Finite Group Databasebeta

← #79499

Subgroups — #79499

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#5555C211yesmaximal, characteristic, complemented
2#5550C211yesmaximal, characteristic, complemented
3C3×#435C211yesmaximal, characteristic, complemented
4#5135C211yesmaximal, characteristic, complemented
5#5133C211yesmaximal, characteristic, complemented
6#5114C211yesmaximal, characteristic, complemented
7C2×#821C211yesmaximal, characteristic, complemented
8#4844C211yesmaximal, characteristic, complemented
9#4842C211yesmaximal, characteristic, complemented
10#4735C211yesmaximal, characteristic, complemented
11#4734C211yesmaximal, characteristic, complemented
12#4732C211yesmaximal, characteristic, complemented
13#4721C211yesmaximal, characteristic, complemented
14#4714C211yesmaximal, characteristic, complemented
15#4711C211yesmaximal, characteristic, complemented
16#3306—31nomaximal, Sylow
17C2×#237—21no—
18C2×#235C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C2×C3×M16C2211yescharacteristic, complemented
21C22×C3×C8C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic
23#937—21no—
24#930C2211yescharacteristic
25#930C2211yescharacteristic
26#930C2211yescharacteristic, complemented
27C2×C4⋊Dic3C2211yescharacteristic
28C2×C4⋊Dic3C2211yescharacteristic, complemented
29#924C2211yescharacteristic
30C2×D24C2211yescharacteristic
31C2×D24C2211yescharacteristic
32C2×#204C2211yescharacteristic, complemented
33#897—21no—
34#894—21no—
35#886—21no—
36#886—21no—
37#884—21no—
38D3×C22⋊C4—21no—
39#875C2211yescharacteristic
40C4×D12C2211yescharacteristic
41C4×D12C2211yescharacteristic, complemented
42#869C2211yescharacteristic, complemented
43C3×C4⋊C8C2211yescharacteristic, complemented
44C3×C22⋊C8C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic, complemented
46C3×#97C2211yescharacteristic, complemented
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53#821C2211yescharacteristic, complemented
54#817C2211yescharacteristic, complemented
55#817C2211yescharacteristic, complemented
56#817C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#816C2211yescharacteristic, complemented
60#547—31no—
61#546—31no—
62#489—31no—
63#483—31no—
64#481—31no—
65#476—31no—
66#469—31no—
67#465—31no—
68#463—31no—
69#449—31no—
70#447—31no—
71#442—31no—
72#435D311yescharacteristic, complemented
73#417—31no—
74C2×#102—31no—
75C23×D3—21no—
76C2×C3×D4D411yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84C22×Dic3—21no—
85#237—44no—
86#235—22no—
87#235—41no—
88C2×D12—22no—
89C2×D12—21no—
90C2×D12—41no—
91C2×D12C2311yescharacteristic
92C2×D12C2311yescharacteristic
93C2×D12C2311yescharacteristic
94C2×C4×D3—21no—
95C2×C4×D3—21no—
96C2×C4×D3—21no—
97C2×Dic6—21no—
98C2×Dic6C2311yescharacteristic
99C3×M16—41no—
100C2×C3×C8—41no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
105C3×C4⋊C4D411yescharacteristic
106C3×C22⋊C4D411yescharacteristic
107C3×C22⋊C4D411yescharacteristic
108C3×C42C2311yescharacteristic
109#217—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
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⋊Dic3—21no—
120C4⋊Dic3C2311yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122C4⋊Dic3C2311yescharacteristic
123C4⋊Dic3C2311yescharacteristic
124Dic3⋊C4—21no—
125D24—22no—
126D24—41no—
127#204—41no—
128C2×#41—31no—
129C2×#41—61no—
130C22×D4—31no—
131C2×SD16—31no—
132C2×D8—31no—
133C2×D8—31no—
134C2×M16C2×D311yescharacteristic, complemented
135C22×C8C2×D311yescharacteristic
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—31no—
145#121—31no—
146#121—61no—
147#120—61no—
148C4×D4—31no—
149C4×D4—31no—
150C4×D4C2×D311yescharacteristic
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C22⋊C4—61no—
154#106—31no—
155#106—31no—
156#106—31no—
157C4⋊C8C2×D311yescharacteristic, complemented
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—
167#102—31no—
168C22⋊C8C2×D311yescharacteristic
169C22⋊C8C2×D311yescharacteristic, complemented
170#97C2×D311yescharacteristic, complemented
171C23×C3C2×D411yescharacteristic
