Finite Group Databasebeta

← #72303386

Subgroups — #72303386

Order 2048.

At least 9183 conjugacy classes of subgroups; only the 1406 normal subgroups are shown.

The subgroup lattice has 400 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#72303310C211yesmaximal, characteristic, complemented
2#72303134C218yesmaximal, complemented
3#72303110C212yesmaximal, complemented
4#72303051C214yesmaximal, complemented
5#72280954C211yesmaximal, characteristic, complemented
6#72279188C212yesmaximal, complemented
7#72267980C212yesmaximal, complemented
8#72258211C211yesmaximal, characteristic, complemented
9#72256657C212yesmaximal, complemented
10#72238718C218yesmaximal, complemented
11C23×#585C2211yescharacteristic, complemented
12#10586228C2211yescharacteristic, complemented
13C2×#60398C2214yes—
14C2×#60387C2212yes—
15#7628649C2211yescharacteristic
16#7628649C2212yes—
17#7622875C22116yescomplemented
18C2×#33358C2212yescomplemented
19C2×#33347C2214yescomplemented
20#6628893C2218yescomplemented
21#6628892C2218yescomplemented
22#6628869C2214yescomplemented
23#6628867C2214yescomplemented
24#6499894C2214yescomplemented
25#6499892C2214yescomplemented
26#6374127C2211yescharacteristic, complemented
27#6374127C2212yescomplemented
28#6370746C2212yescomplemented
29#6370734C2214yescomplemented
30C2×#20591C2211yescharacteristic, complemented
31C2×#20584C2212yescomplemented
32#492899C2212yescomplemented
33#492886C2214yescomplemented
34#147767C22116yescomplemented
35#147752C22132yescomplemented
36#147570C2214yescomplemented
37#147569C2214yescomplemented
38#147568C2218yescomplemented
39#147567C2218yescomplemented
40C22×#585C2312yes—
41C22×#585C2312yes—
42C23×H(5,𝔽2)C2312yes—
43C24×#41C2311yescharacteristic
44#62862C2314yescomplemented
45C2×#3449D412yescomplemented
46C2×#3449D411yescharacteristic, complemented
47C2×#3449C2311yescharacteristic
48C2×#3449C2312yes—
49C22×#525D412yescomplemented
50C22×#525D411yescharacteristic, complemented
51C2×#3028D418yescomplemented
52C2×#3027D418yescomplemented
53C2×#3026C23116yes—
54C2×#3025D4116yescomplemented
55C2×#3022D414yescomplemented
56C2×#3021D414yescomplemented
57C2×#3019C2318yes—
58C2×#3018D418yescomplemented
59C22×#413C2312yes—
60C22×#410C2314yes—
61#35242D412yescomplemented
62#35222C2312yes—
63#35149D412yescomplemented
64#35149C2312yes—
65#35088D414yescomplemented
66#35088C2314yes—
67#34980D414yescomplemented
68#34980C2314yes—
69#34044D411yescharacteristic, complemented
70#34044D412yescomplemented
71#34044C2311yescharacteristic
72#34044C2312yes—
73C2×#2280D411yescharacteristic, complemented
74C2×#2280D412yescomplemented
75C2×#2127C2318yescomplemented
76C2×#2121C23116yescomplemented
77#21750C2314yescomplemented
78#21749C2318yescomplemented
79#20616C2318yes—
80#20609C23116yes—
81C2×#1785C2312yes—
82C2×#1783C2312yes—
83C2×#1782C2314yes—
84C2×#1781C2314yes—
85#12582C2318yescomplemented
86#12574C23116yescomplemented
87C2×#585C2×D418yes—
88C2×#585C2×D418yes—
89C2×#585C2×D414yes—
90C2×#585C2×D414yes—
91C22×H(5,𝔽2)C2414yes—
92C23×#41C2×D411yescharacteristic
93C23×#41C2×D411yescharacteristic
94C23×#41C2×D411yescharacteristic
95C23×#41C2×D412yes—
96C23×#41C2×D412yes—
97C23×#41C2×D412yes—
98C23×#41C2411yescharacteristic
99C23×#41C2411yescharacteristic
100C24×Q8C2×D411yescharacteristic
