Finite Group Databasebeta

← #80341

Subgroups — #80341

Order 384.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5715C211yesmaximal, characteristic, complemented
2#5561C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5158C211yesmaximal, characteristic, complemented
5#5156C211yesmaximal, characteristic, complemented
6#5155C211yesmaximal, characteristic, complemented
7#5154C211yesmaximal, characteristic, complemented
8#5075C211yesmaximal, characteristic, complemented
9#5016C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4783C211yesmaximal, characteristic, complemented
13#4776C211yesmaximal, characteristic, complemented
14#4765C211yesmaximal, characteristic, complemented
15#4760C211yesmaximal, characteristic, complemented
16#3224—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic, complemented
21C3×#127C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic, complemented
23#940C2211yescharacteristic
24#939—21no—
25#939—21no—
26#938C2211yescharacteristic
27#937—21no—
28#937—21no—
29#936C2211yescharacteristic
30#936C2211yescharacteristic, complemented
31C2×#214C2211yescharacteristic
32C2×#214C2211yescharacteristic, complemented
33C2×#214C2211yescharacteristic, complemented
34C2×#213C2211yescharacteristic
35#929—21no—
36#924—21no—
37C2×Dic12C2211yescharacteristic, complemented
38C2×#204C2211yescharacteristic
39#897C2211yescharacteristic
40#888C2211yescharacteristic, complemented
41#885—21no—
42#879—21no—
43#875C2211yescharacteristic
44#872C2211yescharacteristic
45C4×Dic6C2211yescharacteristic, complemented
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#102C2211yescharacteristic
48C3×#102C2211yescharacteristic, complemented
49#832C2211yescharacteristic
50#832C2211yescharacteristic, complemented
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#816C2211yescharacteristic, complemented
55#815C2211yescharacteristic, complemented
56C8×Dic3C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#809C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic, complemented
60#542—31no—
61#535—31no—
62#496—31no—
63#495—31no—
64#489—31no—
65#486—31no—
66#464—31no—
67#463—31no—
68#460—31no—
69#459D311yescharacteristic, complemented
70#452—31no—
71#450—31no—
72#449—31no—
73#447—31no—
74#433—31no—
75C23×D3—21no—
76C2×C3×D4—21no—
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C22×Dic3—21no—
88#237—41no—
89#237—41no—
90D3×D4—41no—
91D3×D4—41no—
92C2×D12C2311yescharacteristic
93C2×C4×D3C2311yescharacteristic
94C2×Dic6—21no—
95C2×Dic6C2311yescharacteristic
96C2×Dic6C2311yescharacteristic
97C3×D8—41no—
98C3×D8—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102C3×C22⋊C4—21no—
103C3×C42C2311yescharacteristic
104#217—21no—
105#217—21no—
106#217—21no—
107#217—21no—
108#217—21no—
109#214—41no—
110#214—41no—
111#214—41no—
112#213—41no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4D411yescharacteristic
117(C2×D3)⋊C4D411yescharacteristic
118C4⋊Dic3—21no—
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4D411yescharacteristic
123Dic3⋊C4D411yescharacteristic
124C4×Dic3—21no—
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129#204—41no—
130C2×#41—31no—
131C22×D4—31no—
132C2×Q16—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×D8—31no—
138C2×D8C2×D311yescharacteristic
139C2×D8C2×D311yescharacteristic, complemented
140C4⋊Q8—31no—
141#127—31no—
142#127C2×D311yescharacteristic
143#126—61no—
144#124—31no—
145#124—31no—
146#124—31no—
147#124—61no—
148#123—61no—
149C22⋊Q8—31no—
150C22⋊Q8—61no—
151#121—31no—
152#121—61no—
153#121—61no—
154#120—61no—
155#120—61no—
156C4×Q8—31no—
157C4×D4C2×D311yescharacteristic, complemented
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic, complemented
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168#102C2×D311yescharacteristic, complemented
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C22×D3—21no—
177C22×D3—42no—
178C22×D3—81no—
179C22×D3—81no—
180C22×D3C2×D411yescharacteristic
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—81no—
188C3×D4—81no—
189C2×C3×C4—21no—
190C2×C3×C4—81no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—42no—
