Finite Group Databasebeta

← #72338856

Subgroups — #72338856

Order 2048.

At least 6356 conjugacy classes of subgroups; only the 992 normal subgroups are shown.

The subgroup lattice has 398 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72338847C211yesmaximal, characteristic, complemented
2#72338833C211yesmaximal, characteristic, complemented
3#72338781C211yesmaximal, characteristic, complemented
4#72338729C212yesmaximal, complemented
5#72338711C214yesmaximal, complemented
6#72338682C218yesmaximal, complemented
7#72338472C211yesmaximal, characteristic, complemented
8#72319191C211yesmaximal, characteristic, complemented
9#72316740C212yesmaximal, complemented
10#72313412C211yesmaximal, characteristic, complemented
11#72306652C212yesmaximal, complemented
12#72285243C212yesmaximal, complemented
13#72236189C211yesmaximal, characteristic
14#72231271C214yesmaximal, complemented
15#10581738C2211yescharacteristic, complemented
16#8698894C2214yes—
17#8195808C2212yes—
18#7985294C2212yes—
19#7737411C2212yes—
20D43C2212yes—
21#7672751C2211yescharacteristic
22#7667013C2211yescharacteristic
23#7667013C2211yescharacteristic
24#6489749C2212yescomplemented
25#6445403C2211yescharacteristic, complemented
26#6347445C2211yescharacteristic, complemented
27#6346639C2212yescomplemented
28#512008C2214yescomplemented
29#508249C2214yescomplemented
30#496508C2212yescomplemented
31#493957C2211yescharacteristic, complemented
32#486613C2214yescomplemented
33#486593C2218yescomplemented
34#486464C2212yescomplemented
35#486367C2212yescomplemented
36#486347C2218yescomplemented
37#482657C2212yescomplemented
38#482363C2212yescomplemented
39#482234C2212yescomplemented
40#480580C2212yescomplemented
41#480392C2211yescharacteristic, complemented
42#248507C2218yescomplemented
43#248456C2218yescomplemented
44#248000C2214yescomplemented
45#247880C2214yescomplemented
46#201373C2212yescomplemented
47#200564C2211yescharacteristic, complemented
48#193878C2212yescomplemented
49#193788C2211yescharacteristic, complemented
50#185733C2211yescharacteristic, complemented
51#185660C2212yescomplemented
52#130982C2218yescomplemented
53#130773C2214yescomplemented
54#130768C2214yescomplemented
55#130740C2218yescomplemented
56#130403C2214yescomplemented
57#130355C2218yescomplemented
58#130313C2212yescomplemented
59#130260C2218yescomplemented
60#130226C2212yescomplemented
61#129645C2214yescomplemented
62#129607C2214yescomplemented
63#62813C2311yescharacteristic, complemented
64C2×#3498D411yescharacteristic
65C2×#3498C2311yescharacteristic
66C22×D42D412yes—
67#61560C2314yescomplemented
68#60276C2311yescharacteristic, complemented
69#60045D412yes—
70#60045D412yes—
71#59945D412yes—
72#59945C2312yes—
73#59864D411yescharacteristic
74#59864C2311yescharacteristic
75#59816D411yescharacteristic
76#59816C2311yescharacteristic
77#59220D412yes—
78#59041D412yes—
79#58997C2314yes—
80#56384D418yes—
81#52988D414yes—
82#52301D412yes—
83#52301C2312yes—
84#45599C2314yes—
85#45355D414yes—
86#39361D412yes—
87#39361C2312yes—
88#38904D414yes—
89#38904C2314yes—
90C2×#2867D411yescharacteristic
91#36128D412yes—
92D4×#127C2312yes—
93D4×#120D414yes—
94D4×#120C2314yes—
95#35272D411yescharacteristic
96#35272C2311yescharacteristic
97#35237D412yes—
98#35237D412yes—
99#34963C2312yes—
100#34963C2312yes—
101#34943D412yes—
102#34364D412yes—
103C4×D42C2312yes—
104C4×#532D411yescharacteristic
