Finite Group Databasebeta

← #80424

Subgroups — #80424

Order 384.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5732C211yesmaximal, characteristic, complemented
2#5559C211yesmaximal, characteristic, complemented
3C3×#461C211yesmaximal, characteristic, complemented
4#5176C211yesmaximal, characteristic
5#5168C211yesmaximal, characteristic, complemented
6#5166C211yesmaximal, characteristic, complemented
7#5163C211yesmaximal, characteristic, complemented
8#5086C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4828C211yesmaximal, characteristic, complemented
12#4819C211yesmaximal, characteristic, complemented
13#4798C211yesmaximal, characteristic, complemented
14#4789C211yesmaximal, characteristic, complemented
15#4788C211yesmaximal, characteristic, complemented
16#3401—31nomaximal, Sylow
17C2×C3×SD16C2211yescharacteristic
18C2×C3×SD16C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21#947C2211yescharacteristic
22#947C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25C2×#216C2211yescharacteristic
26C2×#216C2211yescharacteristic
27C2×#215C2211yescharacteristic
28C2×#215C2211yescharacteristic
29#940C2211yescharacteristic
30#936C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#929—21no—
33C4×#67—21no—
34C2×D24C2211yescharacteristic
35C2×#204C2211yescharacteristic
36#898—21no—
37#898—21no—
38#897—21no—
39#893C2211yescharacteristic
40#892—21no—
41#887C2211yescharacteristic
42#885—21no—
43#883—21no—
44#875C2211yescharacteristic
45C4×D12C2211yescharacteristic
46C4×Dic6C2211yescharacteristic
47C3×C4⋊C8C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#103C2211yescharacteristic
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52#816C2211yescharacteristic
53#815C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#803C2211yescharacteristic
60#559—31no—
61#556—31no—
62#497—31no—
63#496—31no—
64#495—31no—
65#464—31no—
66#464—31no—
67#463—31no—
68#463—31no—
69#461D311yescharacteristic, complemented
70#460—31no—
71#446—31no—
72#442—31no—
73#441—31no—
74C4×SD16—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×D12C2311yescharacteristic
82C2×D12C2311yescharacteristic
83C2×C4×D3—21no—
84C2×C4×D3—21no—
85C2×Dic6D411yescharacteristic
86C2×Dic6C2311yescharacteristic
87C3×SD16—41no—
88C3×SD16—41no—
89C2×C3×C8C2311yescharacteristic
90C2×C3×C8C2311yescharacteristic
91C3×C4⋊C4—21no—
92C3×C4⋊C4—21no—
93C3×C4⋊C4C2311yescharacteristic
94C3×C22⋊C4—21no—
95C3×C42C2311yescharacteristic
96#217—21no—
97#217—21no—
98#216—22no—
99#216—41no—
100#215—22no—
101#215—41no—
102#213—22no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3—21no—
113C4⋊Dic3D411yescharacteristic
114C4⋊Dic3C2311yescharacteristic
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3C2311yescharacteristic
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125D24—41no—
126#204—41no—
127C2×Q16—31no—
128C2×Q16—31no—
129C2×SD16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×SD16C2×D311yescharacteristic
133C2×SD16C2×D311yescharacteristic
134C2×D8—31no—
135C2×D8—31no—
136C4⋊Q8C2×D311yescharacteristic
137#127—31no—
138#126—61no—
139#126—61no—
140#124—31no—
141#124—31no—
142#124—31no—
143#124—31no—
144#124—61no—
145#123—61no—
146#123—61no—
147C22⋊Q8—61no—
148C4×Q8—31no—
149C4×Q8—31no—
150C4×Q8—31no—
151C4×Q8—31no—
152C4×D4—31no—
153C4×D4—31no—
154C4×D4—61no—
155C4×D4C2×D311yescharacteristic
156#117—61no—
157#106—31no—
158C4⋊C8—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic
161#103—31no—
162#103C2×D311yescharacteristic
163#103C2×D311yescharacteristic
164#102—31no—
165#102—31no—
166#102—31no—
167#97—31no—
168#97—31no—
