Finite Group Databasebeta

← #80606

Subgroups — #80606

Order 384.

400 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
1D3×#124C211yesmaximal, characteristic
2#5578C211yesmaximal, characteristic, complemented
3C3×#464C211yesmaximal, characteristic, complemented
4#5190C211yesmaximal, characteristic, complemented
5#5186C211yesmaximal, characteristic, complemented
6#5169C211yesmaximal, characteristic
7#5167C211yesmaximal, characteristic, complemented
8#5054C211yesmaximal, characteristic, complemented
9#5028C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4835C211yesmaximal, characteristic, complemented
12#4809C211yesmaximal, characteristic, complemented
13#4794C211yesmaximal, characteristic, complemented
14#4776C211yesmaximal, characteristic
15#4765C211yesmaximal, characteristic, complemented
16#3215—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic, complemented
20C2×C3×SD16C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic, complemented
23#947C2211yescharacteristic
24#946C2211yescharacteristic
25#944C2211yescharacteristic, complemented
26C2×#216C2211yescharacteristic, complemented
27C2×#216C2211yescharacteristic, complemented
28C2×#215C2211yescharacteristic
29#936C2211yescharacteristic
30C2×#214C2211yescharacteristic
31C2×Dic12C2211yescharacteristic, complemented
32C2×#204C2211yescharacteristic
33#897C2211yescharacteristic
34#896—21no—
35#889—21no—
36#888C2211yescharacteristic, complemented
37#885—21no—
38#885—21no—
39D3×C22⋊C4—21no—
40D3×C22⋊C4—21no—
41#875C2211yescharacteristic
42#872—21no—
43C42×D3C2211yescharacteristic
44C4×Dic6—21no—
45C4×Dic6C2211yescharacteristic, complemented
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×#102C2211yescharacteristic
49#835C2211yescharacteristic, complemented
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#816C2211yescharacteristic, complemented
55#815C2211yescharacteristic, complemented
56#815C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#809C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic, complemented
60#533—31no—
61C2×#124—31no—
62#497—31no—
63#497—31no—
64#489—31no—
65#489—31no—
66#464—31no—
67#464—31no—
68#464—31no—
69#464D311yescharacteristic, complemented
70#451—31no—
71#451—31no—
72#450—31no—
73#450—31no—
74#432—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×Q8C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C2×#67—21no—
80D3×Q8—41no—
81D3×Q8—41no—
82D3×D4—41no—
83D3×D4—41no—
84C2×D12C2311yescharacteristic
85C2×C4×D3—21no—
86C2×C4×D3D411yescharacteristic
87C2×C4×D3D411yescharacteristic
88C2×C4×D3C2311yescharacteristic
89C2×Dic6—21no—
90C2×Dic6—21no—
91C2×Dic6C2311yescharacteristic
92C2×Dic6C2311yescharacteristic
93C3×Q16—41no—
94C3×SD16—41no—
95C2×C3×C8C2311yescharacteristic
96C2×C3×C8C2311yescharacteristic
97C3×C4⋊C4—21no—
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101C3×C42—21no—
102C3×C42C2311yescharacteristic
103#217—21no—
104#217—21no—
105#216—41no—
106#216—41no—
107#215—41no—
108#214—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—
115C4⋊Dic3—21no—
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122C4×Dic3—21no—
123C4×Dic3D411yescharacteristic
124C4×Dic3D411yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129#204—41no—
130C22×Q8—31no—
131C22×D4—31no—
132C2×Q16—31no—
133C2×Q16—31no—
134C2×Q16—31no—
135C2×Q16C2×D311yescharacteristic, complemented
136C2×SD16—31no—
137C2×SD16—31no—
138C2×SD16—31no—
139C2×SD16C2×D311yescharacteristic
140C4⋊Q8—31no—
141C4⋊Q8—31no—
142#125—61no—
143#124—31no—
144#124—31no—
145#124—31no—
146#124—61no—
147#124—61no—
148#124C2×D311yescharacteristic
149C22⋊Q8—31no—
150C22⋊Q8—31no—
151C22⋊Q8—61no—
152C4×Q8—31no—
153C4×Q8—61no—
154C4×Q8C2×D311yescharacteristic, complemented
155#117—61no—
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C42—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic, complemented
161#103—31no—
162#103—31no—
163#103—31no—
164#103C2×D311yescharacteristic, complemented
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172#97—31no—
173C23×C3—21no—
174C22×D3—21no—
175C22×D3—42no—
176C22×D3—42no—
177C22×D3—42no—
178C22×D3C2×D411yescharacteristic
179C3×Q8—21no—
180C3×Q8—21no—