105C4×#532C2311yescharacteristic
106#24350C2318yescomplemented
107#24314C2312yescomplemented
108#24312C2312yes—
109#20723C2312yescomplemented
110#20637C2311yescharacteristic, complemented
111#18389C2314yescomplemented
112#18205C2314yescomplemented
113#18196C2314yes—
114#17363C2312yescomplemented
115#17349C2311yescharacteristic, complemented
116#12798C2312yescomplemented
117#12797C2312yes—
118#12793C2318yescomplemented
119#11903C2314yescomplemented
120#11899C2314yescomplemented
121#11880C2314yes—
122#11877C2318yescomplemented
123#11873C2314yes—
124#11477C2312yescomplemented
125#11475C2312yes—
126#11471C2318yescomplemented
127#9469C2314yescomplemented
128#9468C2314yes—
129#9397C2314yescomplemented
130#8965C2318yescomplemented
131#8959C2312yes—
132#8957C2312yescomplemented
133#8383C2314yescomplemented
134#8375C2318yescomplemented
135C24×D4C2×D414yes—
136#3501C2412yes—
137C2×#550C2×D412yes—
138C2×#550C2×D412yes—
139C2×D42C2×D412yes—
140C2×D42C2×D414yes—
141C2×D42C2×D412yes—
142C2×D42C2412yes—
143C2×#532C2×D411yescharacteristic
144C2×#532C2×D412yes—
145C2×#532C2×D411yescharacteristic
146C22×#127C2×D412yes—
147C22×#127C2×D411yescharacteristic
148C22×#127C2×D411yescharacteristic
149C22×#121C2×D414yes—
150C2×C4×#41C2×D411yescharacteristic
151C2×C4×#41C2411yescharacteristic
152C22×C4×D4C2×D412yes—
153C22×C4×D4C2×D412yes—
154#2972C2412yes—
155#2867C2×D412yes—
156#2867C2×D412yes—
157#2847C2×D414yes—
158#2847C2×D414yes—
159#2826C2×D418yes—
160#2804C2×D412yes—
161#2804C2×D412yes—
162#2782C2×D414yes—
163#2684C2×D414yes—
164#2679C2×D412yes—
165#2679C2×D412yes—
166#2675C2×D418yes—
167#2668C2×D418yes—
168#2619C2×D414yes—
169#2598C2×D414yes—
170#2598C2×D414yes—
171#2572C2×D414yes—
172#2571C2×D412yes—
173#2548C2×D412yes—
174#2548C2×D412yes—
175#2541C2×D412yes—
176#2541C2412yes—
177#2537C2×D412yes—
178#2504C2×D414yes—
179#2487C2×D412yes—
180#2487C2×D412yes—
181#2487C2×D412yes—
182#2487C2412yes—
183#2482C2×D414yes—
184#2433C2×D414yes—
185#2408C2×D414yes—
186#2408C2×D412yes—
187#2408C2×D412yes—
188#2408C2×D412yes—
189#2408C2412yes—
190#2404C2×D414yes—
191#2403C2×D418yes—
192#2398C2×D414yes—
193#2397C2×D414yes—
194#2366C2×D412yes—
195#2358C2×D414yes—
196D4×C22⋊C4C2×D414yes—
197D4×C22⋊C4C2×D414yes—
198C4×C4⋊Q8C2×D411yescharacteristic
199C4×C4⋊Q8C2×D411yescharacteristic
200C4×#127C2×D411yescharacteristic
201C4×#127C2411yescharacteristic
202C4×#124C2×D411yescharacteristic
203C4×#124C2×D411yescharacteristic
204C4×#124C2×D411yescharacteristic
205C4×#124C2411yescharacteristic
206C4×#120C2×D414yes—
207C4×#120C2414yes—
208#2297C2×D411yescharacteristic
209#2297C2×D411yescharacteristic
210#2293C2×D412yes—
211#2285C2×D411yescharacteristic
212#2285C2×D411yescharacteristic
213#2283C2×D412yes—
214#2283C2×D412yes—
215C42×D4C2×D412yes—
216C2×C43C2×D411yescharacteristic
217#1800C2412yes—
218#1799C2414yes—
219#1460C2412yes—
220#1459C2414yes—
221C22×#41#12011yescharacteristic
222C22×#41C22×D411yescharacteristic
223C23×D4#12012yes—
224C23×D4#12012yes—
225C23×D4#12012yes—
226C23×D4C22×D414yes—
227C23×D4C22×D412yes—
228C24×C4C22×D414yes—
229C2×C4⋊Q8#12011yescharacteristic
230C2×C4⋊Q8#12011yescharacteristic
231C2×#127#12014yes—
232C2×#127#12012yes—
233C2×#127#12012yes—
234C2×#127#12011yescharacteristic
235C2×#127C22×D411yescharacteristic
236C2×#127C22×D412yes—
237C2×#127C22×D412yes—
238C2×#124#12011yescharacteristic
239C2×#124C22×D411yescharacteristic
