Finite Group Databasebeta

← C2×#72225609 #72267315

Subgroups — C2×#72225609 #72267315

Order 2048.

At least 7884 conjugacy classes of subgroups; only the 1864 normal subgroups are shown.

The subgroup lattice has 457 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#106542C211yesmaximal, characteristic, complemented
2C22×#9495C211yesmaximal, characteristic, complemented
3C22×#10336C212yesmaximal, complemented
4C2×#392125C211yesmaximal, characteristic, complemented
5C2×#115133C212yesmaximal, complemented
6C2×#393929C212yesmaximal, complemented
7C2×#111458C212yesmaximal, complemented
8C2×#616960C211yesmaximal, characteristic, complemented
9C22×#35346C211yesmaximal, characteristic, complemented
10#72225609C2116yesmaximal, direct product factor
11C22×#32898C211yesmaximal, characteristic, complemented
12C22×#8541C211yesmaximal, characteristic, complemented
13C24×H(5,𝔽2)C2211yescharacteristic
14C23×#534C2211yescharacteristic, complemented
15C24×#120C2211yescharacteristic
16C2×#35346C2218yescomplemented
17C22×#2403C2211yescharacteristic
18C22×#2403C2212yes—
19C22×#2403C2212yes—
20C22×#2403C2212yescomplemented
21C22×#2403C2211yescharacteristic, complemented
22C22×#2308C2211yescharacteristic
23C2×#32898C2218yescomplemented
24C22×#1781C2211yescharacteristic
25C22×#1781C2211yescharacteristic
26C22×#1781C2212yescomplemented
27C22×#1781C2212yescomplemented
28C22×#1781C2212yescomplemented
29C22×#1781C2212yescomplemented
30C22×#1781C2212yescomplemented
31#616960C2218yescomplemented
32C2×#14335C2211yescharacteristic, complemented
33C2×#13867C2211yescharacteristic
34C2×#13867C2211yescharacteristic, complemented
35C2×#13867C2212yescomplemented
36#393929C22116yescomplemented
37#392125C2218yescomplemented
38C2×#10725C2212yescomplemented
39C2×#10336C22116yescomplemented
40C2×#9495C2218yescomplemented
41C2×#8541C2218yescomplemented
42C22×#1304C2212yescomplemented
43C22×#1304C2212yescomplemented
44#115133C22116yescomplemented
45#111458C22116yescomplemented
46#106542C2218yescomplemented
47C28D411yescharacteristic, complemented
48C23×H(5,𝔽2)D418yescomplemented
49C23×H(5,𝔽2)C2314yes—
50C25×D4C2312yes—
51C25×D4C2311yescharacteristic
52C25×D4C2311yescharacteristic
53C25×D4C2311yescharacteristic
54C25×D4C2311yescharacteristic
55C22×#534C2314yescomplemented
56C23×#120D414yescomplemented
57C23×#120D412yescomplemented
58C23×#120D414yescomplemented
59C23×#120C2314yes—
60C24×C22⋊C4D411yescharacteristic, complemented
61C24×C22⋊C4D411yescharacteristic, complemented
62C24×C22⋊C4D411yescharacteristic, complemented
63C24×C22⋊C4C2311yescharacteristic
64C2×#2403D418yescomplemented
65C2×#2403D4116yescomplemented
66C2×#2403D414yescomplemented
67C2×#2403C2314yes—
68C2×#2403C2318yes—
69C2×#2403C2318yes—
70C2×#2403C2314yescomplemented
71C2×#2403C2318yescomplemented
72C2×#2308D418yescomplemented
73C2×#2308C2314yes—
74C22×#386D414yescomplemented
75C22×#386D412yescomplemented
76C22×#386D414yescomplemented
77C22×#379C2312yes—
78C22×#379C2312yes—
79C22×#379C2312yes—
80C22×#379C2311yescharacteristic
81C22×#379C2311yescharacteristic
