Finite Group Databasebeta

← #72341187

Subgroups — #72341187

Order 2048.

At least 8813 conjugacy classes of subgroups; only the 972 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#72341185C214yesmaximal, complemented
2#72341184C214yesmaximal, complemented
3#72341181C218yesmaximal, complemented
4#72341056C211yesmaximal, characteristic, complemented
5#72340902C212yesmaximal, complemented
6#72338721C212yesmaximal, complemented
7#72262289C212yesmaximal, complemented
8#72262288C212yesmaximal, complemented
9#72236894C211yesmaximal, characteristic, complemented
10#72236894C211yesmaximal, characteristic, complemented
11#72236834C211yesmaximal, characteristic, complemented
12#72236108C211yesmaximal, characteristic, complemented
13#72236108C211yesmaximal, characteristic, complemented
14#72236058C211yesmaximal, characteristic, complemented
15#9039188C2214yescomplemented
16#8933866C2211yescharacteristic, complemented
17#8933771C2211yescharacteristic, complemented
18#8933771C2211yescharacteristic, complemented
19#8933771C2211yescharacteristic, complemented
20#8751772C2214yes—
21#7643002C2211yescharacteristic
22#7643002C2211yescharacteristic
23#7643002C2211yescharacteristic
24C2×#35527C2211yescharacteristic
25#6628892C2212yescomplemented
26#6628892C2212yescomplemented
27#6628867C2212yescomplemented
28#500037C2218yescomplemented
29#500033C2218yescomplemented
30#499838C2218yescomplemented
31#499835C2218yescomplemented
32#496231C2214yescomplemented
33#496231C2214yescomplemented
34#496194C2214yescomplemented
35#449527C2212yes—
36#449492C2212yes—
37#417468C2214yescomplemented
38#382394C2211yescharacteristic
39#379759C2212yescomplemented
40#245255C2214yescomplemented
41#245255C2214yescomplemented
42#245234C2212yes—
43#245234C2212yes—
44#245233C2212yes—
45#245233C2212yes—
46#201690C2212yescomplemented
47#201690C2212yescomplemented
48C2×#8546C2211yescharacteristic
49C2×#8546C2211yescharacteristic
50#133936C2212yescomplemented
51#133885C2214yescomplemented
52#133835C2212yescomplemented
53#133500C2218yescomplemented
54#133498C2218yescomplemented
55#133359C2218yescomplemented
56#133358C2218yescomplemented
57#133311C2214yescomplemented
58#133311C2214yescomplemented
59#133309C2212yescomplemented
60#133309C2212yescomplemented
61#133308C2212yescomplemented
62#133308C2212yescomplemented
63C23×H(5,𝔽2)C2311yescharacteristic
64C2×#3484D411yescharacteristic, complemented
65C2×#3484D411yescharacteristic, complemented
66C2×#3484D411yescharacteristic, complemented
67C23×#127C2311yescharacteristic
68C2×#3019C2314yes—
69#52012C2314yes—
70#51992C2314yes—
71#51940C2314yes—
72#51895C2314yes—
73#50856D414yescomplemented
74#50646D414yescomplemented
75#48949D414yescomplemented
76#48773D414yescomplemented
77#48143D414yescomplemented
78#48113D414yescomplemented
79#48058D414yescomplemented
80#47846D414yescomplemented
81#37232D412yescomplemented
82#37232D412yescomplemented
83#37232C2312yes—
84#37232C2312yes—
85#36626D412yescomplemented
86#35554D411yescharacteristic, complemented
87#35554D411yescharacteristic, complemented
88#35554D411yescharacteristic, complemented
89#35554C2311yescharacteristic
90#35554C2311yescharacteristic
91#35554C2311yescharacteristic
92#35528D412yescomplemented
93#35528C2312yes—
94#35527D411yescharacteristic, complemented
95#35527D411yescharacteristic, complemented
96#35527D411yescharacteristic, complemented
97#35527C2311yescharacteristic
98#35527C2311yescharacteristic
99#35527C2311yescharacteristic
100#35527C2312yes—