240C2×#121#12012yes—
241C2×#121C22×D414yes—
242C2×#120#12014yes—
243C2×#120C22×D414yes—
244C2×C4×Q8#12011yescharacteristic
245C2×C4×Q8C22×D411yescharacteristic
246C2×C4×Q8C22×D411yescharacteristic
247C2×C4×Q8C22×D411yescharacteristic
248C2×C4×D4#12011yescharacteristic
249C2×C4×D4#12014yes—
250C2×C4×D4#12014yes—
251C2×C4×D4#12014yes—
252C2×C4×D4C22×D411yescharacteristic
253C2×C4×D4C22×D412yes—
254C2×C4×D4C22×D412yes—
255C2×C4×D4C22×D412yes—
256C2×C4×D4C22×D411yescharacteristic
257C2×C4×D4C2511yescharacteristic
258C2×#117#12011yescharacteristic
259C2×#117C22×D411yescharacteristic
260C2×#117C22×D412yes—
261C22×C4⋊C4#12012yes—
262C22×C4⋊C4C22×D412yes—
263C22×C22⋊C4#12014yes—
264C22×C42#12011yescharacteristic
265C22×C42#12011yescharacteristic
266C22×C42C22×D412yes—
267C22×C42C22×D411yescharacteristic
268#400#12014yes—
269#400C22×D414yes—
270#399#12014yes—
271#399C22×D414yes—
272#394#12018yes—
273#394#12014yes—
274#394C22×D414yes—
275#392#12014yes—
276#392C22×D418yes—
277#392C22×D414yes—
278#391#12011yescharacteristic
279#391#12011yescharacteristic
280#391#12011yescharacteristic
281#391C22×D411yescharacteristic
282#390#12011yescharacteristic
283#390C22×D411yescharacteristic
284#390C22×D411yescharacteristic
285#390C22×D411yescharacteristic
286#386#12014yes—
287#386C22×D414yes—
288#385#12014yes—
289#385C22×D414yes—
290#384#12011yescharacteristic
291#384C22×D411yescharacteristic
292#383#12014yes—
293#382#12011yescharacteristic
294#382#12012yes—
295#382C22×D411yescharacteristic
296#381#12014yes—
297#380#12012yes—
298#380#12012yes—
299#380#12012yes—
300#380C22×D412yes—
301#380C22×D412yes—
302#379#12014yes—
303#379C22×D414yes—
304C4×C4⋊C4C22×D412yes—
305C4×C22⋊C4C22×D414yes—
306#376#12014yes—
307C43#12011yescharacteristic
308C43#12011yescharacteristic
309C43C22×D411yescharacteristic
310C43C22×D411yescharacteristic
311C25C2×#12014yes—
312C22×Q8C2×#12011yescharacteristic
313C22×Q8C2×#12011yescharacteristic
314C22×D4C2×#12012yes—
315C22×D4C2×#12014yes—
316C22×D4C2×#12011yescharacteristic
317C22×D4C2×#12012yes—
318C22×D4C2×#12014yes—
319C22×D4C2×#12014yes—
320C22×D4C23×D411yescharacteristic
321C23×C4C2×#12011yescharacteristic
322C23×C4C2×#12012yes—
323C23×C4C2×#12012yes—
324C23×C4C2×#12011yescharacteristic
325C23×C4C2×#12011yescharacteristic
326C23×C4C2×#12014yes—
327C23×C4C23×D412yes—
328#127B(4,𝔽2)12yes—
329#127B(4,𝔽2)12yes—
330#127#57714yes—
331C2×C4⋊C4C2×#12012yes—
332C2×C4⋊C4C2×#12014yes—
333C2×C4⋊C4C2×#12011yescharacteristic
334C2×C4⋊C4C2×#12011yescharacteristic
335C2×C4⋊C4C2×#12012yes—
336C2×C4⋊C4C2×#12012yes—
337C2×C4⋊C4C2×#12011yescharacteristic
338C2×C4⋊C4C23×D411yescharacteristic
339C2×C22⋊C4C2×#12014yes—
340C2×C22⋊C4C2×#12014yes—
341C2×C22⋊C4C2×#12012yes—
342C2×C22⋊C4C2×#12012yes—
343C2×C22⋊C4C23×D412yes—
344C2×C42C2×#12011yescharacteristic
345C2×C42C2×#12011yescharacteristic
346C2×C42C2×#12012yes—
347C2×C42C2×#12012yes—
348C2×C42C2×#12012yes—
349C2×C42C2×#12011yescharacteristic
350C2×C42C2×#12011yescharacteristic
351C2×C42C2×#12011yescharacteristic
352C2×C42C2×#12011yescharacteristic
353C2×C42C2×#12011yescharacteristic
354C2×C42C23×D411yescharacteristic
355#95C2×#12014yes—
356#95C2×#12014yes—
357#95C2×#12014yes—
358C24#284611yescharacteristic
359C24C22×#12012yes—
360C2×D4#300912yes—
361C2×D4C2×B(4,𝔽2)14yes—
362C2×D4C2×#57712yes—
363C2×D4#358512yes—
364C22×C4#284612yes—
365C22×C4#284611yescharacteristic