82C2×#1781D418yescomplemented
83C2×#1781C2314yes—
84C2×#1781C2314yes—
85C2×#1781C2318yescomplemented
86C2×#1781C2318yescomplemented
87C2×#1781C2318yescomplemented
88C2×#1781C2318yescomplemented
89C2×#1781C2318yescomplemented
90#14335C2314yescomplemented
91#13867D414yescomplemented
92#13867C2314yes—
93#13867C2314yescomplemented
94#13867C2318yescomplemented
95#10725C2318yescomplemented
96C2×#1304C2318yescomplemented
97C2×#1304C2318yescomplemented
98C27C2×D411yescharacteristic
99C27C2×D411yescharacteristic
100C27C2×D411yescharacteristic
101C27C2×D412yes—
102C27C2×D411yescharacteristic, complemented
103C27C2×D414yescomplemented
104C27C2411yescharacteristic
105C22×H(5,𝔽2)C2×D4116yescomplemented
106C24×D4C2×D414yes—
107C24×D4C2×D412yes—
108C24×D4C2×D412yes—
109C24×D4C2×D414yes—
110C24×D4C2×D418yes—
111C24×D4C2×D412yes—
112C24×D4C2×D414yes—
113C24×D4C2×D412yes—
114C24×D4C2×D414yes—
115C24×D4C2×D418yes—
116C24×D4C2×D418yes—
117C24×D4C2×D414yes—
118C24×D4C2×D414yes—
119C24×D4C2×D412yes—
120C24×D4C2×D412yes—
121C24×D4C2×D412yes—
122C24×D4C2×D412yescomplemented
123C24×D4#4114yes—
124C24×D4C2414yes—
125C24×D4C2412yes—
126C24×D4C2412yes—
127C24×D4C2412yes—
128C24×D4C2412yes—
129C25×C4C2×D411yescharacteristic
130C25×C4C2×D412yes—
131C25×C4C2×D411yescharacteristic
132C25×C4C2×D411yescharacteristic
133C25×C4C2×D411yescharacteristic, complemented
134C25×C4C2×D412yescomplemented
135C22×#120C2×D4116yescomplemented
136C22×#120C2×D418yescomplemented
137C23×C22⋊C4C2×D414yes—
138C23×C22⋊C4C2×D414yescomplemented
139C23×C22⋊C4C2×D414yescomplemented
140C23×C22⋊C4C2×D414yescomplemented
141C23×C22⋊C4C2×D412yescomplemented
142C23×C22⋊C4C2×D412yescomplemented
143C23×C22⋊C4C2×D412yescomplemented
144C23×C22⋊C4C2412yes—
145#2403C2×D4132yescomplemented
146#2308C2×D4116yescomplemented
147C2×#386C2×D4116yescomplemented
148C2×#386C2×D418yescomplemented
149C2×#386C2×D4116yescomplemented
150C2×#379C2×D418yes—
151C2×#379C2×D418yes—
152C2×#379C2×D414yes—
153C2×#379C2×D418yescomplemented
154C2×#379C2×D412yescomplemented
155C2×#379C2×D412yescomplemented
156C2×#379C2414yes—
157C2×#379C2414yes—
158C2×#379C2414yes—
159C2×#379C2412yes—
160C2×#379C2412yes—
161#1781C2×D4116yescomplemented
162C26#12012yes—
163C26#12014yes—
164C26#12012yes—
165C26#12012yes—
166C26#12012yes—
167C26#12014yes—
168C26#12018yescomplemented
169C26#12114yes—
170C26C22×D411yescharacteristic
171C26C22×D412yes—
172C26C22×D412yes—
173C26C22×D412yes—
174C26C22×D411yescharacteristic
175C26C22×D411yescharacteristic
176C26C22×D412yes—
177C26C22×D414yes—
178C26C22×D411yescharacteristic
179C26C22×D412yes—
180C26C22×D412yes—
181C26C22×D412yes—
182C26C22×D414yes—
183C26C22×D411yescharacteristic
184C26C22×D411yescharacteristic
185C26C22×D412yes—
186C26C22×D412yescomplemented
187C26C2×#4112yes—
188C26H(5,𝔽2)12yes—
189C26H(5,𝔽2)12yes—
190C26H(5,𝔽2)14yes—
191C26C2511yescharacteristic
192C23×D4#12018yes—
193C23×D4#12014yes—
194C23×D4#12014yes—
195C23×D4#12018yes—
196C23×D4#12018yes—
197C23×D4#12018yes—
198C23×D4#12018yes—
199C23×D4#120116yescomplemented
200C23×D4#12118yes—
