Finite Group Databasebeta

← #80540

Subgroups — #80540

Order 384.

438 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5668C211yesmaximal, characteristic, complemented
2#5548C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic, complemented
4#5175C211yesmaximal, characteristic, complemented
5#5172C211yesmaximal, characteristic, complemented
6#5158C211yesmaximal, characteristic, complemented
7#5151C211yesmaximal, characteristic, complemented
8#5057C211yesmaximal, characteristic, complemented
9#5016C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4833C211yesmaximal, characteristic, complemented
12#4810C211yesmaximal, characteristic, complemented
13#4792C211yesmaximal, characteristic, complemented
14#4781C211yesmaximal, characteristic, complemented
15#4760C211yesmaximal, characteristic, complemented
16#3207—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×#235—21no—
20C2×C3×SD16C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic
22C3×#124C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic
24#944C2211yescharacteristic
25C2×#215C2211yescharacteristic
26#940C2211yescharacteristic
27#940C2211yescharacteristic
28#938C2211yescharacteristic
29C2×#214C2211yescharacteristic
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic
32C4×#67—21no—
33C2×D24C2211yescharacteristic
34C2×#204C2211yescharacteristic
35#895C2211yescharacteristic
36#888C2211yescharacteristic
37#885—21no—
38#884—21no—
39#884—21no—
40#884—21no—
41#67⋊C4—21no—
42#67⋊C4—21no—
43#875C2211yescharacteristic
44C4×D12C2211yescharacteristic
45C42×D3C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic
47C3×#103C2211yescharacteristic
48C3×#102C2211yescharacteristic
49#835C2211yescharacteristic
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#816C2211yescharacteristic
55C8×Dic3C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#810C2211yescharacteristic
58#809C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#532—31no—
61#532—31no—
62#493—31no—
63#492—31no—
64#487—31no—
65#486—31no—
66#463—31no—
67#463—31no—
68#463—31no—
69#463D311yescharacteristic, complemented
70#449—31no—
71#449—31no—
72#449—31no—
73#449—31no—
74#431—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86#239—41no—
87#239—41no—
88#237—41no—
89#237—41no—
90#235—44no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12C2311yescharacteristic
94C2×D12C2311yescharacteristic
95C2×C4×D3—21no—
96C2×C4×D3—21no—
97C2×C4×D3D411yescharacteristic
98C2×C4×D3D411yescharacteristic
99C2×C4×D3C2311yescharacteristic
100C2×Dic6—21no—
101C2×Dic6C2311yescharacteristic
102C3×SD16—41no—
103C3×D8—41no—
104C2×C3×C8C2311yescharacteristic
105C2×C3×C8C2311yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107C3×C22⋊C4—21no—
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110C3×C42C2311yescharacteristic
111#217—21no—
112#215—41no—
113#214—41no—
114#213—41no—
115#213—41no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120C4⋊Dic3C2311yescharacteristic
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124C4×Dic3D411yescharacteristic
125C4×Dic3D411yescharacteristic
126C4×Dic3C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129D24—41no—
130#204—41no—
131C2×#41—31no—
132C2×#41—31no—
133C2×#41—61no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×SD16C2×D311yescharacteristic
138C2×D8—31no—
139C2×D8—31no—
140C2×D8—31no—
141C2×D8C2×D311yescharacteristic
142C4⋊Q8—31no—
143C4⋊Q8—31no—
144#127—31no—
145#127—31no—
146#124—31no—
147#124—61no—
148#124C2×D311yescharacteristic
149#121—31no—
150#121—31no—
151#121—61no—
152#121—61no—
153#121—61no—
154C4×D4—31no—
155C4×D4—61no—
156C4×D4—61no—
157C4×D4—61no—
158C4×D4C2×D311yescharacteristic
159C2×C42—31no—
160C4⋊C8—31no—
161C4⋊C8C2×D311yescharacteristic
162#103—31no—
163#103—31no—
164#103—31no—
165#103C2×D311yescharacteristic
166#102—31no—
167#102—31no—
168#102—31no—
169#102C2×D311yescharacteristic
170C22⋊C8—31no—
171C22⋊C8—31no—
172C4×C8—31no—
173C4×C8—31no—