195#67—42no—
196#67—81no—
197#67—81no—
198#67—81no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—42no—
203C2×Dic3C2×D411yescharacteristic
204C2×Dic3C2×D411yescharacteristic
205C2×Dic3C2×D411yescharacteristic
206D12—21no—
207D12—21no—
208C4×D3—21no—
209C4×D3—21no—
210Dic6—21no—
211Dic6—21no—
212Dic6—21no—
213Dic6—21no—
214Dic6—81no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—61no—
220#41—121no—
221#41—121no—
222C2×Q8—31no—
223C2×Q8—31no—
224C2×Q8—61no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—31no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—121no—
234C2×D4—121no—
235C2×D4C22×D311yescharacteristic
236C2×D4C22×D311yescharacteristic
237C2×D4C22×D311yescharacteristic
238C22×C4—21no—
239C22×C4—31no—
240C22×C4—61no—
241Q16—121no—
242SD16—121no—
243SD16—121no—
244SD16—121no—
245SD16—121no—
246D8—41no—
247D8—41no—
248D8—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8C22×D311yescharacteristic
252C2×C8C22×D311yescharacteristic
253C4⋊C4—31no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4C22×D311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—31no—
262C22⋊C4—31no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—31no—
272C42—61no—
273C42C22×D311yescharacteristic
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3C22×D411yescharacteristic
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—84no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—41no—
294C3×C4H(5,𝔽2)11yescharacteristic
295C3×C4H(5,𝔽2)11yescharacteristic
296Dic3—21no—
297Dic3—21no—
298Dic3—21no—
299Dic3—41no—
300Dic3—41no—
301Dic3—41no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—31no—
306C23—61no—
307C23—122no—
308C23—241no—
309C23—241no—
310Q8—61no—
311Q8—61no—
312Q8—61no—
313Q8—61no—
314Q8—241no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—61no—
322D4—61no—
323D4—81no—
324D4—81no—
325D4—122no—
326D4—122no—
327D4—241no—
328D4—241no—
329D4—241no—
330C2×C4—21no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—81no—
340C2×C4—122no—
341C2×C4D3×D411yescharacteristic
342C2×C4D3×D411yescharacteristic
343C2×C4C23×D311yescharacteristic
344C8—21no—
345C8—21no—
346C8—61no—
347C8—61no—
348C2×C3—81no—
349C2×C3—81no—
350C2×C3—81no—
351C2×C3#53511yescharacteristic
352C2×C3#57611yescharacteristic
353C2×C3#57711yescharacteristic
354D3—41no—
355D3—81no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—61no—
366C22—61no—
367C22—61no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—244no—
372C22C2×D3×D411yescharacteristic
373C4—21no—
374C4—21no—
375C4—41no—
376C4—61no—
377C4—61no—
378C4—61no—
379C4—121no—
380C4—121no—
381C4—121no—
382C4#100411yescharacteristic
383C4#100911yescharacteristic
384C3#322411yescharacteristic, complemented, Sylow
385C2—81no—
386C2—81no—
387C2—81no—
388C2—121no—
389C2—241no—
390C2#560711yescharacteristic
391C2#575711yescharacteristic
392C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3224
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5715C2semidirect
2. #5561C2semidirect
3. C3×#459C2semidirect
4. #5158C2semidirect
5. #5156C2semidirect
6. #5155C2semidirect
7. #5154C2semidirect
8. #5075C2semidirect
9. #5016C2semidirect
10. #4845C2semidirect
11. #4840C2semidirect
12. #4783C2semidirect
13. #4776C2semidirect
14. #4765C2semidirect
15. #4760C2semidirect
20. C2×C3×D8C22semidirect
22. C3×C4×D4C22semidirect
30. #936C22semidirect
32. C2×#214C22semidirect
33. C2×#214C22semidirect
37. C2×Dic12C22semidirect
40. #888C22semidirect
45. C4×Dic6C22semidirect
46. C3×C4⋊C8C22semidirect
48. C3×#102C22semidirect
50. #832C22semidirect
54. #816C22semidirect
55. #815C22semidirect
56. C8×Dic3C22semidirect
57. #810C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
69. #459D3semidirect
139. C2×D8C2×D3semidirect
157. C4×D4C2×D3semidirect
160. C4⋊C8C2×D3semidirect
168. #102C2×D3semidirect
384. C3#3224semidirect

Maximal subgroups

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

subgroupindex
#57152
#55612
C3×#4592
#51582
#51562
#51552
#51542
#50752
#50162
#48452
#48402
#47832
#47762
#47652
#47602
#32243