101#35527C2312yes—
102#35222C2312yes—
103#34980D412yescomplemented
104#34980D412yescomplemented
105#34980C2312yes—
106#34980C2312yes—
107#34171D411yescharacteristic, complemented
108#34171D411yescharacteristic, complemented
109#34171D411yescharacteristic, complemented
110C2×#2309C2311yescharacteristic
111#32900C2314yes—
112#32900C2314yes—
113#32898C2314yes—
114#32898C2314yes—
115#20616C2314yes—
116C2×#1783C2312yes—
117C2×#1781C2312yes—
118C2×#1781C2312yes—
119#13082C2318yes—
120#13082C2318yes—
121#13080C2318yes—
122#13080C2318yes—
123#12858C2318yes—
124#12696C2318yes—
125#11155C2314yes—
126#10989C2314yes—
127#10981C2312yes—
128#10981C2312yes—
129#9501C2314yes—
130#9501C2314yes—
131#9497C2314yes—
132#9497C2314yes—
133#8546C2312yes—
134#8546C2312yes—
135#8546C2312yes—
136#8546C2312yes—
137C22×H(5,𝔽2)C2×D411yescharacteristic
138C22×H(5,𝔽2)C2×D411yescharacteristic
139C22×H(5,𝔽2)C2×D411yescharacteristic
140C22×H(5,𝔽2)C2412yes—
141C24×D4C2×D411yescharacteristic
142C24×D4C2×D411yescharacteristic
143C24×D4C2×D411yescharacteristic
144C24×D4C2×D411yescharacteristic
145C24×D4C2×D411yescharacteristic
146C24×D4C2411yescharacteristic
147C2×#560C2×D414yescomplemented
148C2×#551C2×D414yescomplemented
149C2×#535C2×D412yes—
150C2×#535C2×D412yes—
151C2×#535C2412yes—
152C2×#534C2×D414yes—
153C2×#534C2×D414yes—
154C2×#534C2×D412yes—
155C2×#534C2×D412yes—
156C22×#127C2×D411yescharacteristic
157C22×#127C2×D411yescharacteristic
158C22×#127C2×D411yescharacteristic
159C22×#127C2×D411yescharacteristic
160C22×#127C2×D411yescharacteristic
161C22×#127C2×D411yescharacteristic
162C22×#127C2411yescharacteristic
163C22×#127C2411yescharacteristic
164C2×#525C2×D412yes—
165C22×#120C2×D412yes—
166C22×#120C2×D412yes—
167C2×#518C2×D411yescharacteristic
168C2×#518C2×D412yes—
169C2×#518C2×D411yescharacteristic
170C2×#518C2×D411yescharacteristic
171C23×C42C2×D411yescharacteristic
172C22×#100C2414yes—
173#2748C2×D414yescomplemented
174#2748C2×D414yescomplemented
175#2724C2×D414yescomplemented
176#2717C2×D414yescomplemented
177#2709C2×D414yescomplemented
178#2684C2×D414yescomplemented
179#2679C2×D414yescomplemented
180#2669C2×D414yescomplemented
181#2435C2×D414yes—
182#2435C2×D414yes—
183#2433C2×D414yes—
184#2433C2×D414yes—
185#2405C2×D414yes—
186#2405C2×D414yes—
187#2405C2×D412yes—
188#2405C2×D412yes—
189#2405C2×D412yes—
190#2405C2×D412yes—
191#2405C2×D412yes—
192#2405C2×D412yes—
193#2405C2412yes—
194#2405C2412yes—
195#2403C2×D412yes—
196#2403C2×D412yes—
197#2403C2×D412yes—
198#2403C2×D412yes—
199#2403C2×D412yes—
200#2403C2×D412yes—
201#2403C2×D414yes—
202#2403C2×D414yes—
203#2403C2412yes—
204#2403C2412yes—
205#2379C2×D414yescomplemented
206#2376C2×D414yescomplemented
207#2375C2×D414yescomplemented
208#2374C2×D414yescomplemented
209#2366C2×D414yescomplemented
210#2358C2×D414yescomplemented
211#2309C2×D411yescharacteristic
212#2309C2×D411yescharacteristic
213#2309C2×D411yescharacteristic
214#2309C2×D411yescharacteristic
215#2309C2×D411yescharacteristic
216#2309C2×D411yescharacteristic
217#2309C2×D412yes—
218#2309C2412yes—
219#2309C2412yes—
220#2308C2×D412yes—
221#2308C2×D412yes—
222#2308C2×D414yes—
223#2308C2×D414yes—
224#2297C2×D412yes—
225#2297C2×D412yes—
226#2296C2×D412yes—
227#2296C2×D412yes—
228#2280C2×D412yes—
229C2×#379C2×D412yes—
230C2×#379C2×D412yes—
231#1781C2414yes—
232#1781C2414yes—
233C26#12011yescharacteristic
234C26#12011yescharacteristic
235C26C22×D411yescharacteristic
236C26C22×D411yescharacteristic