181C3×Q8—21no—
182C3×Q8—21no—
183C3×D4—21no—
184C3×D4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2411yescharacteristic
191#67—42no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198D12—21no—
199D12—21no—
200C4×D3—22no—
201C4×D3—21no—
202C4×D3—21no—
203C4×D3—41no—
204C4×D3—42no—
205Dic6—21no—
206Dic6—21no—
207Dic6—21no—
208Dic6—21no—
209Dic6—42no—
210Dic6—42no—
211C3×C8—21no—
212C3×C8—21no—
213C3⋊C8—21no—
214C3⋊C8—21no—
215C24—61no—
216C2×Q8—31no—
217C2×Q8—31no—
218C2×Q8—61no—
219C2×Q8—61no—
220C2×Q8—121no—
221C2×Q8—121no—
222C2×Q8C22×D311yescharacteristic
223C2×Q8C22×D311yescharacteristic
224C2×D4—31no—
225C2×D4—61no—
226C2×D4—121no—
227C2×D4—121no—
228C2×D4C22×D311yescharacteristic
229C22×C4—31no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—61no—
233Q16—41no—
234Q16—121no—
235Q16—121no—
236Q16—121no—
237SD16—41no—
238SD16—121no—
239SD16—121no—
240SD16—121no—
241C2×C8—31no—
242C2×C8—31no—
243C2×C8C22×D311yescharacteristic
244C2×C8C22×D311yescharacteristic
245C4⋊C4—21no—
246C4⋊C4—31no—
247C4⋊C4—61no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4C22×D311yescharacteristic
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C42—21no—
265C42—31no—
266C42—31no—
267C42—31no—
268C42—61no—
269C42C22×D311yescharacteristic
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3C22×D411yescharacteristic
274C2×D3—21no—
275C2×D3—21no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—84no—
280C2×D3#13712yes—
281C3×C4—21no—
282C3×C4—21no—
283C3×C4—41no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4H(5,𝔽2)11yescharacteristic
287C3×C4H(5,𝔽2)11yescharacteristic
288Dic3—21no—
289Dic3—21no—
290Dic3—21no—
291Dic3—41no—
292Dic3—41no—
293Dic3—41no—
294C23—21no—
295C23—31no—
296C23—61no—
297C23—122no—
298C23—122no—
299C23—122no—
300Q8—21no—
301Q8—21no—
302Q8—21no—
303Q8—21no—
304Q8—61no—
305Q8—61no—
306Q8—61no—
307Q8—61no—
308Q8—122no—
309Q8—122no—
310D4—21no—
311D4—21no—
312D4—61no—
313D4—61no—
314D4—122no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—62no—
322C2×C4—61no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—121no—
328C2×C4—122no—
329C2×C4D3×D411yescharacteristic
330C2×C4D3×D411yescharacteristic
331C2×C4C23×D311yescharacteristic
332C8—21no—
333C8—21no—
334C8—61no—
335C8—61no—
336C2×C3—81no—
337C2×C3#53511yescharacteristic
338C2×C3C2×#13711yescharacteristic
339C2×C3#57511yescharacteristic
340D3—22no—
341D3—81no—
342C22—32no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—61no—
347C22—61no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—244no—
352C22C2×D3×D411yescharacteristic
353C4—21no—
354C4—21no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—61no—
359C4—61no—
360C4—61no—
361C4—121no—
362C4—121no—
363C4—121no—
364C4#100411yescharacteristic
365C4#100911yescharacteristic
366C3#321511yescharacteristic, complemented, Sylow
367C2—62no—
368C2—81no—
369C2—241no—
370C2#560711yescharacteristic
371C2#576811yescharacteristic
372C2D3×#13711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#464C2
socleC22×C3C22×D4
prime radicalC2×D3×D4C22
hypercenter#464D3
hypocenterC3#3215
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. #5578C2semidirect
3. C3×#464C2semidirect
4. #5190C2semidirect
5. #5186C2semidirect
7. #5167C2semidirect
8. #5054C2semidirect
9. #5028C2semidirect
10. #4845C2semidirect
11. #4835C2semidirect
12. #4809C2semidirect
13. #4794C2semidirect
15. #4765C2semidirect
19. C2×C3×Q16C22semidirect
22. C3×C4×Q8C22semidirect
25. #944C22semidirect
26. C2×#216C22semidirect
27. C2×#216C22semidirect
31. C2×Dic12C22semidirect
36. #888C22semidirect
45. C4×Dic6C22semidirect
46. C3×C4⋊C8C22semidirect
47. C3×#103C22semidirect
49. #835C22semidirect
54. #816C22semidirect
55. #815C22semidirect
56. #815C22semidirect
57. #810C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
69. #464D3semidirect
135. C2×Q16C2×D3semidirect
154. C4×Q8C2×D3semidirect
160. C4⋊C8C2×D3semidirect
164. #103C2×D3semidirect
366. C3#3215semidirect

Maximal subgroups

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

subgroupindex
D3×#1242
#55782
C3×#4642
#51902
#51862
#51692
#51672
#50542
#50282
#48452
#48352
#48092
#47942
#47762
#47652
#32153