—
193#67—42no—
194#67—42no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12#4112yescomplemented
205C4×D3—21no—
206C4×D3—21no—
207C4×D3—42no—
208C4×D3—42no—
209C4×D3—81no—
210Dic6—21no—
211Dic6—21no—
212Dic6—42no—
213C3×C8—21no—
214C3×C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217C24—61no—
218#41—62no—
219#41—121no—
220#41—124no—
221C2×Q8—31no—
222C2×Q8—61no—
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—61no—
228C2×D4—62no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—121no—
232C2×D4C22×D311yescharacteristic
233C2×D4C22×D311yescharacteristic
234C22×C4—31no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4C22×D311yescharacteristic
241SD16—121no—
242D8—41no—
243D8—62no—
244M16—41no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—121no—
248C2×C8C22×D311yescharacteristic
249C2×C8C22×D311yescharacteristic
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4C22×D311yescharacteristic
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—31no—
262C22⋊C4—31no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C42—31no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3C22×D411yescharacteristic
280C2×D3—21no—
281C2×D3—21no—
282C2×D3—22no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—84no—
286C2×D3AGL(1,ℤ/8)12yes—
287C3×C4—21no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4C2×#4111yescharacteristic
291C3×C4H(5,𝔽2)11yescharacteristic
292Dic3—21no—
293Dic3—21no—
294Dic3—21no—
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298C23—21no—
299C23—21no—
300C23—31no—
301C23—61no—
302C23—122no—
303C23—122no—
304C23—241no—
305C23D3×D411yescharacteristic
306Q8—61no—
307Q8—61no—
308Q8—122no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—32no—
314D4—42no—
315D4—62no—
316D4—61no—
317D4—81no—
318D4—122no—
319D4—122no—
320D4—122no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—122no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—241no—
338C2×C4D3×D411yescharacteristic
339C2×C4C23×D311yescharacteristic
340C8—21no—
341C8—21no—
342C8—61no—
343C8—61no—
344C2×C3—41no—
345C2×C3—81no—
346C2×C3—81no—
347C2×C3#54611yescharacteristic
348C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
349C2×C3#57711yescharacteristic
350D3—22no—
351D3—42no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—32no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—61no—
363C22—61no—
364C22—62no—
365C22—121no—
366C22—121no—
367C22—244no—
368C22C2×D3×D411yescharacteristic
369C4—21no—
370C4—41no—
371C4—41no—
372C4—61no—
373C4—61no—
374C4—61no—
375C4—121no—
376C4—121no—
377C4—121no—
378C4#100411yescharacteristic
379C4D3×#4111yescharacteristic
380C3#330611yescharacteristic, complemented, Sylow
381C2—41no—
382C2—62no—
383C2—81no—
384C2—81no—
385C2—122no—
386C2#561011yescharacteristic
387C2D3×AGL(1,ℤ/8)11yescharacteristic
388C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3306
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. #5612C2semidirect
2. #5611C2semidirect
3. C3×#469C2semidirect
4. #5156C2semidirect
5. #5153C2semidirect
6. #5133C2semidirect
7. #5127C2semidirect
8. #5037C2semidirect
9. #5034C2semidirect
10. #4870C2semidirect
11. #4862C2semidirect
12. #4785C2semidirect
13. #4778C2semidirect
14. D3×#102C2semidirect
15. #4765C2semidirect
21. C2×C3×M16C22semidirect
23. C3×#121C22semidirect
29. D4×Dic3C22semidirect
30. C2×#214C22semidirect
33. C2×C4⋊Dic3C22semidirect
38. #888C22semidirect
43. C3×#106C22semidirect
45. C3×#102C22semidirect
48. #830C22semidirect
52. #816C22semidirect
53. #816C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
56. #810C22semidirect
58. #808C22semidirect
59. #808C22semidirect
66. #469D3semidirect
134. C2×M16C2×D3semidirect
146. #121C2×D3semidirect
156. #106C2×D3semidirect
167. #102C2×D3semidirect
204. D12#41semidirect
380. C3#3306semidirect

Maximal subgroups

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

subgroupindex
#56122
#56112
C3×#4692
#51562
#51532
#51332
#51272
#50372
#50342
#48702
#48622
#47852
#47782
D3×#1022
#47652
#33063