Finite Group Databasebeta

← #81688

Subgroups — #81688

Order 384.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5733C211yesmaximal, characteristic, complemented
2#5674C211yesmaximal, characteristic, complemented
3C3×#489C211yesmaximal, characteristic, complemented
4#5088C211yesmaximal, characteristic, complemented
5#5059C211yesmaximal, characteristic, complemented
6#4819C211yesmaximal, characteristic, complemented
7#4810C211yesmaximal, characteristic, complemented
8#4808C211yesmaximal, characteristic, complemented
9#4798C211yesmaximal, characteristic, complemented
10#4786C211yesmaximal, characteristic, complemented
11#4776C211yesmaximal, characteristic, complemented
12#4773C211yesmaximal, characteristic, complemented
13#4762C211yesmaximal, characteristic, complemented
14#4712C211yesmaximal, characteristic, complemented
15#4709C211yesmaximal, characteristic, complemented
16#3224—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic, complemented
20C3×#124C2211yescharacteristic
21#947—21no—
22#947C2211yescharacteristic
23#946—21no—
24C2×#216C2211yescharacteristic, complemented
25C2×#215C2211yescharacteristic, complemented
26#940C2211yescharacteristic
27C2×#214C2211yescharacteristic
28C2×#213C2211yescharacteristic
29C2×D24C2211yescharacteristic
30C2×D24C2211yescharacteristic, complemented
31C2×#204C2211yescharacteristic
32C2×#204C2211yescharacteristic, complemented
33#898—21no—
34#897C2211yescharacteristic
35#895C2211yescharacteristic
36#894—21no—
37#893C2211yescharacteristic, complemented
38#887C2211yescharacteristic, complemented
39#884—21no—
40#884—21no—
41#882—21no—
42#882—21no—
43#875C2211yescharacteristic, complemented
44#874C2211yescharacteristic
45#872C2211yescharacteristic
46C3×#103C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×#102C2211yescharacteristic
49C3×#102C2211yescharacteristic
50C3×#97C2211yescharacteristic, complemented
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic, complemented
54Dic3⋊C8C2211yescharacteristic, complemented
55#809C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic, complemented
58#809C2211yescharacteristic, complemented
59C4×C3⋊C8C2211yescharacteristic, complemented
60#542—31no—
61#535—31no—
62#496—31no—
63#495—31no—
64#489D311yescharacteristic, complemented
65#486—31no—
66#464—31no—
67#463—31no—
68#460—31no—
69#459—31no—
70#452—31no—
71#450—31no—
72#449—31no—
73#447—31no—
74#433—31no—
75C23×D3—21no—
76C2×C3×Q8—21no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82#239—41no—
83#239—41no—
84D3×D4—41no—
85D3×D4—41no—
86C2×D12—21no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12C2311yescharacteristic
91C2×D12C2311yescharacteristic
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×Dic6C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C42C2311yescharacteristic
105#216—41no—
106#215—41no—
107#214—41no—
108#213—41no—
109(C2×D3)⋊C4—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)⋊C4D411yescharacteristic
117(C2×D3)⋊C4D411yescharacteristic
118C4⋊Dic3—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4D411yescharacteristic
121Dic3⋊C4D411yescharacteristic
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D24—41no—
127D24—41no—
128#204—41no—
129#204—41no—
130C2×#41—31no—
131C22×D4—31no—
132C2×Q16—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×D8—31no—
138C2×D8—31no—
139C2×D8—31no—
140C4⋊Q8C2×D311yescharacteristic, complemented
141#127—31no—
142#127—31no—
143#126—61no—
144#124—31no—
145#124—31no—
146#124—61no—
147#124C2×D311yescharacteristic
148#123—61no—
149C22⋊Q8—31no—
150C22⋊Q8—61no—
151#121—31no—
152#121—61no—
153#121—61no—
154#120—61no—
155#120—61no—
156C4×Q8—31no—
157C4×D4—31no—
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103C2×D311yescharacteristic, complemented
162#103C2×D311yescharacteristic, complemented
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97C2×D311yescharacteristic, complemented
172C4×C8—31no—
173C23×C3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—42no—
178C22×D3—81no—
179C22×D3—81no—
180C22×D3C2×D411yescharacteristic
181C3×Q8—21no—
182C3×Q8—21no—
183C3×Q8—81no—
184C3×D4—21no—
185C3×D4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2411yescharacteristic
192#67—42no—
193#67—81no—
194#67—81no—
195C2×Dic3—21no—
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199D12—21no—
200D12—21no—
201D12—21no—
202D12—21no—
203D12—21no—
204D12—21no—
205D12—42no—
206D12—81no—
207D12—81no—
208D12—81no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—42no—
212C4×D3—81no—
213Dic6—21no—
214Dic6—21no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—61no—
220#41—121no—
221#41—121no—
222C2×Q8—21no—
223C2×Q8—31no—
224C2×Q8C22×D311yescharacteristic
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—31no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4C22×D311yescharacteristic
238C22×C4—31no—
239C22×C4—61no—
240C22×C4—61no—
241Q16—121no—
242SD16—121no—
243SD16—121no—
244SD16—121no—
245SD16—121no—
246D8—121no—
247D8—121no—
248D8—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8C22×D311yescharacteristic
252C2×C8C22×D311yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4C22×D311yescharacteristic
259C4⋊C4C22×D311yescharacteristic
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—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—31no—
272C42—61no—
273C42C22×D311yescharacteristic
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—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—84no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4—41no—
296C3×C4H(5,𝔽2)11yescharacteristic
297C3×C4H(5,𝔽2)11yescharacteristic
298Dic3—21no—
299Dic3—21no—
300Dic3—21no—
301Dic3—41no—
302C23—21no—
303C23—31no—
304C23—61no—
305C23—61no—
306C23—61no—
307C23—122no—
308C23—241no—
309C23—241no—
310Q8—21no—
311Q8—21no—
312Q8—61no—
313Q8—61no—
314Q8—81no—
315D4—21no—
316D4—21no—
317D4—61no—
318D4—61no—
319D4—61no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—122no—
324D4—122no—
325D4—241no—
326D4—241no—
327D4—241no—
328D4—241no—
329D4—241no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—122no—
340C2×C4—241no—
341C2×C4D3×D411yescharacteristic
342C2×C4D3×D411yescharacteristic
343C2×C4C23×D311yescharacteristic
344C8—21no—
345C8—21no—
346C8—61no—
347C8—61no—
348C2×C3—81no—
349C2×C3#53511yescharacteristic
350C2×C3#57611yescharacteristic
351C2×C3#57711yescharacteristic
352D3—41no—
353D3—81no—
354D3—81no—
355D3—81no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—61no—
360C22—61no—
361C22—61no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—244no—
372C22C2×D3×D411yescharacteristic
373C4—21no—
374C4—21no—
375C4—41no—
376C4—41no—
377C4—41no—
378C4—61no—
379C4—61no—
380C4—61no—
381C4—121no—
382C4#100911yescharacteristic
383C4#100911yescharacteristic
384C3#322411yescharacteristic, complemented, Sylow
385C2—81no—
386C2—121no—
387C2—241no—
388C2—241no—
389C2—241no—
390C2#567411yescharacteristic
391C2#577411yescharacteristic
392C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#489C2
socleC22×C3C22×D4
hypercenter#489D3
hypocenterC3#3224
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #489