201C23×D4#12118yes—
202C23×D4#12118yes—
203C23×D4#12118yes—
204C23×D4#12118yes—
205C23×D4#12718yes—
206C23×D4#12718yes—
207C23×D4C22×D418yes—
208C23×D4C22×D414yes—
209C23×D4C22×D414yes—
210C23×D4C22×D418yes—
211C23×D4C22×D414yes—
212C23×D4C22×D414yes—
213C23×D4C22×D418yes—
214C23×D4C22×D418yes—
215C23×D4C22×D414yes—
216C23×D4C2×#4114yes—
217C24×C4#12012yes—
218C24×C4#12012yes—
219C24×C4#12012yes—
220C24×C4#12014yes—
221C24×C4#12014yes—
222C24×C4#12012yes—
223C24×C4#12018yescomplemented
224C24×C4#12114yes—
225C24×C4C22×D412yes—
226C24×C4C22×D414yes—
227C24×C4C22×D412yes—
228C24×C4C22×D412yes—
229C24×C4C22×D412yescomplemented
230C24×C4C22×D414yescomplemented
231C22×C22⋊C4#120116yescomplemented
232C22×C22⋊C4C22×D418yes—
233C22×C22⋊C4C22×D414yescomplemented
234C22×C22⋊C4C22×D414yescomplemented
235#379#120116yescomplemented
236#379C22×D418yes—
237#379C22×D418yes—
238#379C22×D418yescomplemented
239C25#39218yes—
240C25#39218yes—
241C25B(4,𝔽2)116yes—
242C25B(4,𝔽2)18yescomplemented
243C25C2×#12012yes—
244C25C2×#12012yes—
245C25C2×#12011yescharacteristic
246C25C2×#12011yescharacteristic
247C25C2×#12011yescharacteristic
248C25C2×#12011yescharacteristic
249C25C2×#12014yes—
250C25C2×#12014yes—
251C25C2×#12012yes—
252C25C2×#12014yes—
253C25C2×#12014yes—
254C25C2×#12012yes—
255C25C2×#12011yescharacteristic
256C25C2×#12011yescharacteristic
257C25C2×#12014yes—
258C25C2×#12014yes—
259C25C2×#12012yes—
260C25C2×#12012yes—
261C25C2×#12014yes—
262C25C2×#12014yes—
263C25C2×#12012yes—
264C25C2×#12018yescomplemented
265C25C2×#12112yes—
266C25C2×#12114yes—
267C25C2×#12112yes—
268C25C2×#12112yes—
269C25C2×#12112yes—
270C25C2×#12112yes—
271C25C2×#12112yes—
272C25C2×#12712yes—
273C25C2×#12712yes—
274C25#53411yescharacteristic
275C25#53412yes—
276C25#53411yescharacteristic
277C25#53412yes—
278C25#53414yes—
279C25#53414yes—
280C25#53412yes—
281C25#53514yes—
282C25#53514yes—
283C25D4212yes—
284C25D4212yes—
285C25D4212yes—
286C25D4212yes—
287C25D4214yes—
288C25D4214yes—
289C25D4212yes—
290C25D4214yes—
291C25D4212yes—
292C25D4212yes—
293C25D4212yes—
294C25D4214yes—
295C25D4214yes—
296C25D4214yes—
297C25D4214yes—
298C25D4214yes—
299C25#54614yes—
300C25#54614yes—
301C25#54614yes—
302C25#54614yes—
303C25C23×D411yescharacteristic
304C25C23×D412yes—
305C25C23×D411yescharacteristic
306C25C23×D411yescharacteristic
307C25C23×D411yescharacteristic
308C25C23×D411yescharacteristic
309C25C23×D412yes—
310C25C23×D411yescharacteristic
311C25C22×#4111yescharacteristic
312C25C2×H(5,𝔽2)12yes—
313C25C2×H(5,𝔽2)12yes—
314C22×D4#392116yes—
315C22×D4#392116yes—
316C22×D4C2×#12018yes—
317C22×D4C2×#12118yes—
318C22×D4C2×#12118yes—
319C22×D4C2×#12718yes—
320C23×C4#39218yes—
321C23×C4#39218yes—
322C23×C4C2×#12014yes—
323C23×C4C2×#12014yes—
324C23×C4C2×#12014yes—
325C23×C4C2×#12014yes—
326C23×C4C2×#12018yescomplemented
327C23×C4C2×#12114yes—
328C2×C22⋊C4C2×#120116yescomplemented
329C24#2021132yescomplemented
330C24C2×#39212yes—
331C24C2×#39212yes—
332C24C2×#39212yes—
333C24C2×#39212yes—
334C24C2×#39218yes—
335C24#240311yescharacteristic
