Finite Group Databasebeta

← #80010

Subgroups — #80010

Order 384.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5805C211yesmaximal, characteristic, complemented
2#5608C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#5159C211yesmaximal, characteristic, complemented
6#5158C211yesmaximal, characteristic, complemented
7#5158C211yesmaximal, characteristic, complemented
8#5157C211yesmaximal, characteristic, complemented
9#5035C211yesmaximal, characteristic, complemented
10#4780C211yesmaximal, characteristic, complemented
11#4778C211yesmaximal, characteristic, complemented
12#4777C211yesmaximal, characteristic, complemented
13#4774C211yesmaximal, characteristic, complemented
14#4727C211yesmaximal, characteristic, complemented
15#4723C211yesmaximal, characteristic, complemented
16#3198—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C2×#237C2211yescharacteristic, complemented
19C2×#235C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic
22C3×#121C2211yescharacteristic
23#953—21no—
24#953—21no—
25#940—21no—
26#939—21no—
27#938—21no—
28#938C2211yescharacteristic
29#938C2211yescharacteristic
30#938C2211yescharacteristic
31#938C2211yescharacteristic
32#938C2211yescharacteristic, complemented
33#936—21no—
34C2×#214C2211yescharacteristic
35C2×#214C2211yescharacteristic
36#926C2211yescharacteristic
37#924C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
39C2×D3×C8C2211yescharacteristic, complemented
40#897C2211yescharacteristic, complemented
41D3×C4⋊C4C2211yescharacteristic, complemented
42#884—21no—
43#878—21no—
44C3×#102C2211yescharacteristic
45C3×#102C2211yescharacteristic
46C3×C22⋊C8C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic, complemented
50#832C2211yescharacteristic, complemented
51#830C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54#818C2211yescharacteristic, complemented
55#817C2211yescharacteristic, complemented
56#816C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#537—31no—
61#535—31no—
62#485—31no—
63#482—31no—
64#481—31no—
65#480—31no—
66#469—31no—
67#468—31no—
68#466—31no—
69#465—31no—
70#450—31no—
71#449—31no—
72#449—31no—
73#447D311yescharacteristic, complemented
74#408—31no—
75C24×C3—21no—
76C2×C3×D4—21no—
77C2×C3×D4—21no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67D411yescharacteristic
91C2×#67D411yescharacteristic
92C22×Dic3—21no—
93#237—41no—
94#237—41no—
95#235—41no—
96#235—41no—
97C2×D12D411yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3C2311yescharacteristic
100C2×C4×D3C2311yescharacteristic
101C2×Dic6D411yescharacteristic
102C2×Dic6C2311yescharacteristic
103C3×D8—41no—
104C3×D8—41no—
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C22⋊C4—21no—
109#217—21no—
110#217—21no—
111#217—21no—
112#217—21no—
113#217—21no—
114#217—21no—
115#217—21no—
116#214—41no—
117#214—41no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3C2311yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122C4⋊Dic3C2311yescharacteristic
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125C4×Dic3—21no—
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128ΓL(1,𝔽25)—41no—
129D3×C8—41no—
130C2×#41—31no—
131C2×#41—31no—
132C22×D4C2×D311yescharacteristic
133C2×SD16—31no—
134C2×SD16—31no—
135C2×D8C2×D311yescharacteristic
136C2×D8C2×D311yescharacteristic
137C2×M16—31no—
138C22×C8—31no—
139#127—61no—
140#124—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144#121—31no—
145#121—31no—
146#121—31no—
147#121—31no—
148#121—31no—
149#121—61no—
150#121—61no—
151#121—61no—
152#121C2×D311yescharacteristic
153#120—61no—
154#120—61no—
155#117—31no—
156C2×C4⋊C4—31no—
157#107—31no—
158#107—31no—
159#106—31no—
160#106—31no—
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8C2×D311yescharacteristic
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—42no—
177C23×C3—81no—
178C23×C3—81no—
179C23×C3C2×D411yescharacteristic
180C22×D3C2×D411yescharacteristic
181C22×D3C2×D411yescharacteristic
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—42no—