101C24×D4C2×D412yes—
102C24×D4C2411yescharacteristic
103C25×C4C2×D412yes—
104C25×C4C2×D411yescharacteristic
105C2×#536C2×D412yes—
106C2×#536C2×D414yes—
107C2×#536C2×D412yes—
108C2×#536C2×D412yes—
109C2×#535C2×D412yes—
110C2×#535C2×D414yes—
111C2×#535C2×D412yes—
112C2×#535C2412yes—
113C2×#534C2×D414yes—
114C2×#534C2×D414yes—
115C2×#534C2×D414yes—
116C2×#534C2414yes—
117C2×#532C2×D414yes—
118C2×#525C2×D418yes—
119C2×#525C2×D412yes—
120C2×#525C2×D412yes—
121C22×#123C2×D414yes—
122C22×#120C2×D414yes—
123C22×#117C2×D411yescharacteristic
124C22×#117C2×D411yescharacteristic
125C22×#117C2×D411yescharacteristic
126C22×#117C2×D412yes—
127C22×#117C2×D412yes—
128C22×#117C2×D412yes—
129C2×#458C2×D4132yescomplemented
130C2×B(4,𝔽2)C2×D4132yescomplemented
131C2×#455C2×D4116yes—
132C2×#453C2×D4116yes—
133C2×#413C2×D414yes—
134C2×#413C2×D414yes—
135C22×#100C2414yes—
136C2×#410C2×D418yes—
137C2×#410C2×D418yes—
138C22×#99C2418yes—
139#2310C2×D412yes—
140#2310C2×D412yes—
141#2309C2×D412yes—
142#2309C2412yes—
143#2308C2×D414yes—
144#2308C2×D414yes—
145#2308C2×D414yes—
146#2308C2414yes—
147#2280C2×D412yes—
148#2280C2×D412yes—
149C2×#381C2×D414yes—
150C2×#379C2×D414yes—
151C26#12011yescharacteristic
152C26C22×D412yes—
153#585#12018yes—
154#585#12018yes—
155#585#120116yes—
156C2×H(5,𝔽2)C22×D418yes—
157C2×H(5,𝔽2)C22×D4116yes—
158C22×#41#12012yescomplemented
159C22×#41C22×D412yes—
160C22×#41C22×D414yes—
161C22×#41C22×D414yes—
162C22×#41C22×D412yes—
163C22×#41C22×D412yes—
164C22×#41H(5,𝔽2)14yes—
165C22×#41H(5,𝔽2)14yes—
166C22×#41H(5,𝔽2)12yes—
167C22×#41H(5,𝔽2)12yes—
168C23×Q8#12012yes—
169C23×Q8#12011yescharacteristic, complemented
170C23×Q8#12012yescomplemented
171C23×Q8C22×D411yescharacteristic
172C23×Q8C22×D411yescharacteristic
173C23×D4#12012yes—
174C23×D4#12012yes—
175C23×D4#12012yescomplemented
176C23×D4#12011yescharacteristic, complemented
177C23×D4C22×D412yes—
178C23×D4C22×D412yes—
179C23×D4C22×D414yes—
180C23×D4C22×D412yes—
181C23×D4C22×D411yescharacteristic
182C23×D4C2511yescharacteristic
183C24×C4#12011yescharacteristic
184C24×C4#12012yes—
185C24×C4#12011yescharacteristic
186C24×C4#12012yes—
187C24×C4#12011yescharacteristic
188C24×C4#12012yes—
189C24×C4#12014yescomplemented
190C24×C4C22×D412yes—
191C24×C4C22×D411yescharacteristic
192C24×C4C22×D411yescharacteristic
193C24×C4C22×D412yes—
194C24×C4C22×D411yescharacteristic
195C2×C4⋊Q8#12014yescomplemented
196C2×#127C22×D414yes—
197C2×#124#12014yescomplemented
198C2×#124C22×D414yes—
199C2×#123#12018yescomplemented
200C2×#123C22×D418yes—
201C2×#120#12018yescomplemented
202C2×#120C22×D418yes—
203C2×#117#12014yescomplemented
204C2×#117#12014yescomplemented
205C2×#117#12012yescomplemented
206C2×#117C22×D412yes—
207C2×#117C22×D412yes—
208C2×#117C22×D412yes—
209C22×C22⋊C4C22×D414yes—
210C22×C22⋊C4C22×D414yes—
211C22×C22⋊C4C22×D414yes—
212C22×C42#12012yescomplemented
213C2×#100C22×D418yes—
214C2×#99C22×D4116yes—
215C25C2×#12011yescharacteristic
216C25C2×#12012yes—
217C25C2×#12011yescharacteristic
218C25C2×#12011yescharacteristic
219C25C2×#12014yes—
220C25C23×D412yes—
221H(5,𝔽2)C2×#120116yes—
222C2×#41C2×#12012yes—
223C2×#41C2×#12014yes—
224C2×#41C2×#12012yes—
225C2×#41#53414yes—
226C2×#41#53414yes—
227C2×#41#53414yes—
228C2×#41#53414yes—
229C2×#41#53412yes—
230C2×#41#53412yes—
231C2×#41#53414yes—