336C24#240312yes—
337C24#240314yes—
338C24#240312yes—
339C24#240312yes—
340C24#240312yes—
341C24#240311yescharacteristic
342C24#240311yescharacteristic
343C24#240314yes—
344C24#240314yes—
345C24#240312yes—
346C24#240312yes—
347C24#240314yes—
348C24#240311yescharacteristic
349C24#240814yes—
350C24#240814yes—
351C24#240814yes—
352C24#240814yes—
353C24#241012yes—
354C24#241014yes—
355C24#241012yes—
356C24#241012yes—
357C24#241012yes—
358C24#241014yes—
359C24#241014yes—
360C24#241014yes—
361C24#241014yes—
362C24#241014yes—
363C24#243314yes—
364C24#243314yes—
365C24#243314yes—
366C24#243314yes—
367C24#261312yes—
368C24#261312yes—
369C24#261314yes—
370C24#284612yes—
371C24#284612yes—
372C24#284614yes—
373C24C2×B(4,𝔽2)14yes—
374C24C2×B(4,𝔽2)18yes—
375C24C2×B(4,𝔽2)116yes—
376C24C2×B(4,𝔽2)18yescomplemented
377C24#302614yes—
378C24C22×#12011yescharacteristic
379C24C22×#12011yescharacteristic
380C24C22×#12011yescharacteristic
381C24C22×#12011yescharacteristic
382C24C22×#12012yes—
383C24C22×#12012yes—
384C24C22×#12011yescharacteristic
385C24C22×#12111yescharacteristic
386C24C22×#12111yescharacteristic
387C24C22×#12111yescharacteristic
388C24C22×#12711yescharacteristic
389C24C2×#53411yescharacteristic
390C24C2×#53411yescharacteristic
391C24C2×#53412yes—
392C24C2×#53512yes—
393C24C2×D4212yes—
394C24C2×D4212yes—
395C24C2×D4212yes—
396C24C2×D4212yes—
397C24C2×D4212yes—
398C24C2×D4212yes—
399C24C2×#54612yes—
400C24C2×#54612yes—
401C22×C4C2×#39218yes—
402C23#1471512yes—
403C23#1471514yes—
404C23#1471514yes—
405C23#1471512yes—
406C23#1471512yes—
407C23#1471512yes—
408C23C2×#202118yes—
409C23C2×#2021116yescomplemented
410C23#2389514yes—
411C23#2390018yes—
412C23#2390014yes—
413C23#2390012yes—
414C23#2390014yes—
415C23#2390012yes—
416C23#2390312yes—
417C23#2390312yes—
418C23#2440418yes—
419C23#3289818yes—
420C23C22×#39211yescharacteristic
421C23C2×#261311yescharacteristic
422C23C2×#240312yes—
423C23C2×#240311yescharacteristic
424C23C2×#240311yescharacteristic
425C23C2×#241011yescharacteristic
426C23C2×#241012yes—
427C23C2×#241011yescharacteristic
428C23C2×#240812yes—
429C23C2×#243312yes—
430C23C2×#284611yescharacteristic
431C23C22×B(4,𝔽2)12yes—
432C23C22×B(4,𝔽2)14yes—
433C23C2×#302612yes—
434C22#24522518yes—
435C22#24524618yes—
436C22#39405814yes—
437C22#39405814yes—
438C22#39406414yes—
439C22#39406614yes—
440C22#39406614yes—
441C22C2×#1471511yescharacteristic
442C22C22×#202112yes—
443C22C2×#2390012yes—
444C22C2×#2390012yes—
445C22C2×#2390011yescharacteristic
446C22C2×#2390311yescharacteristic
447C22C2×#2389512yes—
448C22C2×#2440412yes—
449C22C2×#3289812yes—
450C2C2×#24522512yes—
451C2C2×#39405811yescharacteristic
452C2#7222560918yesdirect product factor
453C2C2×#39406611yescharacteristic
454C2C2×#39406411yescharacteristic
455C2C2×#24524612yes—
subgroupG/H
center, socleC24#2403
commutator, Frattini, cosocleC26C25
LCSG ▷ C26 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C27 ◁ G
LECSG ▷ C26 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#106542C2semidirect
2. C22×#9495C2semidirect
3. C22×#10336C2semidirect
