Finite Group Databasebeta

← #80545

Subgroups — #80545

Order 384.

388 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5675C211yesmaximal, characteristic, complemented
2#5562C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic, complemented
4#5175C211yesmaximal, characteristic, complemented
5#5170C211yesmaximal, characteristic, complemented
6#5158C211yesmaximal, characteristic, complemented
7#5154C211yesmaximal, characteristic, complemented
8#5057C211yesmaximal, characteristic, complemented
9#5019C211yesmaximal, characteristic, complemented
10#4841C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4806C211yesmaximal, characteristic, complemented
13#4797C211yesmaximal, characteristic, complemented
14#4781C211yesmaximal, characteristic, complemented
15#4768C211yesmaximal, characteristic, complemented
16#3221—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic
23#944C2211yescharacteristic
24C2×#215C2211yescharacteristic
25#940C2211yescharacteristic
26#938C2211yescharacteristic
27#936C2211yescharacteristic
28C2×#214C2211yescharacteristic
29C2×#214C2211yescharacteristic
30C2×#213C2211yescharacteristic
31#930—21no—
32#929—21no—
33C2×#204C2211yescharacteristic
34C2×#204C2211yescharacteristic
35#895C2211yescharacteristic
36#890C2211yescharacteristic
37#884—21no—
38#883—21no—
39#883—21no—
40#883—21no—
41#878—21no—
42#878—21no—
43C4×D12C2211yescharacteristic
44#872C2211yescharacteristic
45#869C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×#102C2211yescharacteristic
49#835C2211yescharacteristic
50#832C2211yescharacteristic
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53#816C2211yescharacteristic
54#816C2211yescharacteristic
55#815C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#810C2211yescharacteristic
58#810C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#540—31no—
61#536—31no—
62#496—31no—
63#492—31no—
64#491—31no—
65#488—31no—
66#463—31no—
67#463D311yescharacteristic, complemented
68#461—31no—
69#461—31no—
70#449—31no—
71#449—31no—
72#448—31no—
73#448—31no—
74#433—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84D3×Q8—41no—
85D3×Q8—41no—
86#237—41no—
87#237—41no—
88C2×D12—21no—
89C2×D12C2311yescharacteristic
90C2×C4×D3—21no—
91C2×C4×D3—21no—
92C2×C4×D3C2311yescharacteristic
93C2×Dic6—21no—
94C2×Dic6C2311yescharacteristic
95C2×Dic6C2311yescharacteristic
96C3×SD16—41no—
97C3×D8—41no—
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C42C2311yescharacteristic
105#217—21no—
106#217—21no—
107#217—21no—
108#215—41no—
109#214—41no—
110#214—41no—
111#213—41no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4D411yescharacteristic
115(C2×D3)⋊C4D411yescharacteristic
116C4⋊Dic3—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4D411yescharacteristic
124Dic3⋊C4D411yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128#204—41no—
129#204—41no—
130C2×#41—31no—
131C22×Q8—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×SD16C2×D311yescharacteristic
138C2×D8—31no—
139C2×D8C2×D311yescharacteristic
140C4⋊Q8—31no—
141C4⋊Q8—31no—
142#127—31no—
143#125—31no—
144#124—31no—
145#124C2×D311yescharacteristic
146#123—61no—
147#123—61no—
148#123—61no—
149#123—61no—
150C22⋊Q8—61no—
151C22⋊Q8—61no—
152#121—31no—
153#121—31no—
154#121—61no—
155#121—61no—
156C4×D4—31no—
157C4×D4C2×D311yescharacteristic
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic
161#103—31no—
162#103—31no—
163#103—31no—
164#103—31no—
165#103—31no—
166#103C2×D311yescharacteristic
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3—21no—
175C22×D3—21no—
176C22×D3C2×D411yescharacteristic
177C3×Q8—21no—
178C3×Q8—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—81no—
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—42no—
190#67—81no—
191#67—81no—
192#67—81no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—42no—
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200D12—21no—
201D12—21no—
202D12—81no—
203C4×D3—21no—
204C4×D3—21no—
205C4×D3—42no—
206C4×D3—81no—
207Dic6—21no—
208Dic6—21no—
209Dic6—21no—
210Dic6—21no—
211Dic6—42no—
212C3×C8—21no—
213C3×C8—21no—
214C3⋊C8—21no—
215C3⋊C8—21no—
216#41—121no—
217#41—121no—
218C2×Q8—31no—
219C2×Q8—31no—
220C2×Q8—61no—
221C2×Q8—121no—
222C2×Q8—121no—
223C2×Q8C22×D311yescharacteristic
224C2×D4—31no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4C22×D311yescharacteristic
231C2×D4C22×D311yescharacteristic
232C22×C4—21no—
233C22×C4—31no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237SD16—41no—
238SD16—121no—
239SD16—121no—
240SD16—121no—
241SD16—121no—
242SD16—121no—
243D8—41no—
244D8—121no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8C22×D311yescharacteristic
248C2×C8C22×D311yescharacteristic
249C4⋊C4—31no—
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—31no—
263C22⋊C4—31no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C42—31no—
270C42C22×D311yescharacteristic
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3C22×D411yescharacteristic
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4H(5,𝔽2)11yescharacteristic
289C3×C4H(5,𝔽2)11yescharacteristic
290Dic3—21no—
291Dic3—21no—
292Dic3—21no—
293Dic3—41no—
294Dic3—41no—
295Dic3—41no—
296C23—21no—
297C23—21no—
298C23—31no—
299C23—61no—
300Q8—21no—
301Q8—21no—
302Q8—61no—
303Q8—61no—
304Q8—61no—
305Q8—61no—
306Q8—122no—
307D4—21no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—61no—
312D4—61no—
313D4—122no—
314D4—241no—
315D4—241no—
316D4—241no—
317D4—241no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—31no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—81no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—241no—
332C2×C4D3×D411yescharacteristic
333C2×C4D3×D411yescharacteristic
334C2×C4C23×D311yescharacteristic
335C8—21no—
336C8—21no—
337C8—61no—
338C8—61no—
339C2×C3—81no—
340C2×C3—81no—
341C2×C3#53511yescharacteristic
342C2×C3#57711yescharacteristic
343C2×C3#57711yescharacteristic
344D3—41no—
345D3—81no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—61no—
353C22—61no—
354C22—61no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22C2×D3×D411yescharacteristic
359C4—21no—
360C4—21no—
361C4—41no—
362C4—41no—
363C4—61no—
364C4—61no—
365C4—61no—
366C4—121no—
367C4—121no—
368C4—121no—
369C4#100411yescharacteristic
370C4#100911yescharacteristic
371C3#322111yescharacteristic, complemented, Sylow
372C2—81no—
373C2—81no—
374C2—121no—
375C2—241no—
376C2#560711yescharacteristic
377C2#577011yescharacteristic
378C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3221
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #5675C2semidirect
2. #5562C2semidirect
3. C3×#463C2semidirect
4. #5175C2semidirect
5. #5170C2semidirect
6. #5158C2semidirect
7. #5154C2semidirect
8. #5057C2semidirect
9. #5019C2semidirect
10. #4841C2semidirect
11. #4840C2semidirect
12. #4806C2semidirect
13. #4797C2semidirect
14. #4781C2semidirect
15. #4768C2semidirect
67. #463D3semidirect
371. C3#3221semidirect

Maximal subgroups

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

subgroupindex
#56752
#55622
C3×#4632
#51752
#51702
#51582
#51542
#50572
#50192
#48412
#48402
#48062
#47972
#47812
#47682
#32213