Finite Group Databasebeta

← #79992

Subgroups — #79992

Order 384.

406 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5798C211yesmaximal, characteristic, complemented
2#5603C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic
4#5218C211yesmaximal, characteristic, complemented
5#5153C211yesmaximal, characteristic, complemented
6#5153C211yesmaximal, characteristic, complemented
7#5152C211yesmaximal, characteristic, complemented
8Dic3×D8C211yesmaximal, characteristic, complemented
9#5035C211yesmaximal, characteristic, complemented
10#4761C211yesmaximal, characteristic, complemented
11#4759C211yesmaximal, characteristic, complemented
12#4758C211yesmaximal, characteristic, complemented
13#4757C211yesmaximal, characteristic, complemented
14#4723C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3118—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C2×C3×D8C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C3×#121C2211yescharacteristic
21C2×#217—21no—
22C2×#217—21no—
23#936—21no—
24#936C2211yescharacteristic, complemented
25#935—21no—
26D4×Dic3—21no—
27D4×Dic3—21no—
28D4×Dic3C2211yescharacteristic
29D4×Dic3C2211yescharacteristic
30D4×Dic3C2211yescharacteristic
31D4×Dic3C2211yescharacteristic
32D4×Dic3C2211yescharacteristic, complemented
33C2×#214C2211yescharacteristic
34C2×#214C2211yescharacteristic
35#926C2211yescharacteristic
36#926C2211yescharacteristic
37#924C2211yescharacteristic
38#888C2211yescharacteristic, complemented
39#887C2211yescharacteristic, complemented
40#879—21no—
41#878—21no—
42C3×#102C2211yescharacteristic
43C3×#102C2211yescharacteristic
44C3×C22⋊C8C2211yescharacteristic
45#832C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic, complemented
48#832C2211yescharacteristic, complemented
49#830C2211yescharacteristic
50#818C2211yescharacteristic, complemented
51#817C2211yescharacteristic, complemented
52#816C2211yescharacteristic, complemented
53#816C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56Dic3⋊C8C2211yescharacteristic, complemented
57C8×Dic3C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#542—31no—
61#518—31no—
62#485—31no—
63#482—31no—
64#476—31no—
65#474—31no—
66#463—31no—
67#463—31no—
68#450—31no—
69#447D311yescharacteristic
70#442—31no—
71#440—31no—
72C4×SD16—31no—
73C4×D8—31no—
74#433—31no—
75C24×C3—21no—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×Dic3—21no—
85C22×Dic3—21no—
86C22×Dic3—21no—
87C22×Dic3—21no—
88C22×Dic3—21no—
89C2×Dic6—21no—
90C2×Dic6C2311yescharacteristic
91C3×D8—22no—
92C3×D8—22no—
93C2×C3×C8C2311yescharacteristic
94C2×C3×C8C2311yescharacteristic
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97#217—21no—
98#217—21no—
99#217—21no—
100#217—21no—
101#217—21no—
102#217—21no—
103#217—21no—
104#217—21no—
105#217—21no—
106#217D411yescharacteristic
107#217D411yescharacteristic
108#214—22no—
109#214—22no—
110C4⋊Dic3D411yescharacteristic
111C4⋊Dic3D411yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4—21no—
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121C4×Dic3C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C22×D4C2×D311yescharacteristic
125C2×SD16—31no—
126C2×SD16—31no—
127C2×D8C2×D311yescharacteristic
128C2×D8C2×D311yescharacteristic
129C4⋊Q8—31no—
130#126—61no—
131#124—31no—
132#124—61no—
133#123—61no—
134C22⋊Q8—31no—
135C22⋊Q8—61no—
136#121C2×D311yescharacteristic
137C4×Q8—31no—
138C4×D4—31no—
139C4×D4—31no—
140C4×D4—31no—
141C4×D4—31no—
142C4×D4—31no—
143C4×D4—61no—
144C4×D4—61no—
145#117—31no—
146#117—31no—
147C2×C22⋊C4—61no—
148C2×C22⋊C4—61no—
149#107—31no—
150#107—31no—
151#106—31no—
152#106—31no—
153C4⋊C8—31no—
154C4⋊C8—31no—
155#103—31no—
156#103—31no—
157#102—31no—
158#102—31no—
159#102—31no—
160#102—31no—
161#102C2×D311yescharacteristic
162#102C2×D311yescharacteristic
163C22⋊C8—31no—
164C22⋊C8C2×D311yescharacteristic
165#97—31no—
166C4×C8—31no—
167C23×C3—24no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—81no—
172C23×C3—81no—
173C23×C3C2×D411yescharacteristic
