Finite Group Databasebeta

← #83549

Subgroups — #83549

Order 384.

482 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#548C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic, complemented
3C22×#216C211yesmaximal, characteristic, complemented
4#5552C211yesmaximal, characteristic, complemented
5#5239C211yesmaximal, characteristic, complemented
6#5227C211yesmaximal, characteristic, complemented
7#5059C211yesmaximal, characteristic, complemented
8#5054C211yesmaximal, characteristic, complemented
9#5051C211yesmaximal, characteristic, complemented
10#5047C211yesmaximal, characteristic, complemented
11#5040C211yesmaximal, characteristic, complemented
12#5038C211yesmaximal, characteristic, complemented
13C4×#216C211yesmaximal, characteristic, complemented
14#5028C211yesmaximal, characteristic, complemented
15D4×C3⋊C8C211yesmaximal, characteristic, complemented
16#3285—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×Q8C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C3×#124—21no—
22C3×#124C2211yescharacteristic
23C3×C22⋊Q8—21no—
24C3×C22⋊Q8C2211yescharacteristic
25C3×#121—21no—
26C3×#121C2211yescharacteristic
27C3×C4×Q8C2211yescharacteristic
28C3×C4×D4—21no—
29C3×C4×D4C2211yescharacteristic
30#950—24no—
31C2×#216—24no—
32C2×#216C2211yescharacteristic
33C2×#216C2211yescharacteristic
34C2×#216C2211yescharacteristic
35C2×#216C2211yescharacteristic
36C2×#215C2211yescharacteristic
37C2×#215C2211yescharacteristic
38#939—21no—
39C2×#214C2211yescharacteristic
40C2×#214C2211yescharacteristic
41C2×#213C2211yescharacteristic
42#930C2211yescharacteristic
43#924C2211yescharacteristic
44C22×C3⋊C8C2211yescharacteristic
45C22×C3⋊C8C2211yescharacteristic
46#896—21no—
47#885—21no—
48#67⋊C4—21no—
49#875C2211yescharacteristic
50C4×Dic6C2211yescharacteristic
51#835C2211yescharacteristic
52#835C2211yescharacteristic
53#832C2211yescharacteristic
54#830C2211yescharacteristic
55#830C2211yescharacteristic
56#810C2211yescharacteristic
57#810C2211yescharacteristic
58#809C2211yescharacteristic
59#807C2211yescharacteristic
60C4⋊(C3⋊C8)C2211yescharacteristic
61C4×C3⋊C8C2211yescharacteristic
62C2×#135—31no—
63C22×Q16—31no—
64#548—31no—
65#548D311yescharacteristic, complemented
66#495—31no—
67#467—31no—
68#466—31no—
69#464—31no—
70#464—31no—
71#452—31no—
72#452—31no—
73#448—31no—
74#448—31no—
75C4×Q16—31no—
76C8×D4—31no—
77C3×#41—22no—
78C3×#41—41no—
79C2×C3×Q8—21no—
80C2×C3×Q8—22no—
81C2×C3×Q8—41no—
82C2×C3×Q8C2311yescharacteristic
83C2×C3×Q8C2311yescharacteristic
84C2×C3×D4—21no—
85C2×C3×D4—21no—
86C2×C3×D4D411yescharacteristic
87C2×C3×D4C2311yescharacteristic
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C2×#67—21no—
93C2×#67—21no—
94C22×Dic3—21no—
95#235—22no—
96#235—41no—
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×Dic6—22no—
100C2×Dic6—21no—
101C2×Dic6—41no—
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C3×C4⋊C4—21no—
105C3×C4⋊C4—21no—
106C3×C4⋊C4D411yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C22⋊C4D411yescharacteristic
113C3×C22⋊C4D411yescharacteristic
114C3×C42—21no—
115C3×C42C2311yescharacteristic
116#217—21no—
117#216—22no—
118#216—22no—
119#216—22no—
120#216D414yescomplemented
121#215—22no—
122#215—41no—
123#214—22no—
124#214—41no—
125#213—22no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4—21no—
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132C4×Dic3—21no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137C2×C3⋊C8C2311yescharacteristic
138C2×C3⋊C8C2311yescharacteristic
139C2×#41—31no—
140C2×#41C2×D311yescharacteristic
141C22×Q8—31no—
142C22×Q8C2×D311yescharacteristic
143#135—64no—
144C2×Q16—31no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×Q16—31no—
148C2×Q16—64no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×SD16—31no—
153C2×D8—31no—
154C22×C8—31no—
155C22×C8—31no—
156#124—21no—
157#124—31no—
158#124—61no—
159#124C2×D311yescharacteristic
160C22⋊Q8—21no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163C22⋊Q8C2×D311yescharacteristic
164#121—21no—
165#121—31no—
166#121—61no—
167#121C2×D311yescharacteristic
168C4×Q8—31no—
169C4×Q8C2×D311yescharacteristic
170C4×D4—21no—
171C4×D4—61no—
172C4×D4C2×D311yescharacteristic
173#107—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#103—31no—
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8—31no—
183C4×C8—31no—
184C23×C3—21no—
185C23×C3C2×D411yescharacteristic
