Finite Group Databasebeta

← #79796

Subgroups — #79796

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
1C2×#916C211yesmaximal, characteristic, complemented
2C2×#914C211yesmaximal, characteristic, complemented
3#5577C211yesmaximal, characteristic, complemented
4#5554C211yesmaximal, characteristic, complemented
5C3×#440C211yesmaximal, characteristic, complemented
6#5028C211yesmaximal, characteristic, complemented
7#5015C211yesmaximal, characteristic, complemented
8#4886C211yesmaximal, characteristic, complemented
9#4883C211yesmaximal, characteristic, complemented
10#4810C211yesmaximal, characteristic, complemented
11#4808C211yesmaximal, characteristic, complemented
12#4781C211yesmaximal, characteristic, complemented
13#4773C211yesmaximal, characteristic, complemented
14#4712C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3268—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#239C2211yescharacteristic, complemented
19C2×#237C2211yescharacteristic
20C2×D3×D4C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic, complemented
23C3×C4×D4C2211yescharacteristic
24C2×#216C2211yescharacteristic, complemented
25C2×#215C2211yescharacteristic
26C2×#213C2211yescharacteristic
27C2×#213C2211yescharacteristic
28#930—21no—
29C2×(C2×D3)⋊C4—21no—
30#916—24no—
31#914—24no—
32C2×D24C2211yescharacteristic
33C2×D24C2211yescharacteristic
34C2×#204C2211yescharacteristic, complemented
35C2×#204C2211yescharacteristic, complemented
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37C2×D3×C8C2211yescharacteristic, complemented
38#897C2211yescharacteristic
39#895—21no—
40#895—21no—
41#895C2211yescharacteristic
42#895C2211yescharacteristic
43#895C2211yescharacteristic, complemented
44#886—21no—
45#882—21no—
46#875C2211yescharacteristic
47#874—21no—
48#874C2211yescharacteristic, complemented
49C4×D12—21no—
50C4×D12C2211yescharacteristic
51C3×#107C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic
54C3×#97C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#809C2211yescharacteristic
59#809C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61C4⋊(C3⋊C8)C2211yescharacteristic, 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#452—31no—
72#449—31no—
73#449—31no—
74#447—31no—
75#440D311yescharacteristic, complemented
76#435—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84#239—22no—
85#239—22no—
86#239—41no—
87#239—41no—
88#237—22no—
89#237—41no—
90D3×D4—22no—
91D3×D4—41no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12—21no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12—21no—
99C2×D12D411yescharacteristic
100C2×D12C2311yescharacteristic
101C2×D12C2311yescharacteristic
102C2×D12C2311yescharacteristic
103C2×C4×D3—21no—
104C2×C4×D3—21no—
105C2×C4×D3C2311yescharacteristic
106C2×C4×D3C2311yescharacteristic
107C2×Dic6C2311yescharacteristic
108C3×SD16D414yescomplemented
109C2×C3×C8C2311yescharacteristic
110C2×C3×C8C2311yescharacteristic
111C3×C4⋊C4—21no—
112C3×C4⋊C4C2311yescharacteristic
113C3×C4⋊C4C2311yescharacteristic
114C3×C22⋊C4—21no—
115C3×C42—21no—
116C3×C42C2311yescharacteristic
117#216—22no—
118#215—22no—
119#213—22no—
120#213—22no—
121(C2×D3)⋊C4—21no—
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
128(C2×D3)⋊C4D411yescharacteristic
129C4⋊Dic3—21no—
130C4⋊Dic3D411yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133D24—22no—
134D24—41no—
135#204—22no—
136#204—41no—
137ΓL(1,𝔽25)—22no—
138D3×C8—22no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41—31no—
142C22×D4—31no—
143AGL(1,ℤ/8)—64no—
144#135—64no—
145C2×Q16—31no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16C2×D311yescharacteristic
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⋊Q8—31no—
161#121—31no—
162#121—31no—
163#121—31no—
164#121—61no—
165#121—61no—
166#121—61no—
167#120—61no—
168C4×Q8C2×D311yescharacteristic, complemented
169C4×D4—31no—
170C4×D4—61no—
171C4×D4C2×D311yescharacteristic
172C2×C22⋊C4—61no—
173#107C2×D311yescharacteristic, complemented
174C4⋊C8—31no—
175#103—31no—
176#103C2×D311yescharacteristic, complemented
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97C2×D311yescharacteristic, complemented
184C23×C3—21no—
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—81no—
191C22×D3C2×D411yescharacteristic
192C22×D3C2×D411yescharacteristic
193C3×Q8—21no—
194C3×Q8C2×D412yes—
195C3×D4—21no—
196C3×D4C2×D412yes—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—81no—
