Finite Group Databasebeta

← #80348

Subgroups — #80348

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×#912C211yesmaximal, characteristic, complemented
2C2×#911C211yesmaximal, characteristic, complemented
3#5718C211yesmaximal, characteristic, complemented
4#5554C211yesmaximal, characteristic, complemented
5C3×#459C211yesmaximal, characteristic, complemented
6#5158C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5074C211yesmaximal, characteristic, complemented
9#5018C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4783C211yesmaximal, characteristic, complemented
13#4781C211yesmaximal, characteristic, complemented
14#4780C211yesmaximal, characteristic, complemented
15#4775C211yesmaximal, characteristic, complemented
16#3268—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×#237C2211yescharacteristic, complemented
20C2×D3×D4C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic
22C2×C3×D8C2211yescharacteristic
23C3×#127C2211yescharacteristic, complemented
24C3×C4×D4C2211yescharacteristic
25#940—21no—
26#939—21no—
27#939—21no—
28#938C2211yescharacteristic
29#938C2211yescharacteristic, complemented
30D4×Dic3—21no—
31C2×#214C2211yescharacteristic
32C2×#214C2211yescharacteristic, complemented
33C2×#214C2211yescharacteristic, complemented
34C2×#213C2211yescharacteristic
35C2×#213C2211yescharacteristic
36#930—21no—
37C2×(C2×D3)⋊C4—21no—
38#912—24no—
39#911—24no—
40C2×Dic12C2211yescharacteristic, complemented
41C2×#204C2211yescharacteristic
42C2×ΓL(1,𝔽25)C2211yescharacteristic
43C2×D3×C8C2211yescharacteristic, complemented
44#897C2211yescharacteristic
45#895C2211yescharacteristic
46#886—21no—
47#882—21no—
48#875C2211yescharacteristic
49C4×D12C2211yescharacteristic
50C4×Dic6C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#807C2211yescharacteristic, complemented
61#803C2211yescharacteristic, 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#459D311yescharacteristic, complemented
71#452—31no—
72#449—31no—
73#449—31no—
74#447—31no—
75#440—31no—
76#435—31no—
77C23×D3—21no—
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C22×Dic3—21no—
91C22×Dic3—21no—
92C22×Dic3—21no—
93#237—22no—
94#237—22no—
95#237—22no—
96#237—41no—
97#237—41no—
98#237—41no—
99D3×D4—22no—
100D3×D4—41no—
101C2×D12—21no—
102C2×D12D411yescharacteristic
103C2×D12C2311yescharacteristic
104C2×C4×D3C2311yescharacteristic
105C2×C4×D3C2311yescharacteristic
106C2×Dic6C2311yescharacteristic
107C2×Dic6C2311yescharacteristic
108C3×D8—22no—
109C3×D8—22no—
110C2×C3×C8C2311yescharacteristic
111C2×C3×C8C2311yescharacteristic
112C3×C4⋊C4C2311yescharacteristic
113C3×C22⋊C4—21no—
114C3×C42C2311yescharacteristic
115#217—21no—
116#217—21no—
117#214—22no—
118#214—41no—
119#214D414yescomplemented
120#213—22no—
121#213—41no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4D411yescharacteristic
127(C2×D3)⋊C4D411yescharacteristic
128C4⋊Dic3—21no—
129C4⋊Dic3D411yescharacteristic
130C4⋊Dic3C2311yescharacteristic
131Dic3⋊C4—21no—
132C4×Dic3—21no—
133C2×C3⋊C8C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135Dic12—22no—
136#204—22no—
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×SD16—31no—
150C2×D8—31no—
151C2×D8—31no—
152C2×D8C2×D311yescharacteristic
153C2×D8C2×D311yescharacteristic
154C2×M16—31no—
155C22×C8—31no—
156#127—61no—
157#127C2×D311yescharacteristic, complemented
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×Q8—31no—
169C4×D4—31no—
170C4×D4—61no—
171C4×D4C2×D311yescharacteristic
172C2×C22⋊C4—61no—
173#107—31no—
174C4⋊C8C2×D311yescharacteristic, complemented
175#103—31no—
176#103—31no—
177#102—31no—
178#102—31no—
179#102C2×D311yescharacteristic, complemented
180#102C2×D311yescharacteristic, complemented
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97—31no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C22×D3—21no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—81no—
191C22×D3C2×D411yescharacteristic
192C22×D3C2×D411yescharacteristic
193C3×D4—21no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—42no—
199C3×D4—81no—
200C3×D4C2×D412yes—
201C2×C3×C4—21no—
202C2×C3×C4—81no—
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2×D411yescharacteristic
205C2×C3×C4C2411yescharacteristic
206#67—42no—
207#67—42no—
