Finite Group Databasebeta

← #83522

Subgroups — #83522

Order 384.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#547C211yesmaximal, characteristic, complemented
2C2×#950C211yesmaximal, characteristic, complemented
3C2×#949C211yesmaximal, characteristic, complemented
4#5550C211yesmaximal, characteristic, complemented
5#5239C211yesmaximal, characteristic, complemented
6#5238C211yesmaximal, characteristic, complemented
7#5086C211yesmaximal, characteristic, complemented
8#5083C211yesmaximal, characteristic, complemented
9#5052C211yesmaximal, characteristic, complemented
10#5048C211yesmaximal, characteristic, complemented
11#5047C211yesmaximal, characteristic, complemented
12#5046C211yesmaximal, characteristic, complemented
13#5016C211yesmaximal, characteristic, complemented
14#5014C211yesmaximal, characteristic, complemented
15#5011C211yesmaximal, characteristic, complemented
16#3267—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×C3×#41C2211yescharacteristic, complemented
19C2×#235C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C3×C4⋊Q8C2211yescharacteristic, complemented
22C3×#123—21no—
23C3×#123—21no—
24C3×C22⋊Q8C2211yescharacteristic
25C3×C22⋊Q8C2211yescharacteristic, complemented
26C3×#121—21no—
27C3×C4×D4C2211yescharacteristic
28C3×C4×D4C2211yescharacteristic, complemented
29C2×C3×C4⋊C4—21no—
30#950—24no—
31#949—24no—
32C2×#216C2211yescharacteristic, complemented
33C2×#216C2211yescharacteristic, complemented
34C2×#215C2211yescharacteristic
35C2×#215C2211yescharacteristic
36C2×#215C2211yescharacteristic
37C2×#215C2211yescharacteristic
38#937—21no—
39C2×#214C2211yescharacteristic, complemented
40C2×#213C2211yescharacteristic
41C2×#213C2211yescharacteristic
42#930C2211yescharacteristic
43#924C2211yescharacteristic
44C2×#208C2211yescharacteristic
45C22×C3⋊C8C2211yescharacteristic, complemented
46#894—21no—
47#884—21no—
48D3×C22⋊C4—21no—
49#875C2211yescharacteristic
50C4×D12C2211yescharacteristic
51#835C2211yescharacteristic, complemented
52#835C2211yescharacteristic, complemented
53#832C2211yescharacteristic, complemented
54#830C2211yescharacteristic
55#830C2211yescharacteristic, complemented
56#810C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#809C2211yescharacteristic
59#808C2211yescharacteristic, complemented
60C4⋊(C3⋊C8)C2211yescharacteristic, complemented
61#803C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×#135—31no—
64#547D311yescharacteristic, complemented
65#546—31no—
66#497—31no—
67#470—31no—
68#465—31no—
69#463—31no—
70#461—31no—
71#452—31no—
72#452—31no—
73#450—31no—
74#449—31no—
75#442—31no—
76#435—31no—
77C23×D3—21no—
78C3×#41—22no—
79C3×#41—22no—
80C3×#41—41no—
81C3×#41—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×C4—21no—
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C2×#67—21no—
94C2×#67—21no—
95#235—22no—
96#235—41no—
97C2×D12—22no—
98C2×D12—21no—
99C2×D12—41no—
100C2×D12C2311yescharacteristic
101C2×D12C2311yescharacteristic
102C2×C4×D3—21no—
103C2×C4×D3—21no—
104C2×Dic6C2311yescharacteristic
105C3×C4⋊C4—21no—
106C3×C4⋊C4—21no—
107C3×C4⋊C4—21no—
108C3×C4⋊C4—21no—
109C3×C4⋊C4D411yescharacteristic
110C3×C4⋊C4C2311yescharacteristic
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113C3×C22⋊C4—21no—
114C3×C22⋊C4D411yescharacteristic
115C3×C22⋊C4D411yescharacteristic
116C3×C42C2311yescharacteristic
117#217—21no—
118#216—22no—
119#216—41no—
120#215—22no—
121#215—22no—
122#215—22no—
123#215—41no—
124#214—22no—
125#213—22no—
126#213D414yescomplemented
127(C2×D3)⋊C4—21no—
128(C2×D3)⋊C4—21no—
129(C2×D3)⋊C4—21no—
130(C2×D3)⋊C4—21no—
131C4⋊Dic3C2311yescharacteristic
132Dic3⋊C4—21no—
133#208—22no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137C2×C3⋊C8C2311yescharacteristic
138C2×C3⋊C8C2311yescharacteristic
139C2×#41—31no—
140C2×#41C2×D311yescharacteristic
141C2×#41C2×D311yescharacteristic, complemented
142C22×D4—31no—
143AGL(1,ℤ/8)—64no—
144#135—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—31no—
152C2×D8—31no—
153C2×D8—31no—
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8C2×D311yescharacteristic, complemented
157#124—31no—
158#123—21no—
159#123—21no—
160#123—61no—
161C22⋊Q8—31no—
162C22⋊Q8C2×D311yescharacteristic
163C22⋊Q8C2×D311yescharacteristic, complemented
164#121—21no—
165#121—31no—
166#121—61no—
167#120—61no—
168C4×D4—31no—
169C4×D4C2×D311yescharacteristic
170C4×D4C2×D311yescharacteristic, complemented
171C2×C4⋊C4—21no—
172C2×C22⋊C4—61no—
173#106—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#102—31no—
179#102—31no—
180#102—31no—
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97—31no—
184C23×C3—21no—
185C23×C3C2×D411yescharacteristic
186C23×C3C2×D411yescharacteristic