169C4×C8—31no—
170C23×C3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×D4—21no—
178C3×D4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—81no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—42no—
187#67—81no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3C2×D411yescharacteristic
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194D12—21no—
195D12—21no—
196D12—21no—
197D12—21no—
198C4×D3—42no—
199C4×D3—81no—
200Dic6—21no—
201Dic6—21no—
202Dic6—41no—
203C3×C8—21no—
204C3×C8—21no—
205C3⋊C8—21no—
206C3⋊C8#4112yes—
207C2×Q8—31no—
208C2×Q8—31no—
209C2×Q8C22×D311yescharacteristic
210C2×Q8C22×D311yescharacteristic
211C2×D4—31no—
212C2×D4—31no—
213C2×D4—61no—
214C2×D4—61no—
215C2×D4C22×D311yescharacteristic
216C22×C4—21no—
217C22×C4—61no—
218C22×C4—61no—
219Q16—62no—
220Q16—121no—
221SD16—41no—
222SD16—41no—
223SD16—62no—
224SD16—121no—
225SD16—121no—
226D8—62no—
227D8—121no—
228C2×C8—31no—
229C2×C8—31no—
230C2×C8C22×D311yescharacteristic
231C2×C8C22×D311yescharacteristic
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—31no—
235C4⋊C4—31no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4C22×D311yescharacteristic
242C22⋊C4—21no—
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—
252C22⋊C4—61no—
253C42—31no—
254C42—31no—
255C42—61no—
256C42—61no—
257C42—61no—
258C42C22×D311yescharacteristic
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3C22×D411yescharacteristic
263C2×D3—41no—
264C2×D3—41no—
265C2×D3—41no—
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C3×C4—21no—
270C3×C4—21no—
271C3×C4—41no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4C2×#4111yescharacteristic
275C3×C4H(5,𝔽2)11yescharacteristic
276Dic3—21no—
277Dic3—21no—
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282C23—21no—
283C23—61no—
284C23—61no—
285Q8—21no—
286Q8—21no—
287Q8—21no—
288Q8—21no—
289Q8—61no—
290Q8—61no—
291Q8—121no—
292D4—21no—
293D4—21no—
294D4—61no—
295D4—61no—
296D4—61no—
297D4—61no—
298D4—122no—
299D4—241no—
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—81no—
310C2×C4—122no—
311C2×C4—241no—
312C2×C4D3×D411yescharacteristic
313C2×C4D3×D411yescharacteristic
314C2×C4C23×D311yescharacteristic
315C8—21no—
316C8—21no—
317C8—32no—
318C8—61no—
319C2×C3—81no—
320C2×C3#55011yescharacteristic
321C2×C3#57511yescharacteristic
322C2×C3#57711yescharacteristic
323D3—81no—
324D3—81no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—121no—
329C22—121no—
330C22—121no—
331C22—121no—
332C22—121no—
333C22—121no—
334C22C2×D3×D411yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—61no—
341C4—61no—
342C4—61no—
343C4—61no—
344C4—121no—
345C4—121no—
346C4#100411yescharacteristic
347C4D3×#4111yescharacteristic
348C3#340111yescharacteristic, complemented, Sylow
349C2—81no—
350C2—241no—
351C2—241no—
352C2#559911yescharacteristic
353C2#576211yescharacteristic
354C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#461C2
socleC22×C3C22×D4
prime radical#5176C2
hypercenter#461D3
hypocenterC3#3401
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #5732C2semidirect
2. #5559C2semidirect
3. C3×#461C2semidirect
5. #5168C2semidirect
6. #5166C2semidirect
7. #5163C2semidirect
8. #5086C2semidirect
9. #5014C2semidirect
10. #4844C2semidirect
11. #4828C2semidirect
12. #4819C2semidirect
13. #4798C2semidirect
14. #4789C2semidirect
15. #4788C2semidirect
69. #461D3semidirect
348. C3#3401semidirect

Maximal subgroups

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

subgroupindex
#57322
#55592
C3×#4612
#51762
#51682
#51662
#51632
#50862
#50142
#48442
#48282
#48192
#47982
#47892
#47882
#34013