Finite Group Databasebeta

← #79999

Subgroups — #79999

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#5805C211yesmaximal, characteristic, complemented
2#5603C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#5155C211yesmaximal, characteristic, complemented
6#5154C211yesmaximal, characteristic, complemented
7#5153C211yesmaximal, characteristic, complemented
8#5150C211yesmaximal, characteristic, complemented
9#5037C211yesmaximal, characteristic, complemented
10#4767C211yesmaximal, characteristic, complemented
11#4765C211yesmaximal, characteristic, complemented
12#4762C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3224—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C3×#121C2211yescharacteristic
22#953—21no—
23#953—21no—
24#940C2211yescharacteristic
25#940C2211yescharacteristic
26#938—21no—
27#938—21no—
28#936—21no—
29#936C2211yescharacteristic
30#936C2211yescharacteristic
31#936C2211yescharacteristic, complemented
32#935—21no—
33D4×Dic3C2211yescharacteristic, complemented
34C2×#214C2211yescharacteristic
35C2×#214C2211yescharacteristic
36C2×#214C2211yescharacteristic, complemented
37C2×#213C2211yescharacteristic, complemented
38#926C2211yescharacteristic
39#924C2211yescharacteristic
40C2×Dic12C2211yescharacteristic, complemented
41C2×#204C2211yescharacteristic, complemented
42#888C2211yescharacteristic, complemented
43#887C2211yescharacteristic, complemented
44#879—21no—
45#878—21no—
46C3×#102C2211yescharacteristic
47C3×#102C2211yescharacteristic
48C3×C22⋊C8C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#830C2211yescharacteristic
52#821C2211yescharacteristic, complemented
53#816C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56Dic3⋊C8C2211yescharacteristic, complemented
57C8×Dic3C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#809C2211yescharacteristic, 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#459—31no—
70#452—31no—
71#450—31no—
72#449—31no—
73#447D311yescharacteristic, complemented
74#433—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—
89C22×Dic3—21no—
90#235—41no—
91#235—41no—
92C2×D12C2311yescharacteristic
93C2×C4×D3—21no—
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—
103#217—21no—
104#217—21no—
105#217—21no—
106#217—21no—
107#217—21no—
108#217—21no—
109#217—21no—
110#217—21no—
111#217D411yescharacteristic
112#217D411yescharacteristic
113#214—41no—
114#214—41no—
115#214—41no—
116#213—41no—
117C4⋊Dic3D411yescharacteristic
118C4⋊Dic3D411yescharacteristic
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123C4×Dic3—21no—
124C4×Dic3C2311yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129#204—41no—
130C2×#41—31no—
131C22×D4C2×D311yescharacteristic
132C2×Q16—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×D8—31no—
138C2×D8C2×D311yescharacteristic
139C2×D8C2×D311yescharacteristic
140C4⋊Q8—31no—
141#127—31no—
142#127—31no—
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—61no—
152#121—61no—
153#121C2×D311yescharacteristic
154#120—61no—
155#120—61no—
156C4×Q8—31no—
157C4×D4—31no—
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8—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⋊C8C2×D311yescharacteristic
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—42no—
177C23×C3—81no—
178C23×C3—81no—
179C23×C3C2×D411yescharacteristic
180C22×D3—21no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—42no—
188C3×D4—81no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—42no—
195#67—81no—
196#67—81no—
197#67—81no—
198#67—81no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—81no—
203C2×Dic3C2×D411yescharacteristic
204C2×Dic3C2×D411yescharacteristic
205C2×Dic3C2×D411yescharacteristic
206C2×Dic3C2×D411yescharacteristic
207D12—21no—
208D12—21no—
209C4×D3—42no—
210Dic6—21no—
211Dic6—21no—
212Dic6—21no—
213Dic6—21no—
214Dic6—81no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—21no—
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—41no—
229C2×D4—41no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4C22×D311yescharacteristic
236C2×D4C22×D311yescharacteristic
237C2×D4C22×D311yescharacteristic
238C22×C4—61no—
239C22×C4—61no—
240C22×C4C22×D311yescharacteristic
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—31no—
273C42—61no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—84no—
287C22×C3C22×D411yescharacteristic
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C3×C4—21no—
292C3×C4—41no—
293C3×C4H(5,𝔽2)11yescharacteristic
294C3×C4H(5,𝔽2)11yescharacteristic
295Dic3—21no—
296Dic3—21no—
297Dic3—21no—
298Dic3—21no—
299Dic3—41no—
300Dic3—41no—
301Dic3—41no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—42no—
306C23—61no—
307C23—81no—
308C23—81no—
309C23D3×D411yescharacteristic
310Q8—61no—
311Q8—61no—
312Q8—61no—
313Q8—61no—
314Q8—241no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—61no—
323D4—61no—
324D4—81no—
325D4—122no—
326D4—241no—
327D4—241no—
328D4—241no—
329D4—241no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—122no—
341C2×C4—241no—
342C2×C4D3×D411yescharacteristic
343C2×C4C23×D311yescharacteristic
344C8—21no—
345C8—21no—
346C8—61no—
347C8—61no—
348C2×C3—41no—
349C2×C3—81no—
350C2×C3—81no—
351C2×C3—81no—
352C2×C3#53511yescharacteristic
353C2×C3#57611yescharacteristic
354C2×C3#57711yescharacteristic
355D3—81no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22C2×D3×D411yescharacteristic
373C4—21no—
374C4—41no—
375C4—61no—
376C4—61no—
377C4—61no—
378C4—61no—
379C4—121no—
380C4—121no—
381C4—121no—
382C4#100411yescharacteristic
383C4#100411yescharacteristic
384C3#322411yescharacteristic, complemented, Sylow
385C2—41no—
386C2—81no—
387C2—81no—
388C2—81no—
389C2—241no—
390C2#559211yescharacteristic
391C2#575711yescharacteristic
392C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3224
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. #5603C2semidirect
3. C3×#447C2semidirect
4. #5218C2semidirect
5. #5155C2semidirect
6. #5154C2semidirect
7. #5153C2semidirect
8. #5150C2semidirect
9. #5037C2semidirect
10. #4767C2semidirect
11. #4765C2semidirect
12. #4762C2semidirect
13. #4760C2semidirect
14. #4733C2semidirect
15. #4720C2semidirect
31. #936C22semidirect
33. D4×Dic3C22semidirect
36. C2×#214C22semidirect
37. C2×#213C22semidirect
40. C2×Dic12C22semidirect
41. C2×#204C22semidirect
42. #888C22semidirect
43. #887C22semidirect
52. #821C22semidirect
53. #816C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
56. Dic3⋊C8C22semidirect
57. C8×Dic3C22semidirect
58. #810C22semidirect
59. #809C22semidirect
73. #447D3semidirect
384. C3#3224semidirect

Maximal subgroups

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

subgroupindex
#58052
#56032
C3×#4472
#52182
#51552
#51542
#51532
#51502
#50372
#47672
#47652
#47622
#47602
#47332
#47202
#32243