186C23×C3C2×D411yescharacteristic
187C22×D3—21no—
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—21no—
191C3×Q8—42no—
192C3×Q8C2×D412yes—
193C3×D4—22no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—42no—
197C3×D4—42no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4C2×D411yescharacteristic
208C2×C3×C4C2×D411yescharacteristic
209C2×C3×C4C2×D411yescharacteristic
210C2×C3×C4C2×D411yescharacteristic
211C2×C3×C4C2411yescharacteristic
212#67—42no—
213#67—42no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—42no—
218D12—21no—
219D12—21no—
220C4×D3—42no—
221Dic6—21no—
222Dic6—21no—
223Dic6—21no—
224Dic6—42no—
225Dic6C2×D412yes—
226C3⋊C8—21no—
227C3⋊C8—21no—
228C3⋊C8—21no—
229C3⋊C8C2×D412yes—
230#41—22no—
231#41—41no—
232#41—62no—
233#41—121no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—31no—
237C2×Q8—31no—
238C2×Q8—41no—
239C2×Q8—62no—
240C2×Q8—61no—
241C2×Q8—121no—
242C2×Q8C22×D311yescharacteristic
243C2×Q8C22×D311yescharacteristic
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—31no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4#6711yescharacteristic
250C2×D4C22×D311yescharacteristic
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—61no—
254C22×C4—61no—
255C22×C4C22×D311yescharacteristic
256C22×C4C22×D311yescharacteristic
257Q16—34no—
258Q16—62no—
259Q16—62no—
260Q16—62no—
261SD16—62no—
262SD16—62no—
263SD16—121no—
264SD16—121no—
265D8—62no—
266C2×C8—31no—
267C2×C8—31no—
268C2×C8—31no—
269C2×C8—31no—
270C2×C8—62no—
271C2×C8—62no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—31no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4#6711yescharacteristic
278C4⋊C4C22×D311yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—21no—
282C22⋊C4—21no—
283C22⋊C4—61no—
284C22⋊C4—61no—
285C22⋊C4—61no—
286C22⋊C4—61no—
287C22⋊C4#6711yescharacteristic
288C22⋊C4#6711yescharacteristic
289C42—21no—
290C42—61no—
291C42C22×D311yescharacteristic
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3#13512yescomplemented
301C22×C3C22×D411yescharacteristic
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—41no—
305C3×C4—21no—
306C3×C4—21no—
307C3×C4—22no—
308C3×C4—21no—
309C3×C4—21no—
310C3×C4—41no—
311C3×C4—41no—
312C3×C4C22×D411yescharacteristic
313C3×C4H(5,𝔽2)11yescharacteristic
314Dic3—22no—
315Dic3—41no—
316Dic3—41no—
317C23—21no—
318C23—61no—
319C23C2×#6711yescharacteristic
320C23C2×#6711yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—32no—
325Q8—42no—
326Q8—61no—
327Q8—61no—
328Q8—61no—
329Q8—122no—
330Q8D3×D412yes—
331D4—22no—
332D4—21no—
333D4—21no—
334D4—42no—
335D4—42no—
336D4—61no—
337D4—61no—
338D4—122no—
339D4—122no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—122no—
353C2×C4—122no—
354C2×C4C2×#6711yescharacteristic
355C2×C4C2×#6711yescharacteristic
356C2×C4C2×#6711yescharacteristic
357C2×C4C2×#6711yescharacteristic
358C2×C4C23×D311yescharacteristic
359C8—32no—
360C8—61no—
361C8—61no—
362C8—61no—
363C2×C3—22no—
364C2×C3—41no—
365C2×C3—81no—
366C2×C3D4211yescharacteristic
367C2×C3C2×#13511yescharacteristic
368C2×C3#57811yescharacteristic
369D3—81no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22#95012yescomplemented
382C22C22×#6711yescharacteristic
383C4—21no—
384C4—21no—
385C4—22no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—62no—
391C4—121no—
392C4—121no—
393C4C2×D3×D411yescharacteristic
394C4#100411yescharacteristic
395C3#328511yescharacteristic, complemented, Sylow
396C2—22no—
397C2—41no—
398C2—81no—
399C2—241no—
400C2D4×#6711yescharacteristic
401C2C2×#95011yescharacteristic
402C2#583811yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#548C2
socleC22×C3C22×D4
hypercenter#548D3
hypocenterC3#3285
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C3×#548C2semidirect
2. C2×#950C2semidirect
3. C22×#216C2semidirect
4. #5552C2semidirect
5. #5239C2semidirect
6. #5227C2semidirect
7. #5059C2semidirect
8. #5054C2semidirect
9. #5051C2semidirect
10. #5047C2semidirect
11. #5040C2semidirect
12. #5038C2semidirect
13. C4×#216C2semidirect
14. #5028C2semidirect
15. D4×C3⋊C8C2semidirect
65. #548D3semidirect
120. #216D4semidirect
300. C22×C3#135semidirect
381. C22#950semidirect
395. C3#3285semidirect

Maximal subgroups

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

subgroupindex
C3×#5482
C2×#9502
C22×#2162
#55522
#52392
#52272
#50592
#50542
#50512
#50472
#50402
#50382
C4×#2162
#50282
D4×C3⋊C82
#32853