4. C2×#392125C2semidirect
5. C2×#115133C2semidirect
6. C2×#393929C2semidirect
7. C2×#111458C2semidirect
8. C2×#616960C2semidirect
9. C22×#35346C2semidirect
10. #72225609C2direct (N × H)
11. C22×#32898C2semidirect
12. C22×#8541C2semidirect
14. C23×#534C22semidirect
16. C2×#35346C22semidirect
20. C22×#2403C22semidirect
21. C22×#2403C22semidirect
23. C2×#32898C22semidirect
26. C22×#1781C22semidirect
27. C22×#1781C22semidirect
28. C22×#1781C22semidirect
29. C22×#1781C22semidirect
30. C22×#1781C22semidirect
31. #616960C22semidirect
32. C2×#14335C22semidirect
34. C2×#13867C22semidirect
35. C2×#13867C22semidirect
36. #393929C22semidirect
37. #392125C22semidirect
38. C2×#10725C22semidirect
39. C2×#10336C22semidirect
40. C2×#9495C22semidirect
41. C2×#8541C22semidirect
42. C22×#1304C22semidirect
43. C22×#1304C22semidirect
44. #115133C22semidirect
45. #111458C22semidirect
46. #106542C22semidirect
47. C28D4semidirect
48. C23×H(5,𝔽2)D4semidirect
55. C22×#534C23semidirect
56. C23×#120D4semidirect
57. C23×#120D4semidirect
58. C23×#120D4semidirect
60. C24×C22⋊C4D4semidirect
61. C24×C22⋊C4D4semidirect
62. C24×C22⋊C4D4semidirect
64. C2×#2403D4semidirect
65. C2×#2403D4semidirect
66. C2×#2403D4semidirect
70. C2×#2403C23semidirect
71. C2×#2403C23semidirect
72. C2×#2308D4semidirect
74. C22×#386D4semidirect
75. C22×#386D4semidirect
76. C22×#386D4semidirect
82. C2×#1781D4semidirect
85. C2×#1781C23semidirect
86. C2×#1781C23semidirect
87. C2×#1781C23semidirect
88. C2×#1781C23semidirect
89. C2×#1781C23semidirect
90. #14335C23semidirect
91. #13867D4semidirect
93. #13867C23semidirect
94. #13867C23semidirect
95. #10725C23semidirect
96. C2×#1304C23semidirect
97. C2×#1304C23semidirect
102. C27C2×D4semidirect
103. C27C2×D4semidirect
105. C22×H(5,𝔽2)C2×D4semidirect
122. C24×D4C2×D4semidirect
133. C25×C4C2×D4semidirect
134. C25×C4C2×D4semidirect
135. C22×#120C2×D4semidirect
136. C22×#120C2×D4semidirect
138. C23×C22⋊C4C2×D4semidirect
139. C23×C22⋊C4C2×D4semidirect
140. C23×C22⋊C4C2×D4semidirect
141. C23×C22⋊C4C2×D4semidirect
142. C23×C22⋊C4C2×D4semidirect
143. C23×C22⋊C4C2×D4semidirect
145. #2403C2×D4semidirect
146. #2308C2×D4semidirect
147. C2×#386C2×D4semidirect
148. C2×#386C2×D4semidirect
149. C2×#386C2×D4semidirect
153. C2×#379C2×D4semidirect
154. C2×#379C2×D4semidirect
155. C2×#379C2×D4semidirect
161. #1781C2×D4semidirect
168. C26#120semidirect
186. C26C22×D4semidirect
199. C23×D4#120semidirect
223. C24×C4#120semidirect
229. C24×C4C22×D4semidirect
230. C24×C4C22×D4semidirect
231. C22×C22⋊C4#120semidirect
233. C22×C22⋊C4C22×D4semidirect
234. C22×C22⋊C4C22×D4semidirect
235. #379#120semidirect
238. #379C22×D4semidirect
242. C25B(4,𝔽2)semidirect
264. C25C2×#120semidirect
326. C23×C4C2×#120semidirect
328. C2×C22⋊C4C2×#120semidirect
329. C24#2021semidirect
376. C24C2×B(4,𝔽2)semidirect
409. C23C2×#2021semidirect
452. C2#72225609direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#1065422
C22×#94952
C22×#103362
C2×#3921252
C2×#1151332
C2×#3939292
C2×#1114582
C2×#6169602
C22×#353462
#722256092
C22×#328982
C22×#85412