Complements

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

NH ≅ G/Ntype
1. #5733C2semidirect
2. #5674C2semidirect
3. C3×#489C2semidirect
4. #5088C2semidirect
5. #5059C2semidirect
6. #4819C2semidirect
7. #4810C2semidirect
8. #4808C2semidirect
9. #4798C2semidirect
10. #4786C2semidirect
11. #4776C2semidirect
12. #4773C2semidirect
13. #4762C2semidirect
14. #4712C2semidirect
15. #4709C2semidirect
19. C3×C4⋊Q8C22semidirect
24. C2×#216C22semidirect
25. C2×#215C22semidirect
30. C2×D24C22semidirect
32. C2×#204C22semidirect
37. #893C22semidirect
38. #887C22semidirect
43. #875C22semidirect
46. C3×#103C22semidirect
47. C3×#103C22semidirect
50. C3×#97C22semidirect
53. Dic3⋊C8C22semidirect
54. Dic3⋊C8C22semidirect
57. #809C22semidirect
58. #809C22semidirect
59. C4×C3⋊C8C22semidirect
64. #489D3semidirect
140. C4⋊Q8C2×D3semidirect
161. #103C2×D3semidirect
162. #103C2×D3semidirect
171. #97C2×D3semidirect
384. C3#3224semidirect

Maximal subgroups

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

subgroupindex
#57332
#56742
C3×#4892
#50882
#50592
#48192
#48102
#48082
#47982
#47862
#47762
#47732
#47622
#47122
#47092
#32243