Finite Group Databasebeta

← #80266

Subgroups — #80266

Order 384.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5834C211yesmaximal, characteristic, complemented
2C2×#949C211yesmaximal, characteristic, complemented
3C2×#919C211yesmaximal, characteristic, complemented
4#5630C211yesmaximal, characteristic, complemented
5C3×#452C211yesmaximal, characteristic, complemented
6#5185C211yesmaximal, characteristic, complemented
7#5176C211yesmaximal, characteristic, complemented
8#5172C211yesmaximal, characteristic, complemented
9#5049C211yesmaximal, characteristic, complemented
10#4817C211yesmaximal, characteristic, complemented
11#4810C211yesmaximal, characteristic, complemented
12#4786C211yesmaximal, characteristic, complemented
13#4775C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#4728C211yesmaximal, characteristic, complemented
16#3268—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#239C2211yescharacteristic
19C2×#239C2211yescharacteristic, complemented
20C22×D12C2211yescharacteristic
21C2×C3×Q16C2211yescharacteristic, complemented
22C2×C3×SD16C2211yescharacteristic, complemented
23C3×C22⋊Q8C2211yescharacteristic
24#949—24no—
25#947C2211yescharacteristic
26Q8×Dic3C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic
28C2×#215C2211yescharacteristic
29C2×#215C2211yescharacteristic, complemented
30#940—21no—
31#940C2211yescharacteristic, complemented
32#938—21no—
33#938C2211yescharacteristic, complemented
34C2×#213C2211yescharacteristic
35C2×#213C2211yescharacteristic
36#930—21no—
37#930C2211yescharacteristic
38C4×#67—21no—
39C4×#67C2211yescharacteristic
40C2×#208C2211yescharacteristic
41#919—24no—
42C2×D24C2211yescharacteristic
43C2×D24C2211yescharacteristic
44C2×D3×C8C2211yescharacteristic, complemented
45#895—21no—
46#895C2211yescharacteristic
47#894—21no—
48#893C2211yescharacteristic
49#882—21no—
50D3×C22⋊C4—21no—
51C3×#103C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic
54#835C2211yescharacteristic, complemented
55#821C2211yescharacteristic
56#821C2211yescharacteristic
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58#815C2211yescharacteristic, complemented
59Dic3⋊C8C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×#135—31no—
64#550—31no—
65#546—31no—
66#496—31no—
67#468—31no—
68#466—31no—
69#464—31no—
70#459—31no—
71#452D311yescharacteristic, complemented
72#449—31no—
73#449—31no—
74#447—31no—
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C3×#41—22no—
79C3×#41—41no—
80C2×C3×Q8C2311yescharacteristic
81C2×C3×Q8C2311yescharacteristic
82C2×C3×D4—21no—
83C2×C3×D4C2311yescharacteristic
84C22×C3×C4—21no—
85C22×C3×C4C2311yescharacteristic
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67D411yescharacteristic
91#239—22no—
92#239—22no—
93#239—41no—
94#239—41no—
95C2×D12—22no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12—21no—
100C2×D12—41no—
101C2×D12C2311yescharacteristic
102C2×D12C2311yescharacteristic
103C2×D12C2311yescharacteristic
104C2×C4×D3—21no—
105C2×C4×D3—21no—
106C2×C4×D3—21no—
107C2×C4×D3C2311yescharacteristic
108C3×Q16—22no—
109C3×SD16—41no—
110C2×C3×C8C2311yescharacteristic
111C2×C3×C8C2311yescharacteristic
112C3×C4⋊C4—21no—
113C3×C4⋊C4C2311yescharacteristic
114C3×C22⋊C4—21no—
115#217—21no—
116#217D411yescharacteristic
117#215—22no—
118#215—22no—
119#215D414yescomplemented
120#213—22no—
121#213—22no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4D411yescharacteristic
128C4⋊Dic3—21no—
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4D411yescharacteristic
131C4×Dic3—21no—
132C4×Dic3C2311yescharacteristic
133#208—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136D24—22no—
137D24—41no—
138D3×C8—22no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41C2×D311yescharacteristic
142C22×D4—31no—
143AGL(1,ℤ/8)—64no—
144#135—64no—
145C2×Q16C2×D311yescharacteristic, complemented
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16C2×D311yescharacteristic, complemented
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×M16—31no—
155C22×C8—31no—
156#127—31no—
157#127—61no—
158#124—31no—
159#123—61no—
160C22⋊Q8C2×D311yescharacteristic
161#121—31no—
162#121—31no—
163#121—31no—
164#121—61no—
165#121—61no—
166#121—61no—
167#120—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—31no—
171C4×D4—61no—
172C2×C22⋊C4—61no—
173#107—31no—
174C4⋊C8—31no—
175#103—31no—
176#103C2×D311yescharacteristic, complemented
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic, complemented
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic
183#97—31no—
184C23×C3—21no—
185C23×C3C2×D411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—81no—
192C22×D3C2×D411yescharacteristic
193C3×Q8—21no—