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—81no—
192C3×Q8—21no—
193C3×Q8—21no—
194C3×Q8—21no—
195C3×Q8—21no—
196C3×D4—21no—
197C3×D4—41no—
198C3×D4—42no—
199C3×D4—42no—
200C3×D4C2×D412yes—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4C2×D411yescharacteristic
212C2×C3×C4C2×D411yescharacteristic
213C2×C3×C4C2×D411yescharacteristic
214C2×C3×C4C2×D411yescharacteristic
215C2×C3×C4C2411yescharacteristic
216#67—42no—
217#67—81no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220D12—21no—
221D12—21no—
222D12—21no—
223D12—42no—
224D12C2×D412yes—
225C4×D3—42no—
226C4×D3—81no—
227Dic6—21no—
228Dic6—21no—
229C3⋊C8—21no—
230C3⋊C8—21no—
231C3⋊C8—21no—
232C3⋊C8C2×D412yes—
233C24—61no—
234#41—22no—
235#41—22no—
236#41—41no—
237#41—41no—
238#41—62no—
239#41—121no—
240C2×Q8—31no—
241C2×Q8C22×D311yescharacteristic
242C2×Q8C22×D311yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—62no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—121no—
252C2×D4#6711yescharacteristic
253C2×D4C22×D311yescharacteristic
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4C22×D311yescharacteristic
260C22×C4C22×D311yescharacteristic
261Q16—62no—
262Q16—121no—
263SD16—62no—
264SD16—62no—
265SD16—62no—
266SD16—62no—
267SD16—121no—
268D8—34no—
269D8—62no—
270M16—62no—
271C2×C8—31no—
272C2×C8—31no—
273C2×C8—31no—
274C2×C8—31no—
275C2×C8—62no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—31no—
281C4⋊C4—61no—
282C4⋊C4#6711yescharacteristic
283C4⋊C4C22×D311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4#6711yescharacteristic
293C22⋊C4#6711yescharacteristic
294C42C22×D311yescharacteristic
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—22no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3C22×D411yescharacteristic
305C2×D3—22no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—84no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—21no—
315C3×C4—21no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4C22×D411yescharacteristic
320C3×C4H(5,𝔽2)11yescharacteristic
321Dic3—41no—
322Dic3—41no—
323C23—21no—
324C23—61no—
325C23—61no—
326C23—122no—
327C23—122no—
328C23—241no—
329C23C2×#6711yescharacteristic
330C23C2×#6711yescharacteristic
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—21no—
335Q8—61no—
336Q8—61no—
337D4—21no—
338D4—32no—
339D4—41no—
340D4—42no—
341D4—42no—
342D4—61no—
343D4—61no—
344D4—61no—
345D4—122no—
346D4—122no—
347D4—241no—
348D4D3×D412yes—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—122no—
362C2×C4—241no—
363C2×C4C2×#6711yescharacteristic
364C2×C4C2×#6711yescharacteristic
365C2×C4C2×#6711yescharacteristic
366C2×C4C2×#6711yescharacteristic
367C2×C4C23×D311yescharacteristic
368C8—32no—
369C8—61no—
370C8—61no—
371C8—61no—
372C2×C3—41no—
373C2×C3—41no—
374C2×C3—42no—
375C2×C3D4211yescharacteristic
376C2×C3#57511yescharacteristic
377C2×C3#57711yescharacteristic
378D3—42no—
379D3—81no—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—41no—
388C22—41no—
389C22—62no—
390C22—121no—
391C22—121no—
392C22—121no—
393C22—121no—
394C22—121no—
395C22—244no—
396C22C22×#6711yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—121no—
405C4—121no—
406C4C2×D3×D411yescharacteristic
407C4#100411yescharacteristic
408C3#326711yescharacteristic, complemented, Sylow
409C2—41no—
410C2—41no—
411C2—42no—
412C2—122no—
413C2—241no—
414C2D4×#6711yescharacteristic
415C2#582211yescharacteristic
416C2#583711yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#547C2
socleC22×C3C22×D4
hypercenter#547D3
hypocenterC3#3267
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C3×#547C2semidirect
2. C2×#950C2semidirect
3. C2×#949C2semidirect
4. #5550C2semidirect
5. #5239C2semidirect
6. #5238C2semidirect
7. #5086C2semidirect
8. #5083C2semidirect
9. #5052C2semidirect
10. #5048C2semidirect
11. #5047C2semidirect
12. #5046C2semidirect
13. #5016C2semidirect
14. #5014C2semidirect
15. #5011C2semidirect
18. C2×C3×#41C22semidirect
21. C3×C4⋊Q8C22semidirect
25. C3×C22⋊Q8C22semidirect
28. C3×C4×D4C22semidirect
32. C2×#216C22semidirect
33. C2×#216C22semidirect
39. C2×#214C22semidirect
45. C22×C3⋊C8C22semidirect
51. #835C22semidirect
52. #835C22semidirect
53. #832C22semidirect
55. #830C22semidirect
56. #810C22semidirect
59. #808C22semidirect
60. C4⋊(C3⋊C8)C22semidirect
61. #803C22semidirect
64. #547D3semidirect
126. #213D4semidirect
141. C2×#41C2×D3semidirect
156. C4⋊Q8C2×D3semidirect
163. C22⋊Q8C2×D3semidirect
170. C4×D4C2×D3semidirect
408. C3#3267semidirect

Maximal subgroups

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

subgroupindex
C3×#5472
C2×#9502
C2×#9492
#55502
#52392
#52382
#50862
#50832
#50522
#50482
#50472
#50462
#50162
#50142
#50112
#32673