174C23×C3—21no—
175C23×C3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3C2×D411yescharacteristic
179C3×Q8—21no—
180C3×Q8—21no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—42no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2411yescharacteristic
191#67—42no—
192#67—42no—
193#67—42no—
194#67—42no—
195#67—42no—
196#67—42no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—42no—
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12—21no—
205D12—21no—
206D12—21no—
207D12—42no—
208D12—42no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—22no—
212C4×D3—22no—
213C4×D3—42no—
214C4×D3—42no—
215Dic6—21no—
216Dic6—21no—
217Dic6—42no—
218C3×C8—21no—
219C3×C8—21no—
220C3⋊C8—21no—
221C3⋊C8—21no—
222#41—124no—
223#41—121no—
224#41—121no—
225#41—121no—
226#41—121no—
227C2×Q8—31no—
228C2×Q8—61no—
229C2×Q8C22×D311yescharacteristic
230C2×D4—31no—
231C2×D4—31no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4C22×D311yescharacteristic
241C2×D4C22×D311yescharacteristic
242C22×C4—21no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—31no—
246C22×C4—61no—
247C22×C4—61no—
248C22×C4—61no—
249SD16—41no—
250SD16—121no—
251SD16—121no—
252SD16—121no—
253D8—41no—
254D8—121no—
255D8—121no—
256D8—121no—
257C2×C8—31no—
258C2×C8—31no—
259C2×C8C22×D311yescharacteristic
260C2×C8C22×D311yescharacteristic
261C4⋊C4—31no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4C22×D311yescharacteristic
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C42—31no—
275C42—31no—
276C42—31no—
277C42C22×D311yescharacteristic
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3C22×D411yescharacteristic
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—21no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4H(5,𝔽2)11yescharacteristic
299C3×C4H(5,𝔽2)11yescharacteristic
300Dic3—21no—
301Dic3—41no—
302Dic3—41no—
303Dic3#13512yes—
304Dic3AGL(1,ℤ/8)12yes—
305C23—21no—
306C23—21no—
307C23—31no—
308C23—61no—
309C23—61no—
310Q8—21no—
311Q8—21no—
312Q8—61no—
313Q8—61no—
314Q8—122no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—61no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—122no—
324D4—122no—
325D4—122no—
326D4—122no—
327D4—122no—
328D4—122no—
329D4—122no—
330D4—122no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—42no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—122no—
344C2×C4—122no—
345C2×C4—122no—
346C2×C4D3×D411yescharacteristic
347C2×C4D3×D411yescharacteristic
348C2×C4C23×D311yescharacteristic
349C8—21no—
350C8—21no—
351C8—61no—
352C8—61no—
353C2×C3—81no—
354C2×C3—81no—
355C2×C3#53511yescharacteristic
356C2×C3C2×#13511yescharacteristic
357C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
358D3—41no—
359D3—81no—
360D3—81no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—61no—
368C22—61no—
369C22—61no—
370C22—121no—
371C22—121no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—121no—
376C22C2×D3×D411yescharacteristic
377C4—21no—
378C4—21no—
379C4—32no—
380C4—32no—
381C4—41no—
382C4—41no—
383C4—61no—
384C4—121no—
385C4—121no—
386C4#100411yescharacteristic
387C4#100911yescharacteristic
388C3#320711yescharacteristic, complemented, Sylow
389C2—81no—
390C2—81no—
391C2—121no—
392C2—241no—
393C2—241no—
394C2#560711yescharacteristic
395C2D3×#13511yescharacteristic
396C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3207
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #5668C2semidirect
2. #5548C2semidirect
3. C3×#463C2semidirect
4. #5175C2semidirect
5. #5172C2semidirect
6. #5158C2semidirect
7. #5151C2semidirect
8. #5057C2semidirect
9. #5016C2semidirect
10. #4844C2semidirect
11. #4833C2semidirect
12. #4810C2semidirect
13. #4792C2semidirect
14. #4781C2semidirect
15. #4760C2semidirect
69. #463D3semidirect
388. C3#3207semidirect

Maximal subgroups

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

subgroupindex
#56682
#55482
C3×#4632
#51752
#51722
#51582
#51512
#50572
#50162
#48442
#48332
#48102
#47922
#47812
#47602
#32073