Finite Group Databasebeta

← #80923

Subgroups — #80923

Order 384.

472 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#918C211yesmaximal, characteristic, complemented
2C2×#909C211yesmaximal, characteristic, complemented
3#5632C211yesmaximal, characteristic, complemented
4#5631C211yesmaximal, characteristic
5C3×#470C211yesmaximal, characteristic, complemented
6#5183C211yesmaximal, characteristic, complemented
7#5128C211yesmaximal, characteristic, complemented
8#5050C211yesmaximal, characteristic, complemented
9#5047C211yesmaximal, characteristic, complemented
10#4867C211yesmaximal, characteristic, complemented
11#4852C211yesmaximal, characteristic, complemented
12#4818C211yesmaximal, characteristic, complemented
13#4809C211yesmaximal, characteristic, complemented
14#4808C211yesmaximal, characteristic, complemented
15#4795C211yesmaximal, characteristic, complemented
16#3276—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×D3×Q8C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic
22C2×C3×M16C2211yescharacteristic
23C3×C22⋊Q8C2211yescharacteristic
24C3×C22⋊Q8C2211yescharacteristic
25#944C2211yescharacteristic
26#944C2211yescharacteristic
27C2×#216C2211yescharacteristic
28C2×#216C2211yescharacteristic
29C2×#216C2211yescharacteristic
30C2×#215C2211yescharacteristic
31C4×#67C2211yescharacteristic
32#918—24no—
33#909—24no—
34C2×Dic12C2211yescharacteristic
35C2×Dic12C2211yescharacteristic
36C2×#204C2211yescharacteristic
37C2×#204C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic
39#897C2211yescharacteristic
40#897C2211yescharacteristic
41#896—21no—
42#894—21no—
43#893C2211yescharacteristic
44D3×C4⋊C4—21no—
45#888—21no—
46#887C2211yescharacteristic
47#884—21no—
48#883—21no—
49#67⋊C4—21no—
50#878—21no—
51C3×#106C2211yescharacteristic
52C3×#103C2211yescharacteristic
53C3×#103C2211yescharacteristic
54#830C2211yescharacteristic
55(C2×D3)⋊C8C2211yescharacteristic
56Dic3⋊C8C2211yescharacteristic
57C8×Dic3C2211yescharacteristic
58#810C2211yescharacteristic
59#810C2211yescharacteristic
60#810C2211yescharacteristic
61#809C2211yescharacteristic
62C2×#137—31no—
63C2×#137—31no—
64D4×Q8—31no—
65#547—31no—
66#494—31no—
67#470—31no—
68#470D311yescharacteristic, complemented
69#462—31no—
70#461—31no—
71#452—31no—
72#452—31no—
73#451—31no—
74#451—31no—
75C4×SD16—31no—
76#436—31no—
77C2×C3×Q8C2311yescharacteristic
78C2×C3×Q8C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67D411yescharacteristic
82C22×Dic3—21no—
83#239—22no—
84#239—41no—
85D3×Q8—22no—
86D3×Q8—41no—
87#235—22no—
88#235—41no—
89C2×D12—21no—
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C2×Dic6—22no—
97C2×Dic6—21no—
98C2×Dic6—21no—
99C2×Dic6—41no—
100C2×Dic6C2311yescharacteristic
101C2×Dic6C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C3×Q16—22no—
104C3×M16—22no—
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4C2311yescharacteristic
110C3×C4⋊C4C2311yescharacteristic
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113#217D411yescharacteristic
114#216—22no—
115#216—41no—
116#216—41no—
117#215—22no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4D411yescharacteristic
122C4⋊Dic3—21no—
123C4⋊Dic3—21no—
124Dic3⋊C4—21no—
125Dic3⋊C4—21no—
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128Dic3⋊C4—21no—
129Dic3⋊C4D411yescharacteristic
130C4×Dic3—21no—
131C4×Dic3C2311yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133C2×C3⋊C8C2311yescharacteristic
134Dic12—22no—
135Dic12—22no—
136#204—22no—
137#204D414yescomplemented
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C22×Q8—31no—
142C22×Q8—31no—
143#137—64no—
144#137—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×Q16—31no—
148C2×Q16—31no—
149C2×Q16—31no—
150C2×Q16C2×D311yescharacteristic
151C2×SD16—31no—
152C2×SD16—31no—
153C2×SD16—31no—
154C2×M16—31no—
155C2×M16C2×D311yescharacteristic
156C4⋊Q8—31no—
157C4⋊Q8—31no—
158C4⋊Q8—61no—
159#123—61no—
160#123—61no—
161C22⋊Q8—31no—
162C22⋊Q8—31no—
163C22⋊Q8—61no—
164C22⋊Q8—61no—
165C22⋊Q8C2×D311yescharacteristic
166C22⋊Q8C2×D311yescharacteristic
167#121—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—31no—
171C4×D4—61no—
172C2×C4⋊C4—61no—
173#106C2×D311yescharacteristic