208#67—42no—
209#67—42no—
210#67—81no—
211#67—81no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—42no—
216C2×Dic3—42no—
217C2×Dic3—42no—
218C2×Dic3C2×D411yescharacteristic
219C2×Dic3C2×D411yescharacteristic
220D12—21no—
221D12—21no—
222D12—41no—
223D12—81no—
224C4×D3—21no—
225C4×D3—21no—
226C4×D3—21no—
227C4×D3—21no—
228Dic6—21no—
229Dic6—21no—
230Dic6—21no—
231Dic6C2×D412yes—
232C3×C8—21no—
233C3×C8—21no—
234C3⋊C8—21no—
235C3⋊C8C2×D412yes—
236C24—61no—
237#41—62no—
238#41—62no—
239#41—62no—
240#41—121no—
241#41—121no—
242#41—121no—
243C2×Q8—31no—
244C2×Q8—31no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—31no—
248C2×D4—31no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—62no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—121no—
258C2×D4C22×D311yescharacteristic
259C2×D4C22×D311yescharacteristic
260C2×D4C22×D311yescharacteristic
261C22×C4—21no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267Q16—62no—
268SD16—34no—
269SD16—62no—
270SD16—62no—
271SD16—121no—
272D8—22no—
273D8—22no—
274D8—62no—
275D8—121no—
276M16—62no—
277C2×C8—31no—
278C2×C8—31no—
279C2×C8—62no—
280C2×C8C22×D311yescharacteristic
281C2×C8C22×D311yescharacteristic
282C4⋊C4—31no—
283C4⋊C4—31no—
284C4⋊C4—61no—
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—61no—
297C42C22×D311yescharacteristic
298C22×C3—22no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3C22×D411yescharacteristic
308C2×D3—21no—
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—21no—
312C2×D3—21no—
313C2×D3—21no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—84no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—41no—
321C3×C4C22×D411yescharacteristic
322C3×C4H(5,𝔽2)11yescharacteristic
323Dic3—22no—
324Dic3—21no—
325Dic3—21no—
326Dic3—41no—
327Dic3—41no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—31no—
332C23—31no—
333C23—61no—
334C23—122no—
335C23—122no—
336C23—241no—
337Q8—32no—
338Q8—61no—
339Q8—61no—
340Q8—61no—
341D4—21no—
342D4—21no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—42no—
347D4—61no—
348D4—61no—
349D4—81no—
350D4—121no—
351D4—122no—
352D4—122no—
353D4—122no—
354D4—122no—
355D4—241no—
356D4—241no—
357D4—241no—
358D4D3×D412yes—
359C2×C4—21no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—61no—
363C2×C4—61no—
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—21no—
378C8—32no—
379C8—61no—
380C2×C3—42no—
381C2×C3—81no—
382C2×C3—81no—
383C2×C3D4211yescharacteristic
384C2×C3#57511yescharacteristic
385C2×C3#57611yescharacteristic
386D3—41no—
387D3—41no—
388D3—81no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—61no—
399C22—61no—
400C22—61no—
401C22—61no—
402C22—61no—
403C22—61no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—244no—
408C22C2×D3×D411yescharacteristic
409C4—21no—
410C4—21no—
411C4—41no—
412C4—61no—
413C4—61no—
414C4—62no—
415C4—121no—
416C4—121no—
417C4C2×D3×D411yescharacteristic
418C4#100911yescharacteristic
419C3#326811yescharacteristic, complemented, Sylow
420C2—42no—
421C2—81no—
422C2—81no—
423C2—121no—
424C2—121no—
425C2—241no—
426C2#560911yescharacteristic
427C2#575711yescharacteristic
428C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3268
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. C2×#912C2semidirect
2. C2×#911C2semidirect
3. #5718C2semidirect
4. #5554C2semidirect
5. C3×#459C2semidirect
6. #5158C2semidirect
7. #5156C2semidirect
8. #5074C2semidirect
9. #5018C2semidirect
10. #4845C2semidirect
11. #4836C2semidirect
12. #4783C2semidirect
13. #4781C2semidirect
14. #4780C2semidirect
15. #4775C2semidirect
19. C2×#237C22semidirect
23. C3×#127C22semidirect
29. #938C22semidirect
32. C2×#214C22semidirect
33. C2×#214C22semidirect
40. C2×Dic12C22semidirect
43. C2×D3×C8C22semidirect
50. C4×Dic6C22semidirect
51. C3×C4⋊C8C22semidirect
52. C3×#102C22semidirect
53. C3×#102C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. #816C22semidirect
59. #810C22semidirect
60. #807C22semidirect
61. #803C22semidirect
70. #459D3semidirect
119. #214D4semidirect
157. #127C2×D3semidirect
174. C4⋊C8C2×D3semidirect
179. #102C2×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
C2×#9122
C2×#9112
#57182
#55542
C3×#4592
#51582
#51562
#50742
#50182
#48452
#48362
#47832
#47812
#47802
#47752
#32683