Finite Group Databasebeta

← #81839

Subgroups — #81839

Order 384.

406 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5718C211yesmaximal, characteristic, complemented
2#5671C211yesmaximal, characteristic, complemented
3C3×#496C211yesmaximal, characteristic, complemented
4#5175C211yesmaximal, characteristic, complemented
5#5164C211yesmaximal, characteristic, complemented
6#5163C211yesmaximal, characteristic, complemented
7#5162C211yesmaximal, characteristic, complemented
8#5154C211yesmaximal, characteristic, complemented
9#5153C211yesmaximal, characteristic, complemented
10#5150C211yesmaximal, characteristic, complemented
11Dic3×D8C211yesmaximal, characteristic, complemented
12#5075C211yesmaximal, characteristic, complemented
13#5057C211yesmaximal, characteristic, complemented
14#4710C211yesmaximal, characteristic, complemented
15#4701C211yesmaximal, characteristic, complemented
16#3397—31nomaximal, Sylow
17C2×#237—21no—
18C2×C3×SD16C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic
22C3×#127C2211yescharacteristic
23C3×#124C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25#944C2211yescharacteristic
26C2×#215C2211yescharacteristic
27#940C2211yescharacteristic
28#940C2211yescharacteristic
29#939—21no—
30#939—21no—
31#938—21no—
32#936C2211yescharacteristic
33D4×Dic3C2211yescharacteristic
34D4×Dic3C2211yescharacteristic
35D4×Dic3C2211yescharacteristic
36C2×#214C2211yescharacteristic
37C2×#214C2211yescharacteristic
38C2×#213C2211yescharacteristic
39C2×#204C2211yescharacteristic
40#879—21no—
41#879—21no—
42#878—21no—
43#877—21no—
44C4×D12C2211yescharacteristic
45C4×Dic6C2211yescharacteristic
46C4×Dic6C2211yescharacteristic
47C3×#97C2211yescharacteristic
48#835C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#832C2211yescharacteristic
52#821C2211yescharacteristic
53#818C2211yescharacteristic
54#816C2211yescharacteristic
55#815C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57Dic3⋊C8C2211yescharacteristic
58C8×Dic3C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#556—31no—
61#550—31no—
62#496—31no—
63#496D311yescharacteristic, complemented
64#492—31no—
65#463—31no—
66#463—31no—
67#461—31no—
68#460—31no—
69#460—31no—
70#459—31no—
71#446—31no—
72#440—31no—
73#440—31no—
74C4×D8—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4—21no—
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86C22×Dic3—21no—
87C22×Dic3—21no—
88#237—44no—
89C2×D12C2311yescharacteristic
90C2×C4×D3—21no—
91C2×Dic6D411yescharacteristic
92C2×Dic6C2311yescharacteristic
93C3×SD16—22no—
94C3×SD16—41no—
95C3×D8—22no—
96C3×D8—41no—
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101C3×C42C2311yescharacteristic
102#217—21no—
103#217—21no—
104#217—21no—
105#217—21no—
106#217—21no—
107#215—41no—
108#214—41no—
109#214—41no—
110#213—41no—
111(C2×D3)⋊C4—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3D411yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115C4⋊Dic3C2311yescharacteristic
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4D411yescharacteristic
120Dic3⋊C4D411yescharacteristic
121C4×Dic3—21no—
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C4×Dic3C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127#204—22no—
128C2×#41—61no—
129C2×SD16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16C2×D311yescharacteristic
134C2×SD16C2×D311yescharacteristic
135C2×D8—31no—
136C2×D8C2×D311yescharacteristic
137C2×D8C2×D311yescharacteristic
138C4⋊Q8—31no—
139#127—31no—
140#127—31no—
141#127C2×D311yescharacteristic
142#126—61no—
143#126—61no—
144#124—31no—
145#124C2×D311yescharacteristic
146C22⋊Q8—61no—
147#121—61no—
148#121—61no—
149#121—61no—
150C4×Q8—31no—
151C4×Q8—31no—
152C4×Q8—31no—
153C4×D4—31no—
154C4×D4—31no—
155C4×D4—31no—
156C4×D4—31no—
157#117—61no—
