Finite Group Databasebeta

← C2×#72231712 #72310702

Subgroups — C2×#72231712 #72310702

Order 2048.

At least 6641 conjugacy classes of subgroups; only the 1160 normal subgroups are shown.

The subgroup lattice has 418 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486342C211yesmaximal, characteristic, complemented
2C2×#500005C211yesmaximal, characteristic, complemented
3C2×#130847C211yesmaximal, characteristic, complemented
4C2×#10584532C211yesmaximal, characteristic, complemented
5C2×#511937C211yesmaximal, characteristic, complemented
6C2×#129596C211yesmaximal, characteristic, complemented
7C2×#130234C211yesmaximal, characteristic, complemented
8C2×#486345C212yesmaximal, complemented
9C2×#130296C212yesmaximal, complemented
10C2×#130852C212yesmaximal, complemented
11C2×#500001C212yesmaximal, complemented
12#72231712C2116yesmaximal, direct product factor
13#10584532C2218yescomplemented
14C2×#62334C2212yes—
15C2×#62328C2211yescharacteristic
16C2×#61288C2211yescharacteristic
17C2×#61293C2212yes—
18C2×#60320C2211yescharacteristic
19C2×#33533C2211yescharacteristic, complemented
20C2×#33531C2212yescomplemented
21C2×#33388C2211yescharacteristic, complemented
22C2×#24288C2212yescomplemented
23C2×#24287C2211yescharacteristic, complemented
24#511937C2218yescomplemented
25#500005C2218yescomplemented
26#500001C22116yescomplemented
27C2×#13079C2212yescomplemented
28C2×#13076C2211yescharacteristic, complemented
29C2×#12988C2212yescomplemented
30C2×#12982C2211yescharacteristic, complemented
31C2×#12874C2211yescharacteristic, complemented
32C2×#12874C2212yescomplemented
33C2×#12810C2212yescomplemented
34C2×#12809C2211yescharacteristic, complemented
35#486345C22116yescomplemented
36#486342C2218yescomplemented
37C2×#11872C2212yescomplemented
38C2×#11870C2212yescomplemented
39C2×#11870C2212yescomplemented
40C2×#11486C2212yescomplemented
41C2×#11485C2211yescharacteristic, complemented
42#130852C22116yescomplemented
43#130847C2218yescomplemented
44#130296C22116yescomplemented
45#130234C2218yescomplemented
46#129596C2218yescomplemented
47C2×#3584C2311yescharacteristic
48C2×#3483D411yescharacteristic
49C2×#3407C2311yescharacteristic
50#62334D418yes—
51#62334C2318yes—
52#62328D414yes—
53#62328C2314yes—
54#61293D418yes—
55#61293C2318yes—
56#61288D414yes—
57#61288C2314yes—
58C2×#3256D412yes—
59C2×#3253D412yes—
60C2×#3253D411yescharacteristic
61C2×#3253D411yescharacteristic
62C2×#3204D412yes—
63C2×#3199D412yes—
64C2×#3199D412yes—
65C2×#3162D412yes—
66C22×#498D411yescharacteristic
67C2×#3060D411yescharacteristic
68C2×#3060C2311yescharacteristic
69C2×#3058D412yes—
70C2×#3058C2312yes—
71#60320D414yes—
72#60320C2314yes—
73C2×#2980D412yes—
74C2×#2979D411yescharacteristic
75C22×#442D412yes—
76C2×C4×#135D411yescharacteristic
77C2×#2932C2312yes—
78C2×#2931C2311yescharacteristic
79C2×#2912D411yescharacteristic
80C2×#2912D412yes—
81C2×#2912C2311yescharacteristic
82#33533C2314yescomplemented
83#33531C2318yescomplemented
84#33388C2314yescomplemented
85C2×#2115C2311yescharacteristic
86C2×#2115C2312yes—
87#24288C2318yescomplemented
88#24287C2314yescomplemented
89C2×#1813C2311yescharacteristic
90C2×#1813C2312yes—
91#13079C2318yescomplemented
92#13076C2314yescomplemented
93#12988C2318yescomplemented
94#12982C2314yescomplemented
95#12874C2318yescomplemented
96#12874C2314yescomplemented
97#12810C2318yescomplemented
98#12809C2314yescomplemented
99#11872C2318yescomplemented
100#11870C2318yescomplemented
101#11870C2318yescomplemented
102#11486C2318yescomplemented
103#11485C2314yescomplemented
104#3584C2412yes—
105C22×#135C2×D411yescharacteristic
106C23×D8C2×D412yes—
107C22×#131C2×D411yescharacteristic
108C22×#131C2×D412yes—
109C22×#131C2411yescharacteristic
110#3483D814yes—
111#3483C2×D414yes—
112C2×Q82D811yescharacteristic