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#103C2×D311yescharacteristic
179#103C2×D311yescharacteristic
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8—31no—
183C4×C8—31no—
184C23×C3C2×D411yescharacteristic
185C22×D3—21no—
186C22×D3C2×D411yescharacteristic
187C3×Q8—21no—
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2411yescharacteristic
199#67—22no—
200#67—42no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—42no—
205C2×Dic3C2×D411yescharacteristic
206C2×Dic3C2×D411yescharacteristic
207C2×Dic3C2×D411yescharacteristic
208D12—21no—
209D12—42no—
210D12C2×D412yes—
211C4×D3—21no—
212C4×D3—21no—
213C4×D3—42no—
214C4×D3—42no—
215C4×D3—42no—
216C4×D3—42no—
217Dic6—21no—
218Dic6—21no—
219Dic6—21no—
220Dic6—21no—
221Dic6—21no—
222Dic6—42no—
223Dic6—42no—
224Dic6C2×D412yes—
225C3×C8—21no—
226C3×C8C2×D412yes—
227C3⋊C8—21no—
228C3⋊C8—21no—
229#41—62no—
230#41—62no—
231#41—121no—
232#41—121no—
233C2×Q8—31no—
234C2×Q8—31no—
235C2×Q8—31no—
236C2×Q8—61no—
237C2×Q8—62no—
238C2×Q8—61no—
239C2×Q8—62no—
240C2×Q8—121no—
241C2×Q8—121no—
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—61no—
247C2×D4—61no—
248C22×C4—31no—
249C22×C4—61no—
250C22×C4—61no—
251C22×C4—61no—
252C22×C4—61no—
253C22×C4—61no—
254C22×C4C22×D311yescharacteristic
255Q16—22no—
256Q16—62no—
257Q16—62no—
258Q16—62no—
259Q16—121no—
260Q16—121no—
261SD16—34no—
262SD16—62no—
263SD16—62no—
264M16—22no—
265M16—62no—
266C2×C8—31no—
267C2×C8—31no—
268C2×C8C22×D311yescharacteristic
269C2×C8C22×D311yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—31no—
273C4⋊C4—61no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4C22×D311yescharacteristic
281C4⋊C4C22×D311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—31no—
285C22⋊C4—31no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C42—31no—
290C42—61no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3C22×D411yescharacteristic
295C2×D3—21no—
296C2×D3—22no—
297C2×D3—21no—
298C2×D3—21no—
299C2×D3—41no—
300C2×D3—41no—
301C3×C4—21no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4C22×D411yescharacteristic
307C3×C4H(5,𝔽2)11yescharacteristic
308Dic3—22no—
309Dic3—21no—
310Dic3—21no—
311Dic3—41no—
312Dic3—41no—
313Dic3#13712yescomplemented
314C23—31no—
315C23—61no—
316C23D3×D411yescharacteristic
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—21no—
321Q8—32no—
322Q8—61no—
323Q8—61no—
324Q8—61no—
325Q8—61no—
326Q8—61no—
327Q8—122no—
328Q8—122no—
329D4—32no—
330D4—62no—
331D4—61no—
332D4—122no—
333D4—122no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—122no—
349C2×C4—122no—
350C2×C4—122no—
351C2×C4—122no—
352C2×C4—122no—
353C2×C4D3×D411yescharacteristic
354C2×C4C23×D311yescharacteristic
355C8—21no—
356C8—61no—
357C8—61no—
358C8D3×D412yes—
359C2×C3—41no—
360C2×C3D4211yescharacteristic
361C2×C3C2×#13711yescharacteristic
362C2×C3#57811yescharacteristic
363D3—41no—
364D3—42no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—61no—
369C22—61no—
370C22—61no—
371C22—62no—
372C22—121no—
373C22—121no—
374C22C2×D3×D411yescharacteristic
375C4—21no—
376C4—32no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—61no—
382C4—61no—
383C4—62no—
384C4—121no—
385C4—121no—
386C4C2×D3×D411yescharacteristic
387C4#100911yescharacteristic
388C3#327611yescharacteristic, complemented, Sylow
389C2—41no—
390C2—121no—
391C2—122no—
392C2#560911yescharacteristic
393C2D3×#13711yescharacteristic
394C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#470C2
socleC22×C3C22×D4
prime radical#5631C2
hypercenter#470D3
hypocenterC3#3276
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#918C2semidirect
2. C2×#909C2semidirect
3. #5632C2semidirect
5. C3×#470C2semidirect
6. #5183C2semidirect
7. #5128C2semidirect
8. #5050C2semidirect
9. #5047C2semidirect
10. #4867C2semidirect
11. #4852C2semidirect
12. #4818C2semidirect
13. #4809C2semidirect
14. #4808C2semidirect
15. #4795C2semidirect
68. #470D3semidirect
137. #204D4semidirect
313. Dic3#137semidirect
388. C3#3276semidirect

Maximal subgroups

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

subgroupindex
C2×#9182
C2×#9092
#56322
#56312
C3×#4702
#51832
#51282
#50502
#50472
#48672
#48522
#48182
#48092
#48082
#47952
#32763