194C3×Q8—21no—
195C3×Q8—21no—
196C3×Q8C2×D412yes—
197C3×D4—21no—
198C3×D4—21no—
199C3×D4—42no—
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×C4C2×D411yescharacteristic
207C2×C3×C4C2411yescharacteristic
208#67—41no—
209#67—42no—
210#67—81no—
211#67—81no—
212#67—81no—
213C2×Dic3—21no—
214C2×Dic3C2×D411yescharacteristic
215C2×Dic3C2×D411yescharacteristic
216C2×Dic3C2×D411yescharacteristic
217D12—21no—
218D12—21no—
219D12—21no—
220D12—21no—
221D12—21no—
222D12—42no—
223D12—42no—
224D12—42no—
225D12—81no—
226D12C2×D412yes—
227C4×D3—21no—
228C4×D3—21no—
229C4×D3—42no—
230C4×D3—42no—
231C4×D3—81no—
232C3×C8—21no—
233C3×C8—21no—
234C3⋊C8—21no—
235C3⋊C8C2×D412yes—
236C24—61no—
237#41—22no—
238#41—41no—
239#41—62no—
240#41—62no—
241#41—121no—
242#41—121no—
243C2×Q8C22×D311yescharacteristic
244C2×Q8C22×D311yescharacteristic
245C2×D4—21no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—31no—
249C2×D4—31no—
250C2×D4—62no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—121no—
260C2×D4C22×D311yescharacteristic
261C22×C4—21no—
262C22×C4—31no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4C22×D311yescharacteristic
267Q16—22no—
268SD16—34no—
269SD16—41no—
270SD16—62no—
271SD16—62no—
272D8—62no—
273D8—62no—
274D8—62no—
275D8—121no—
276M16—62no—
277C2×C8—31no—
278C2×C8—31no—
279C2×C8—62no—
280C2×C8C22×D311yescharacteristic
281C2×C8C22×D311yescharacteristic
282C4⋊C4—21no—
283C4⋊C4—31no—
284C4⋊C4—31no—
285C4⋊C4—61no—
286C4⋊C4C22×D311yescharacteristic
287C22⋊C4—21no—
288C22⋊C4—31no—
289C22⋊C4—31no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C42—31no—
297C42—61no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3C22×D411yescharacteristic
305C2×D3—21no—
306C2×D3—22no—
307C2×D3—21no—
308C2×D3—21no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—84no—
318C3×C4—22no—
319C3×C4—21no—
320C3×C4—41no—
321C3×C4—41no—
322C3×C4C22×D411yescharacteristic
323C3×C4H(5,𝔽2)11yescharacteristic
324Dic3—21no—
325Dic3—21no—
326Dic3—21no—
327Dic3—41no—
328C23—21no—
329C23—31no—
330C23—61no—
331C23—61no—
332C23—61no—
333C23—122no—
334C23—122no—
335C23—241no—
336C23D3×D411yescharacteristic
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8D3×D412yes—
341D4—21no—
342D4—21no—
343D4—32no—
344D4—42no—
345D4—61no—
346D4—61no—
347D4—61no—
348D4—61no—
349D4—61no—
350D4—122no—
351D4—122no—
352D4—122no—
353D4—121no—
354D4—122no—
355D4—241no—
356D4—241no—
357D4—241no—
358D4—241no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—31no—
367C2×C4—42no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—241no—
374C2×C4D3×D411yescharacteristic
375C2×C4C23×D311yescharacteristic
376C8—21no—
377C8—21no—
378C8—32no—
379C8—61no—
380C2×C3—41no—
381C2×C3—81no—
382C2×C3D4211yescharacteristic
383C2×C3#57511yescharacteristic
384C2×C3#57611yescharacteristic
385D3—41no—
386D3—42no—
387D3—81no—
388D3—81no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—61no—
396C22—61no—
397C22—61no—
398C22—62no—
399C22—121no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—244no—
408C22C2×D3×D411yescharacteristic
409C4—21no—
410C4—22no—
411C4—41no—
412C4—41no—
413C4—61no—
414C4—61no—
415C4—61no—
416C4—121no—
417C4C2×D3×D411yescharacteristic
418C4#100411yescharacteristic
419C3#326811yescharacteristic, complemented, Sylow
420C2—41no—
421C2—81no—
422C2—121no—
423C2—122no—
424C2—241no—
425C2—241no—
426C2#559011yescharacteristic
427C2#576911yescharacteristic
428C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
hypercenter#452D3
hypocenterC3#3268
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #5834C2semidirect
2. C2×#949C2semidirect
3. C2×#919C2semidirect
4. #5630C2semidirect
5. C3×#452C2semidirect
6. #5185C2semidirect
7. #5176C2semidirect
8. #5172C2semidirect
9. #5049C2semidirect
10. #4817C2semidirect
11. #4810C2semidirect
12. #4786C2semidirect
13. #4775C2semidirect
14. #4732C2semidirect
15. #4728C2semidirect
19. C2×#239C22semidirect
21. C2×C3×Q16C22semidirect
22. C2×C3×SD16C22semidirect
26. Q8×Dic3C22semidirect
29. C2×#215C22semidirect
31. #940C22semidirect
33. #938C22semidirect
44. C2×D3×C8C22semidirect
51. C3×#103C22semidirect
52. C3×#102C22semidirect
54. #835C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. #815C22semidirect
59. Dic3⋊C8C22semidirect
60. #809C22semidirect
61. #807C22semidirect
71. #452D3semidirect
119. #215D4semidirect
145. C2×Q16C2×D3semidirect
149. C2×SD16C2×D3semidirect
176. #103C2×D3semidirect
180. #102C2×D3semidirect
419. C3#3268semidirect

Maximal subgroups

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

subgroupindex
#58342
C2×#9492
C2×#9192
#56302
C3×#4522
#51852
#51762
#51722
#50492
#48172
#48102
#47862
#47752
#47322
#47282
#32683