237C22×#41C22×D412yes—
238C23×D4#12011yescharacteristic
239C23×D4#12011yescharacteristic
240C23×D4#12011yescharacteristic
241C23×D4#12011yescharacteristic
242C23×D4#12011yescharacteristic
243C23×D4#12011yescharacteristic
244C23×D4#12011yescharacteristic
245C23×D4#12011yescharacteristic
246C23×D4#12012yes—
247C23×D4#12012yes—
248C23×D4#12011yescharacteristic
249C23×D4#12011yescharacteristic
250C23×D4#12012yes—
251C23×D4#12012yes—
252C23×D4C22×D411yescharacteristic
253C23×D4C22×D411yescharacteristic
254C23×D4C22×D411yescharacteristic
255C23×D4C22×D411yescharacteristic
256C23×D4C22×D411yescharacteristic
257C23×D4C22×D411yescharacteristic
258C23×D4C22×D411yescharacteristic
259C23×D4C22×D411yescharacteristic
260C23×D4C22×D412yes—
261C23×D4C22×D412yes—
262C23×D4C22×D414yes—
263C23×D4C22×D414yes—
264C23×D4C22×D411yescharacteristic
265C23×D4C22×D411yescharacteristic
266C23×D4C22×D411yescharacteristic
267C23×D4C22×D411yescharacteristic
268C23×D4C22×D411yescharacteristic
269C23×D4C2511yescharacteristic
270C24×C4#12011yescharacteristic
271C24×C4#12011yescharacteristic
272C24×C4C22×D411yescharacteristic
273C2×#127#12014yes—
274C2×#127C22×D414yes—
275C2×#124#12014yes—
276C2×#124C22×D414yes—
277C2×#123#12014yes—
278C2×#123#12014yes—
279C2×#123C22×D414yes—
280C2×#123C22×D414yes—
281C2×C22⋊Q8#12012yes—
282C2×C22⋊Q8#12012yes—
283C2×#121#12012yes—
284C2×#121#12012yes—
285C2×#120#12014yes—
286C2×#120#12014yes—
287C2×#120C22×D414yes—
288C2×#120C22×D414yes—
289C2×#117C22×D412yes—
290C2×#117C22×D412yes—
291C2×#117C22×D412yes—
292C22×C22⋊C4#12012yes—
293C22×C22⋊C4#12012yes—
294C22×C22⋊C4#12012yes—
295C22×C22⋊C4#12012yes—
296C22×C22⋊C4#12014yes—
297C22×C22⋊C4#12014yes—
298C22×C22⋊C4C22×D412yes—
299C22×C22⋊C4C22×D412yes—
300C22×C42#12011yescharacteristic
301C22×C42#12011yescharacteristic
302C22×C42C22×D411yescharacteristic
303C22×C42C22×D411yescharacteristic
304#390#12014yes—
305#390#12014yes—
306#390C22×D414yes—
307#390C22×D414yes—
308#386#12014yes—
309#386#12014yes—
310#386#12014yes—
311#386#12014yes—
312#386C22×D414yes—
313#386C22×D414yes—
314#386C22×D414yes—
315#386C22×D414yes—
316#379#12014yes—
317#379#12014yes—
318#379C22×D412yes—
319#379C22×D412yes—
320#379C22×D412yes—
321#379C22×D412yes—
322C4×C22⋊C4#12014yes—
323C4×C22⋊C4#12014yes—
324C25B(4,𝔽2)12yes—
325C25B(4,𝔽2)12yes—
326C25C2×#12011yescharacteristic
327C25C2×#12011yescharacteristic
328C25C2×#12011yescharacteristic
329C25C2×#12011yescharacteristic
330C25C2×#12012yes—
331C25C2×#12012yes—
332C25C2×#12012yes—
333C25C2×#12012yes—
334C25C2×#12011yescharacteristic
335C25C2×#12011yescharacteristic
336C25C2×#12012yes—
337C25C2×#12012yes—
338C25C2×#12011yescharacteristic
339C25C23×D411yescharacteristic
340C25C23×D411yescharacteristic
341C22×Q8C2×#12012yes—
342C22×D4C2×#12014yes—
343C22×D4C2×#12014yes—
344C22×D4C2×#12012yes—
345C22×D4C2×#12014yes—
346C22×D4C23×D414yes—
347C23×C4C2×#12012yes—
348C23×C4C2×#12012yes—
349C23×C4C2×#12011yescharacteristic
350C23×C4C2×#12011yescharacteristic
351C23×C4C2×#12011yescharacteristic
352C23×C4C2×#12011yescharacteristic
353C23×C4C2×#12011yescharacteristic
354C23×C4C2×#12012yes—
355C23×C4C2×#12012yes—
356C23×C4C2×#12012yes—
357C23×C4C2×#12012yes—
358C23×C4C2×#12012yes—
359C23×C4C2×#12012yes—
360C23×C4C23×D411yescharacteristic
361C2×C4⋊C4C2×#12012yes—
362C2×C4⋊C4C2×#12012yes—
363C2×C22⋊C4C2×#12014yes—
364C2×C22⋊C4C2×#12014yes—
365C2×C22⋊C4C2×#12014yes—
366C2×C22⋊C4C2×#12014yes—
367C2×C42C2×#12014yes—
368C2×C42C2×#12014yes—
369C24#284612yes—