174C3×D4—24no—
175C3×D4—21no—
176C3×D4—21no—
177C3×D4—21no—
178C3×D4—21no—
179C3×D4—81no—
180C3×D4#4112yes—
181C3×D4#4112yes—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2411yescharacteristic
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193C2×Dic3—81no—
194C2×Dic3—81no—
195C2×Dic3—81no—
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200Dic6—21no—
201Dic6—21no—
202Dic6—81no—
203C3×C8—21no—
204C3×C8—21no—
205C3⋊C8—21no—
206C3⋊C8—21no—
207C24—21no—
208C2×Q8—31no—
209C2×Q8—61no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—21no—
213C2×D4—21no—
214C2×D4C22×D311yescharacteristic
215C2×D4C22×D311yescharacteristic
216C2×D4C22×D311yescharacteristic
217C22×C4—61no—
218C22×C4—61no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4—61no—
222C22×C4C22×D311yescharacteristic
223SD16—62no—
224SD16—62no—
225D8—22no—
226D8—22no—
227C2×C8—31no—
228C2×C8—31no—
229C2×C8C22×D311yescharacteristic
230C2×C8C22×D311yescharacteristic
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—31no—
235C4⋊C4—31no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4—61no—
239C4⋊C4C22×D311yescharacteristic
240C22⋊C4—21no—
241C22⋊C4—31no—
242C22⋊C4—31no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C42—31no—
253C42—31no—
254C42—61no—
255C42—61no—
256C22×C3—21no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—84no—
269C22×C3C22×D411yescharacteristic
270C3×C4—21no—
271C3×C4—41no—
272C3×C4C2×#4111yescharacteristic
273C3×C4C2×#4111yescharacteristic
274Dic3—21no—
275Dic3—21no—
276Dic3—21no—
277Dic3—21no—
278Dic3—41no—
279Dic3—41no—
280Dic3—41no—
281Dic3—41no—
282C23—24no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—81no—
287C23—81no—
288C23D3×D411yescharacteristic
289Q8—61no—
290Q8—61no—
291Q8—241no—
292D4—24no—
293D4—21no—
294D4—21no—
295D4—21no—
296D4—21no—
297D4—81no—
298D4#23712yes—
299D4#23712yes—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—31no—
304C2×C4—31no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—61no—
308C2×C4—61no—
309C2×C4—61no—
310C2×C4—61no—
311C2×C4—241no—
312C2×C4—241no—
313C2×C4—241no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4D3×D411yescharacteristic
317C2×C4C23×D311yescharacteristic
318C8—21no—
319C8—21no—
320C8—61no—
321C8—61no—
322C2×C3—41no—
323C2×C3—42no—
324C2×C3—42no—
325C2×C3—81no—
326C2×C3#52511yescharacteristic
327C2×C3#57611yescharacteristic
328C2×C3#57711yescharacteristic
329C22—21no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—22no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—84no—
342C22C2×D3×D411yescharacteristic
343C4—21no—
344C4—41no—
345C4—61no—
346C4—61no—
347C4—61no—
348C4—61no—
349C4—121no—
350C4—121no—
351C4—121no—
352C4—121no—
353C4C2×#23711yescharacteristic
354C4C2×#23711yescharacteristic
355C3#311811yescharacteristic, complemented, Sylow
356C2—41no—
357C2—42no—
358C2—42no—
359C2—81no—
360C2#558611yescharacteristic
361C2#575711yescharacteristic
362C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
Fitting, prime radicalC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3118
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #5798C2semidirect
2. #5603C2semidirect
4. #5218C2semidirect
5. #5153C2semidirect
6. #5153C2semidirect
7. #5152C2semidirect
8. Dic3×D8C2semidirect
9. #5035C2semidirect
10. #4761C2semidirect
11. #4759C2semidirect
12. #4758C2semidirect
13. #4757C2semidirect
14. #4723C2semidirect
15. #4720C2semidirect
24. #936C22semidirect
32. D4×Dic3C22semidirect
38. #888C22semidirect
39. #887C22semidirect
47. #832C22semidirect
48. #832C22semidirect
50. #818C22semidirect
51. #817C22semidirect
52. #816C22semidirect
53. #816C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
56. Dic3⋊C8C22semidirect
57. C8×Dic3C22semidirect
58. #808C22semidirect
59. #807C22semidirect
355. C3#3118semidirect

Maximal subgroups

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

subgroupindex
#57982
#56032
C3×#4472
#52182
#51532
#51522
Dic3×D82
#50352
#47612
#47592
#47582
#47572
#47232
#47202
#31183