232C2×#41D4212yes—
233C2×#41D4214yes—
234C2×#41D4212yes—
235C2×#41D4214yes—
236C2×#41D4214yes—
237C2×#41D4214yes—
238C2×#41D4214yes—
239C2×#41H(3,𝔽4)12yes—
240C2×#41H(3,𝔽4)12yes—
241C2×#41H(3,𝔽4)14yes—
242C22×Q8C2×#12012yes—
243C22×Q8C2×#12012yes—
244C22×Q8#53412yes—
245C22×Q8D4214yes—
246C22×Q8D4214yes—
247C22×D4C2×#12014yes—
248C22×D4C2×#12012yes—
249C22×D4#53414yes—
250C22×D4D4214yes—
251C22×D4D4214yes—
252C22×D4D4214yes—
253C22×D4D4214yes—
254C22×D4C23×D412yes—
255C22×D4C23×D414yes—
256C22×D4C2×H(5,𝔽2)12yes—
257C22×D4C2×H(5,𝔽2)14yes—
258C23×C4B(4,𝔽2)11yescharacteristic
259C23×C4B(4,𝔽2)11yescharacteristic
260C23×C4B(4,𝔽2)11yescharacteristic
261C23×C4B(4,𝔽2)11yescharacteristic
262C23×C4B(4,𝔽2)12yes—
263C23×C4B(4,𝔽2)14yes—
264C23×C4B(4,𝔽2)12yes—
265C23×C4B(4,𝔽2)14yes—
266C23×C4C2×#12011yescharacteristic
267C23×C4C2×#12012yes—
268C23×C4C2×#12011yescharacteristic
269C23×C4C2×#12012yes—
270C23×C4C2×#12012yes—
271C23×C4C2×#12012yes—
272C23×C4C2×#12011yescharacteristic
273C23×C4C2×#12011yescharacteristic
274C23×C4C2×#12014yes—
275C23×C4C2×#12011yescharacteristic
276C23×C4#53412yes—
277C23×C4#53412yes—
278C23×C4#53411yescharacteristic
279C23×C4#53411yescharacteristic
280C23×C4D4214yes—
281C23×C4D4214yes—
282C23×C4D4214yes—
283C23×C4D4214yes—
284C23×C4D4214yes—
285C23×C4D4214yes—
286C23×C4C23×D411yescharacteristic
287C2×C22⋊C4C2×#12018yes—
288C2×C42C2×#12014yes—
289C24#240314yes—
290C24#240311yescharacteristic
291C24#240311yescharacteristic
292C24C2×B(4,𝔽2)11yescharacteristic
293C24C2×B(4,𝔽2)11yescharacteristic
294C24C2×B(4,𝔽2)14yes—
295C24C2×B(4,𝔽2)12yes—
296C24C22×#12012yes—
297C24C22×#12011yescharacteristic
298C24C2×#53412yes—
299C24C2×D4214yes—
300C24C2×D4214yes—
301C2×Q8#240312yes—
302C2×Q8#240314yes—
303C2×Q8#240312yes—
304C2×Q8#267912yes—
305C2×Q8#267914yes—
306C2×Q8#267912yes—
307C2×D4#240314yes—
308C2×D4#240314yes—
309C2×D4#240314yes—
310C2×D4#240314yes—
311C2×D4#267914yes—
312C2×D4#267914yes—
313C2×D4#267914yes—
314C2×D4#267914yes—
315C2×D4C22×#12012yes—
316C2×D4C2×#53412yes—
317C2×D4C2×#53414yes—
318C2×D4C2×D4214yes—
319C2×D4C2×D4212yes—
320C2×D4C2×H(3,𝔽4)12yes—
321C22×C4#240314yes—
322C22×C4#240314yes—
323C22×C4#240312yes—
324C22×C4#240312yes—
325C22×C4#240314yes—
326C22×C4#240314yes—
327C22×C4#240314yes—
328C22×C4#240314yes—
329C22×C4#240312yes—
330C22×C4#240312yes—
331C22×C4#240314yes—
332C22×C4#240311yescharacteristic
333C22×C4#240312yes—
334C22×C4#240311yescharacteristic
335C22×C4#240312yes—
336C22×C4#267914yes—
337C22×C4#267914yes—
338C22×C4#267914yes—
339C22×C4#267914yes—
340C22×C4#267914yes—
341C22×C4#267912yes—
342C22×C4#267912yes—
343C22×C4C2×B(4,𝔽2)11yescharacteristic
344C22×C4C2×B(4,𝔽2)11yescharacteristic
345C22×C4C2×B(4,𝔽2)14yes—
346C22×C4C2×B(4,𝔽2)12yes—
347C22×C4C2×B(4,𝔽2)11yescharacteristic
348C22×C4C2×B(4,𝔽2)11yescharacteristic
349C22×C4C2×B(4,𝔽2)12yes—
350C22×C4C2×B(4,𝔽2)11yescharacteristic
351C22×C4C2×B(4,𝔽2)11yescharacteristic
352C22×C4C2×B(4,𝔽2)12yes—
353C22×C4C22×#12011yescharacteristic
354C22×C4C2×#53411yescharacteristic
355C22×C4C2×D4214yes—
356C23#1594712yes—
357C23#1594712yes—
358C23C2×#240314yes—
359C23C2×#240311yescharacteristic
360C23C2×#240312yes—
361C23C2×#267914yes—
362C23C22×B(4,𝔽2)11yescharacteristic
363C23C22×B(4,𝔽2)11yescharacteristic
364C23C22×B(4,𝔽2)12yes—