370C24#284612yes—
371C24#284611yescharacteristic
372C24#284611yescharacteristic
373C24C2×B(4,𝔽2)12yes—
374C24C2×B(4,𝔽2)12yes—
375C24C2×B(4,𝔽2)11yescharacteristic
376C24C2×B(4,𝔽2)11yescharacteristic
377C24#302611yescharacteristic
378C24#302611yescharacteristic
379C24C22×#12012yes—
380C24C22×#12012yes—
381C24C22×#12011yescharacteristic
382C22×C4#284612yes—
383C22×C4C22×#12012yes—
384C23#3289811yescharacteristic
385C23#3289811yescharacteristic
386C23#3289811yescharacteristic
387C23#3289811yescharacteristic
388C23C2×#284611yescharacteristic
389C23C22×B(4,𝔽2)11yescharacteristic
390C23C2×#302611yescharacteristic
391C2×C4#6039914yes—
392C22C2×#3289811yescharacteristic
393C22#662889612yes—
394C2#7222571211yescharacteristic
395C2#7223513011yescharacteristic
396C2#7223513011yescharacteristic
subgroupG/H
center, socleC22C2×#32898
commutator, Frattini, cosocleC23×D4C25
LCSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ C23×#127 ◁ G
LECSG ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #72341185C2semidirect
2. #72341184C2semidirect
3. #72341181C2semidirect
4. #72341056C2semidirect
5. #72340902C2semidirect
6. #72338721C2semidirect
7. #72262289C2semidirect
8. #72262288C2semidirect
9. #72236894C2semidirect
10. #72236894C2semidirect
11. #72236834C2semidirect
12. #72236108C2semidirect
13. #72236108C2semidirect
14. #72236058C2semidirect
15. #9039188C22semidirect
16. #8933866C22semidirect
17. #8933771C22semidirect
18. #8933771C22semidirect
19. #8933771C22semidirect
25. #6628892C22semidirect
26. #6628892C22semidirect
27. #6628867C22semidirect
28. #500037C22semidirect
29. #500033C22semidirect
30. #499838C22semidirect
31. #499835C22semidirect
32. #496231C22semidirect
33. #496231C22semidirect
34. #496194C22semidirect
37. #417468C22semidirect
39. #379759C22semidirect
40. #245255C22semidirect
41. #245255C22semidirect
46. #201690C22semidirect
47. #201690C22semidirect
50. #133936C22semidirect
51. #133885C22semidirect
52. #133835C22semidirect
53. #133500C22semidirect
54. #133498C22semidirect
55. #133359C22semidirect
56. #133358C22semidirect
57. #133311C22semidirect
58. #133311C22semidirect
59. #133309C22semidirect
60. #133309C22semidirect
61. #133308C22semidirect
62. #133308C22semidirect
64. C2×#3484D4semidirect
65. C2×#3484D4semidirect
66. C2×#3484D4semidirect
73. #50856D4semidirect
74. #50646D4semidirect
75. #48949D4semidirect
76. #48773D4semidirect
77. #48143D4semidirect
78. #48113D4semidirect
79. #48058D4semidirect
80. #47846D4semidirect
81. #37232D4semidirect
82. #37232D4semidirect
85. #36626D4semidirect
86. #35554D4semidirect
87. #35554D4semidirect
88. #35554D4semidirect
92. #35528D4semidirect
94. #35527D4semidirect
95. #35527D4semidirect
96. #35527D4semidirect
103. #34980D4semidirect
104. #34980D4semidirect
107. #34171D4semidirect
108. #34171D4semidirect
109. #34171D4semidirect
147. C2×#560C2×D4semidirect
148. C2×#551C2×D4semidirect
173. #2748C2×D4semidirect
174. #2748C2×D4semidirect
175. #2724C2×D4semidirect
176. #2717C2×D4semidirect
177. #2709C2×D4semidirect
178. #2684C2×D4semidirect
179. #2679C2×D4semidirect
180. #2669C2×D4semidirect
205. #2379C2×D4semidirect
206. #2376C2×D4semidirect
207. #2375C2×D4semidirect
208. #2374C2×D4semidirect
209. #2366C2×D4semidirect
210. #2358C2×D4semidirect

Maximal subgroups

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

subgroupindex
#723411852
#723411842
#723411812
#723410562
#723409022
#723387212
#722622892
#722622882
#722368942
#722368342
#722361082
#722360582