172C23×C3C2×D411yescharacteristic
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—42no—
177C22×D3—42no—
178C22×D3—81no—
179C3×D4—22no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—42no—
190#67—42no—
191#67—42no—
192#67—81no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200D12—21no—
201D12—21no—
202D12—21no—
203D12—21no—
204D12—21no—
205D12—42no—
206D12#4112yescomplemented
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—31no—
224C2×D4—31no—
225C2×D4—31no—
226C2×D4—62no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—121no—
233C2×D4D1211yescharacteristic
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241SD16—121no—
242D8—62no—
243D8—121no—
244M16—41no—
245C2×C8—41no—
246C2×C8C22×D311yescharacteristic
247C2×C8C22×D311yescharacteristic
248C2×C8C22×D311yescharacteristic
249C2×C8C22×D311yescharacteristic
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—31no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4D1211yescharacteristic
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4D1211yescharacteristic
268C22⋊C4D1211yescharacteristic
269C42C22×D311yescharacteristic
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—21no—
275C22×C3AGL(1,ℤ/8)12yes—
276C22×C3C22×D411yescharacteristic
277C2×D3—22no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—84no—
287C3×C4—21no—
288C3×C4—21no—
289C3×C4—21no—
290C3×C4—21no—
291C3×C4C2×#4111yescharacteristic
292C3×C4H(5,𝔽2)11yescharacteristic
293Dic3—41no—
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23—122no—
302C23—122no—
303C23—241no—
304C23C2×D1211yescharacteristic
305C23C2×D1211yescharacteristic
306Q8—61no—
307Q8—61no—
308Q8—122no—
309D4—22no—
310D4—32no—
311D4—61no—
312D4—61no—
313D4—61no—
314D4—61no—
315D4—61no—
316D4—122no—
317D4—122no—
318D4—122no—
319D4—122no—
320D4—241no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—241no—
335C2×C4C2×D1211yescharacteristic
336C2×C4C2×D1211yescharacteristic
337C2×C4C2×D1211yescharacteristic
338C2×C4C2×D1211yescharacteristic
339C2×C4C23×D311yescharacteristic
340C8—21no—
341C8—21no—
342C8—21no—
343C8—21no—
344C2×C3—22no—
345C2×C3—41no—
346C2×C3#54611yescharacteristic
347C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
348C2×C3#57711yescharacteristic
349D3—42no—
350D3—81no—
351D3—81no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—62no—
358C22—121no—
359C22—121no—
360C22—121no—
361C22—121no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—244no—
367C22#90812yes—
368C22C22×D1211yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—121no—
374C4—121no—
375C4—121no—
376C4—121no—
377C4—121no—
378C4#100411yescharacteristic
379C4D3×#4111yescharacteristic
380C3#330611yescharacteristic, complemented, Sylow
381C2—22no—
382C2—41no—
383C2—122no—
384C2—241no—
385C2—241no—
386C2#555411yescharacteristic
387C2C2×#90811yescharacteristic
388C2#575111yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
hypercenter#435D3
hypocenterC3#3306
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #5555C2semidirect
2. #5550C2semidirect
3. C3×#435C2semidirect
4. #5135C2semidirect
5. #5133C2semidirect
6. #5114C2semidirect
7. C2×#821C2semidirect
8. #4844C2semidirect
9. #4842C2semidirect
10. #4735C2semidirect
11. #4734C2semidirect
12. #4732C2semidirect
13. #4721C2semidirect
14. #4714C2semidirect
15. #4711C2semidirect
20. C2×C3×M16C22semidirect
26. #930C22semidirect
28. C2×C4⋊Dic3C22semidirect
32. C2×#204C22semidirect
41. C4×D12C22semidirect
42. #869C22semidirect
43. C3×C4⋊C8C22semidirect
45. C3×C22⋊C8C22semidirect
46. C3×#97C22semidirect
53. #821C22semidirect
54. #817C22semidirect
55. #817C22semidirect
56. #817C22semidirect
57. #816C22semidirect
58. #816C22semidirect
59. #816C22semidirect
72. #435D3semidirect
134. C2×M16C2×D3semidirect
157. C4⋊C8C2×D3semidirect
169. C22⋊C8C2×D3semidirect
170. #97C2×D3semidirect
206. 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
#55552
#55502
C3×#4352
#51352
#51332
#51142
C2×#8212
#48442
#48422
#47352
#47342
#47322
#47212
#47142
#47112
#33063