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204#67—42no—
205#67—42no—
206C2×Dic3—21no—
207C2×Dic3—42no—
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210D12—21no—
211D12—21no—
212D12—21no—
213D12—21no—
214D12—21no—
215D12—21no—
216D12—41no—
217D12—42no—
218D12—42no—
219D12—42no—
220D12—81no—
221D12—81no—
222D12—81no—
223D12—81no—
224C4×D3—21no—
225C4×D3—21no—
226C4×D3—21no—
227C4×D3—21no—
228C4×D3—42no—
229C4×D3—42no—
230Dic6—21no—
231Dic6—21no—
232C3×C8—21no—
233C3×C8C2×D412yes—
234C3⋊C8—21no—
235C3⋊C8—21no—
236C24—61no—
237#41—62no—
238#41—62no—
239#41—62no—
240#41—121no—
241#41—121no—
242#41—121no—
243C2×Q8—31no—
244C2×Q8C22×D311yescharacteristic
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—31no—
248C2×D4—31no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—62no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—61no—
259C2×D4—121no—
260C2×D4C22×D311yescharacteristic
261C22×C4—21no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267Q16—62no—
268SD16—62no—
269SD16—62no—
270SD16—121no—
271SD16D1214yescomplemented
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—61no—
285C4⋊C4C22×D311yescharacteristic
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—21no—
297C42C22×D311yescharacteristic
298C22×C3—22no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3C22×D411yescharacteristic
302C2×D3—21no—
303C2×D3—21no—
304C2×D3—21no—
305C2×D3—21no—
306C2×D3—21no—
307C2×D3—21no—
308C2×D3—41no—
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—21no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4C22×D411yescharacteristic
324C3×C4H(5,𝔽2)11yescharacteristic
325Dic3—21no—
326Dic3—21no—
327Dic3—41no—
328C23—21no—
329C23—31no—
330C23—31no—
331C23—61no—
332C23—61no—
333C23—61no—
334C23—122no—
335C23—122no—
336C23—241no—
337Q8—21no—
338Q8—61no—
339Q8—61no—
340Q8C2×D1212yes—
341D4—21no—
342D4—61no—
343D4—61no—
344D4—61no—
345D4—61no—
346D4—61no—
347D4—61no—
348D4—121no—
349D4—122no—
350D4—122no—
351D4—122no—
352D4—122no—
353D4—122no—
354D4—241no—
355D4—241no—
356D4—241no—
357D4—241no—
358D4C2×D1212yes—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—81no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4D3×D411yescharacteristic
374C2×C4D3×D411yescharacteristic
375C2×C4C23×D311yescharacteristic
376C8—21no—
377C8—61no—
378C8—61no—
379C8C2×D1212yes—
380C2×C3—42no—
381C2×C3D4211yescharacteristic
382C2×C3#57511yescharacteristic
383C2×C3#57611yescharacteristic
384D3—41no—
385D3—41no—
386D3—81no—
387D3—81no—
388D3—81no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—61no—
393C22—61no—
394C22—61no—
395C22—61no—
396C22—61no—
397C22—61no—
398C22—121no—
399C22—121no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—244no—
408C22C2×D3×D411yescharacteristic
409C4—21no—
410C4—21no—
411C4—22no—
412C4—41no—
413C4—41no—
414C4—61no—
415C4—61no—
416C4—121no—
417C4C22×D1211yescharacteristic
418C4#100911yescharacteristic
419C3#326811yescharacteristic, complemented, Sylow
420C2—42no—
421C2—121no—
422C2—121no—
423C2—241no—
424C2—241no—
425C2—241no—
426C2D4×D1211yescharacteristic
427C2#577411yescharacteristic
428C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
hypercenter#440D3
hypocenterC3#3268
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#916C2semidirect
2. C2×#914C2semidirect
3. #5577C2semidirect
4. #5554C2semidirect
5. C3×#440C2semidirect
6. #5028C2semidirect
7. #5015C2semidirect
8. #4886C2semidirect
9. #4883C2semidirect
10. #4810C2semidirect
11. #4808C2semidirect
12. #4781C2semidirect
13. #4773C2semidirect
14. #4712C2semidirect
15. #4706C2semidirect
18. C2×#239C22semidirect
22. C3×C4×Q8C22semidirect
24. C2×#216C22semidirect
34. C2×#204C22semidirect
35. C2×#204C22semidirect
37. C2×D3×C8C22semidirect
43. #895C22semidirect
48. #874C22semidirect
51. C3×#107C22semidirect
52. C3×#103C22semidirect
54. C3×#97C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #810C22semidirect
59. #809C22semidirect
60. #809C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
75. #440D3semidirect
108. C3×SD16D4semidirect
168. C4×Q8C2×D3semidirect
173. #107C2×D3semidirect
176. #103C2×D3semidirect
183. #97C2×D3semidirect
271. SD16D12semidirect
419. C3#3268semidirect

Maximal subgroups

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

subgroupindex
C2×#9162
C2×#9142
#55772
#55542
C3×#4402
#50282
#50152
#48862
#48832
#48102
#48082
#47812
#47732
#47122
#47062
#32683