189C3×D4—81no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2411yescharacteristic
195#67—41no—
196#67—41no—
197#67—42no—
198#67—81no—
199#67—81no—
200#67—81no—
201#67—81no—
202#67—81no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—42no—
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210D12—41no—
211C4×D3—21no—
212C4×D3—21no—
213C4×D3—21no—
214C4×D3—21no—
215C4×D3—81no—
216Dic6—21no—
217Dic6—21no—
218Dic6—41no—
219C3×C8—21no—
220C3×C8—21no—
221C3⋊C8—21no—
222C3⋊C8—21no—
223C24—21no—
224#41—121no—
225#41—121no—
226#41—121no—
227#41—121no—
228C2×Q8—31no—
229C2×Q8—31no—
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—31no—
233C2×D4—31no—
234C2×D4—31no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4C22×D311yescharacteristic
244C2×D4C22×D311yescharacteristic
245C2×D4C22×D311yescharacteristic
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4C22×D311yescharacteristic
251SD16—121no—
252SD16—121no—
253D8—41no—
254D8—41no—
255M16—121no—
256C2×C8—31no—
257C2×C8—31no—
258C2×C8—121no—
259C2×C8C22×D311yescharacteristic
260C2×C8C22×D311yescharacteristic
261C4⋊C4—31no—
262C4⋊C4—31no—
263C4⋊C4—31no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4—61no—
267C4⋊C4C22×D311yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C42—61no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—84no—
291C22×C3C22×D411yescharacteristic
292C2×D3—21no—
293C2×D3—21no—
294C2×D3—21no—
295C2×D3—21no—
296C2×D3—21no—
297C2×D3—21no—
298C3×C4—21no—
299C3×C4—41no—
300C3×C4H(5,𝔽2)11yescharacteristic
301C3×C4H(5,𝔽2)11yescharacteristic
302Dic3—21no—
303Dic3—21no—
304Dic3—41no—
305Dic3—41no—
306Dic3—41no—
307Dic3—41no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—31no—
312C23—31no—
313C23—42no—
314C23—81no—
315C23—81no—
316C23D3×D411yescharacteristic
317Q8—61no—
318Q8—61no—
319Q8—121no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—81no—
328D4—121no—
329D4—121no—
330D4—121no—
331D4—122no—
332D4—241no—
333D4—241no—
334D4—241no—
335D4—241no—
336D4—241no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—122no—
351C2×C4—241no—
352C2×C4D3×D411yescharacteristic
353C2×C4C23×D311yescharacteristic
354C8—21no—
355C8—21no—
356C8—61no—
357C8—61no—
358C2×C3—41no—
359C2×C3—81no—
360C2×C3—81no—
361C2×C3—81no—
362C2×C3#53411yescharacteristic
363C2×C3#57611yescharacteristic
364C2×C3#57711yescharacteristic
365D3—41no—
366D3—41no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—61no—
380C22—61no—
381C22—61no—
382C22—61no—
383C22—61no—
384C22—61no—
385C22—84no—
386C22C2×D3×D411yescharacteristic
387C4—21no—
388C4—41no—
389C4—61no—
390C4—61no—
391C4—121no—
392C4—121no—
393C4—121no—
394C4—121no—
395C4#100411yescharacteristic
396C4#100411yescharacteristic
397C3#319811yescharacteristic, complemented, Sylow
398C2—41no—
399C2—81no—
400C2—81no—
401C2—81no—
402C2—121no—
403C2—121no—
404C2#558911yescharacteristic
405C2#575711yescharacteristic
406C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3198
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. #5805C2semidirect
2. #5608C2semidirect
3. C3×#447C2semidirect
4. #5218C2semidirect
5. #5159C2semidirect
6. #5158C2semidirect
7. #5158C2semidirect
8. #5157C2semidirect
9. #5035C2semidirect
10. #4780C2semidirect
11. #4778C2semidirect
12. #4777C2semidirect
13. #4774C2semidirect
14. #4727C2semidirect
15. #4723C2semidirect
18. C2×#237C22semidirect
32. #938C22semidirect
38. C2×ΓL(1,𝔽25)C22semidirect
39. C2×D3×C8C22semidirect
40. #897C22semidirect
41. D3×C4⋊C4C22semidirect
49. #832C22semidirect
50. #832C22semidirect
52. (C2×D3)⋊C8C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. #818C22semidirect
55. #817C22semidirect
56. #816C22semidirect
57. #816C22semidirect
58. #808C22semidirect
59. #807C22semidirect
73. #447D3semidirect
397. C3#3198semidirect

Maximal subgroups

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

subgroupindex
#58052
#56082
C3×#4472
#52182
#51592
#51582
#51572
#50352
#47802
#47782
#47772
#47742
#47272
#47232
#31983