158#107—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161C4⋊C8—31no—
162#103—31no—
163#103—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#97—31no—
169#97C2×D311yescharacteristic
170C4×C8—31no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C22×D3—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—42no—
184C3×D4—81no—
185C2×C3×C4—21no—
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2×D411yescharacteristic
188C2×C3×C4C2411yescharacteristic
189#67—42no—
190#67—42no—
191#67—81no—
192#67—81no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12—21no—
205C4×D3—42no—
206Dic6—22no—
207Dic6—21no—
208Dic6—21no—
209C3×C8—21no—
210C3×C8#4112yescomplemented
211C3⋊C8—21no—
212C3⋊C8—21no—
213#41—124no—
214C2×Q8—31no—
215C2×Q8—31no—
216C2×Q8C22×D311yescharacteristic
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—31no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4C22×D311yescharacteristic
225C2×D4C22×D311yescharacteristic
226C2×D4C22×D311yescharacteristic
227C22×C4—61no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231SD16—22no—
232SD16—41no—
233SD16—62no—
234SD16—121no—
235SD16—121no—
236SD16—121no—
237D8—22no—
238D8—41no—
239D8—121no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—31no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—31no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C42—31no—
262C42—31no—
263C42—61no—
264C42—61no—
265C42C22×D311yescharacteristic
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3C22×D411yescharacteristic
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4C2×#4111yescharacteristic
283C3×C4H(5,𝔽2)11yescharacteristic
284Dic3—21no—
285Dic3—21no—
286Dic3—21no—
287Dic3—41no—
288Dic3—41no—
289Dic3—41no—
290Dic3AGL(1,ℤ/8)12yes—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—61no—
295Q8—21no—
296Q8—21no—
297Q8—62no—
298Q8—61no—
299Q8—61no—
300D4—21no—
301D4—21no—
302D4—21no—
303D4—21no—
304D4—21no—
305D4—21no—
306D4—42no—
307D4—61no—
308D4—61no—
309D4—81no—
310D4—122no—
311D4—122no—
312D4—241no—
313D4—241no—
314C2×C4—21no—
315C2×C4—31no—
316C2×C4—31no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—61no—
320C2×C4—61no—
321C2×C4—61no—
322C2×C4—122no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4D3×D411yescharacteristic
327C2×C4D3×D411yescharacteristic
328C2×C4C23×D311yescharacteristic
329C8—21no—
330C8—61no—
331C8—61no—
332C8#23712yescomplemented
333C2×C3—81no—
334C2×C3—81no—
335C2×C3—81no—
336C2×C3#55011yescharacteristic
337C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
338C2×C3#57711yescharacteristic
339D3—81no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22C2×D3×D411yescharacteristic
353C4—21no—
354C4—21no—
355C4—32no—
356C4—41no—
357C4—61no—
358C4—61no—
359C4—61no—
360C4—121no—
361C4—121no—
362C4—121no—
363C4C2×#23711yescharacteristic
364C4#100411yescharacteristic
365C3#339711yescharacteristic, complemented, Sylow
366C2—81no—
367C2—81no—
368C2—81no—
369C2—241no—
370C2#571811yescharacteristic
371C2D3×AGL(1,ℤ/8)11yescharacteristic
372C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#496C2
socleC22×C3C22×D4
hypercenter#496D3
hypocenterC3#3397
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. #5718C2semidirect
2. #5671C2semidirect
3. C3×#496C2semidirect
4. #5175C2semidirect
5. #5164C2semidirect
6. #5163C2semidirect
7. #5162C2semidirect
8. #5154C2semidirect
9. #5153C2semidirect
10. #5150C2semidirect
11. Dic3×D8C2semidirect
12. #5075C2semidirect
13. #5057C2semidirect
14. #4710C2semidirect
15. #4701C2semidirect
63. #496D3semidirect
210. C3×C8#41semidirect
332. C8#237semidirect
365. C3#3397semidirect

Maximal subgroups

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

subgroupindex
#57182
#56712
C3×#4962
#51752
#51642
#51632
#51622
#51542
#51532
#51502
Dic3×D82
#50752
#50572
#47102
#47012
#33973