365C2×C4#1594714yes—
366C2×C4#1594714yes—
367C2×C4#1594711yescharacteristic
368C2×C4#1594712yes—
369C2×C4#1594711yescharacteristic
370C2×C4#2390012yes—
371C2×C4#2390011yescharacteristic
372C2×C4#2390011yescharacteristic
373C2×C4#2390012yes—
374C2×C4#2390012yes—
375C2×C4#2390011yescharacteristic
376C2×C4#2390012yes—
377C2×C4#2390011yescharacteristic
378C2×C4#2390012yes—
379C2×C4#2390012yes—
380C2×C4#2390012yes—
381C2×C4#2390012yes—
382C2×C4#2390011yescharacteristic
383C2×C4#2390011yescharacteristic
384C2×C4#2390012yes—
385C2×C4C2×#240312yes—
386C2×C4C2×#267912yes—
387C22C2×#1594711yescharacteristic
388C22C2×#2390012yes—
389C22C2×#2390011yescharacteristic
390C22C2×#2390011yescharacteristic
391C22C2×#2390012yes—
392C4#44947512yes—
393C4#44947511yescharacteristic
394C4#44947511yescharacteristic
395C4#44947511yescharacteristic
396C4#44947512yes—
397C4#44947511yescharacteristic
398C2C2×#44947511yescharacteristic
subgroupG/H
centerC4#449475
commutator, Frattini, cosocleC23×D4C25
socleC2C2×#449475
LCSG ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C23×C4 ◁ C24×#41 ◁ G
LECSG ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72303310C2semidirect
2. #72303134C2semidirect
3. #72303110C2semidirect
4. #72303051C2semidirect
5. #72280954C2semidirect
6. #72279188C2semidirect
7. #72267980C2semidirect
8. #72258211C2semidirect
9. #72256657C2semidirect
10. #72238718C2semidirect
11. C23×#585C22semidirect
12. #10586228C22semidirect
17. #7622875C22semidirect
18. C2×#33358C22semidirect
19. C2×#33347C22semidirect
20. #6628893C22semidirect
21. #6628892C22semidirect
22. #6628869C22semidirect
23. #6628867C22semidirect
24. #6499894C22semidirect
25. #6499892C22semidirect
26. #6374127C22semidirect
27. #6374127C22semidirect
28. #6370746C22semidirect
29. #6370734C22semidirect
30. C2×#20591C22semidirect
31. C2×#20584C22semidirect
32. #492899C22semidirect
33. #492886C22semidirect
34. #147767C22semidirect
35. #147752C22semidirect
36. #147570C22semidirect
37. #147569C22semidirect
38. #147568C22semidirect
39. #147567C22semidirect
44. #62862C23semidirect
45. C2×#3449D4semidirect
46. C2×#3449D4semidirect
49. C22×#525D4semidirect
50. C22×#525D4semidirect
51. C2×#3028D4semidirect
52. C2×#3027D4semidirect
54. C2×#3025D4semidirect
55. C2×#3022D4semidirect
56. C2×#3021D4semidirect
58. C2×#3018D4semidirect
61. #35242D4semidirect
63. #35149D4semidirect
65. #35088D4semidirect
67. #34980D4semidirect
69. #34044D4semidirect
70. #34044D4semidirect
73. C2×#2280D4semidirect
74. C2×#2280D4semidirect
75. C2×#2127C23semidirect
76. C2×#2121C23semidirect
77. #21750C23semidirect
78. #21749C23semidirect
85. #12582C23semidirect
86. #12574C23semidirect
129. C2×#458C2×D4semidirect
130. C2×B(4,𝔽2)C2×D4semidirect
158. C22×#41#120semidirect
169. C23×Q8#120semidirect
170. C23×Q8#120semidirect
175. C23×D4#120semidirect
176. C23×D4#120semidirect
189. C24×C4#120semidirect
195. C2×C4⋊Q8#120semidirect
197. C2×#124#120semidirect
199. C2×#123#120semidirect
201. C2×#120#120semidirect
203. C2×#117#120semidirect
204. C2×#117#120semidirect
205. C2×#117#120semidirect
212. C22×C42#120semidirect

Maximal subgroups

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

subgroupindex
#723033102
#723031342
#723031102
#723030512
#722809542
#722791882
#722679802
#722582112
#722566572
#722387182