366C22×C4#284611yescharacteristic
367C22×C4#284612yes—
368C22×C4#284611yescharacteristic
369C22×C4C22×#12011yescharacteristic
370C22×C4C22×#12011yescharacteristic
371C22×C4C22×#12012yes—
372C22×C4C22×#12011yescharacteristic
373C42#300912yes—
374C42C2×B(4,𝔽2)11yescharacteristic
375C42C2×B(4,𝔽2)11yescharacteristic
376C42#302611yescharacteristic
377C42#302611yescharacteristic
378C23#3289612yes—
379C23C2×#284611yescharacteristic
380C23#6037314yes—
381C2×C4#3289811yescharacteristic
382C2×C4#3289811yescharacteristic
383C2×C4#3289811yescharacteristic
384C2×C4#3289811yescharacteristic
385C2×C4C2×#300911yescharacteristic
386C2×C4#6036711yescharacteristic
387C2×C4C22×B(4,𝔽2)11yescharacteristic
388C2×C4C2×#302611yescharacteristic
389C22#662885211yescharacteristic
390C22C2×#3289611yescharacteristic
391C22C2×#3289811yescharacteristic
392C4#51181912yes—
393C4#762288012yes—
394C2C2×#51181911yescharacteristic
395C2#7223818411yescharacteristic
396C2#7225198811yescharacteristic
subgroupG/H
center, socleC22C2×#32898
commutator, Frattini, cosocleC2×C4×D4C25
prime radical#72236189C2
LCSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#3498 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72338847C2semidirect
2. #72338833C2semidirect
3. #72338781C2semidirect
4. #72338729C2semidirect
5. #72338711C2semidirect
6. #72338682C2semidirect
7. #72338472C2semidirect
8. #72319191C2semidirect
9. #72316740C2semidirect
10. #72313412C2semidirect
11. #72306652C2semidirect
12. #72285243C2semidirect
14. #72231271C2semidirect
15. #10581738C22semidirect
24. #6489749C22semidirect
25. #6445403C22semidirect
26. #6347445C22semidirect
27. #6346639C22semidirect
28. #512008C22semidirect
29. #508249C22semidirect
30. #496508C22semidirect
31. #493957C22semidirect
32. #486613C22semidirect
33. #486593C22semidirect
34. #486464C22semidirect
35. #486367C22semidirect
36. #486347C22semidirect
37. #482657C22semidirect
38. #482363C22semidirect
39. #482234C22semidirect
40. #480580C22semidirect
41. #480392C22semidirect
42. #248507C22semidirect
43. #248456C22semidirect
44. #248000C22semidirect
45. #247880C22semidirect
46. #201373C22semidirect
47. #200564C22semidirect
48. #193878C22semidirect
49. #193788C22semidirect
50. #185733C22semidirect
51. #185660C22semidirect
52. #130982C22semidirect
53. #130773C22semidirect
54. #130768C22semidirect
55. #130740C22semidirect
56. #130403C22semidirect
57. #130355C22semidirect
58. #130313C22semidirect
59. #130260C22semidirect
60. #130226C22semidirect
61. #129645C22semidirect
62. #129607C22semidirect
63. #62813C23semidirect
67. #61560C23semidirect
68. #60276C23semidirect
106. #24350C23semidirect
107. #24314C23semidirect
109. #20723C23semidirect
110. #20637C23semidirect
111. #18389C23semidirect
112. #18205C23semidirect
114. #17363C23semidirect
115. #17349C23semidirect
116. #12798C23semidirect
118. #12793C23semidirect
119. #11903C23semidirect
120. #11899C23semidirect
122. #11877C23semidirect
124. #11477C23semidirect
126. #11471C23semidirect
127. #9469C23semidirect
129. #9397C23semidirect
130. #8965C23semidirect
132. #8957C23semidirect
133. #8383C23semidirect
134. #8375C23semidirect

Maximal subgroups

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

subgroupindex
#723388472
#723388332
#723387812
#723387292
#723387112
#723386822
#723384722
#723191912
#723167402
#723134122
#723066522
#722852432
#722361892
#722312712