113C2×#555D812yes—
114C2×#555D811yescharacteristic
115C2×#532C2×D412yes—
116C2×#532C2×D411yescharacteristic
117#3407C2412yes—
118#3256C2×D418yes—
119#3253C2×D414yes—
120#3253C2×D414yes—
121#3253C2×D418yes—
122#3204C2×D418yes—
123#3199C2×D418yes—
124#3199C2×D418yes—
125#3162C2×D418yes—
126C2×#498C2×D414yes—
127#3060C2×D414yes—
128#3060C2×D414yes—
129#3060C2×D412yes—
130#3060C2412yes—
131#3058C2×D414yes—
132#3058C2×D414yes—
133#3058C2×D418yes—
134#3058C2×D414yes—
135#3058C2414yes—
136C2×#470C2×D412yes—
137C2×#469C2×D412yes—
138C2×#469C2×D412yes—
139C2×#469C2×D412yes—
140C2×#469C2×D412yes—
141C2×#469C2×D412yes—
142#2980C2×D418yes—
143#2979C2×D414yes—
144C2×#442C2×D418yes—
145C4×#135C2×D414yes—
146#2932C2×D414yes—
147#2932C2×D414yes—
148#2932C2×D414yes—
149#2932C2414yes—
150#2931C2×D412yes—
151#2931C2×D412yes—
152#2931C2×D412yes—
153#2931C2412yes—
154#2912C2×D418yes—
155#2912C2×D414yes—
156#2912C2×D414yes—
157#2912C2×D412yes—
158#2912C2412yes—
159C2×#427C2×D412yes—
160C2×#427C2×D412yes—
161C2×#427C2×D411yescharacteristic
162C22×#106C2×D411yescharacteristic
163C22×#106C2×D411yescharacteristic
164C22×#106C2×D412yes—
165C2×#423C2×D412yes—
166C2×#423C2×D411yescharacteristic
167C2×#408C2×D412yes—
168C2×#408C2×D411yescharacteristic
169C2×#408C2×D411yescharacteristic
170C2×#408C2×D412yes—
171C22×#97C2×D412yes—
172C22×C4×C8C2×D411yescharacteristic
173#2115C2412yes—
174#2115C2414yes—
175#1813C2414yes—
176#1813C2412yes—
177C22×#41#12012yes—
178C22×#41#12011yescharacteristic
179C22×#41C22×D411yescharacteristic
180C2×#135#12012yes—
181C2×#135C22×D412yes—
182C22×Q16#12011yescharacteristic
183C22×D8#12011yescharacteristic
184C22×D8#12012yes—
185C22×D8#12012yes—
186C22×D8#12014yes—
187C22×D8C22×D414yes—
188C2×#131#12012yes—
189C2×#131#12014yes—
190C2×#131C22×D412yes—
191C2×#131C22×D414yes—
192C2×#131C22×D411yescharacteristic
193C2×#131C22×D412yes—
194C2×#131C22×D411yescharacteristic
195C2×#131C22×D412yes—
196C2×#131C22×D411yescharacteristic
197C2×#131C2511yescharacteristic
198C22×M16#12012yes—
199C22×M16#12011yescharacteristic
200C22×M16C22×D412yes—
201C22×M16C22×D411yescharacteristic
202C23×C8#12011yescharacteristic
203C23×C8#12012yes—
204C23×C8C22×D411yescharacteristic
205C23×C8C22×D412yes—
206Q82C2×D814yes—
207#555C2×D814yes—
208#555C2×D818yes—
209#532#12012yes—
210#532#12014yes—
211#532C2×D814yes—
212#532C2×D812yes—
213#532AGL(1,ℤ/8)14yes—
214#532AGL(1,ℤ/8)12yes—
215#532C22×D412yes—
216#532C22×D414yes—
217C2×C4⋊Q8C2×D811yescharacteristic
218C2×C4⋊Q8C2×D812yes—
219C2×#127AGL(1,ℤ/8)11yescharacteristic
220C2×#127AGL(1,ℤ/8)12yes—
221C2×#124C2×D812yes—
222C2×#124C2×D811yescharacteristic
223C2×#124AGL(1,ℤ/8)12yes—
224C2×#124AGL(1,ℤ/8)11yescharacteristic
225C22×C42#12011yescharacteristic
226C22×C42#12012yes—
227#470#12014yes—
228#470C22×D414yes—
229#469#12014yes—
230#469#12014yes—
231#469#12014yes—
232#469#12014yes—
233#469#12014yes—
234#469C22×D414yes—
235#469C22×D414yes—
236#469C22×D414yes—
237#469C22×D414yes—
238#469C22×D414yes—
239#427#12014yes—
240#427#12014yes—
241#427#12012yes—
242#427C22×D414yes—
243#427C22×D414yes—
244#427C22×D412yes—
245C2×#106#12012yes—
246C2×#106#12012yes—
247C2×#106#12014yes—
248C2×#106#12012yes—
249C2×#106#12012yes—
250C2×#106#12012yes—
251C2×#106#12012yes—
252C2×#106#12011yescharacteristic
253C2×#106#12011yescharacteristic
254C2×#106C22×D414yes—
255C2×#106C22×D412yes—
256C2×#106C22×D412yes—
257#423#12014yes—
258#423#12012yes—
259#423C22×D414yes—
260#423C22×D412yes—
261#408#12014yes—
262#408#12014yes—
263#408#12012yes—
264#408#12012yes—
265#408C22×D414yes—
266#408C22×D412yes—
267#408C22×D414yes—
268#408C22×D412yes—
269C2×#97#12014yes—
270C2×#97C22×D414yes—
271C2×C4×C8#12012yes—
272C2×C4×C8C22×D412yes—
273C2×#41#44714yes—
274C2×#41#44712yes—
275C2×#41C2×#12014yes—
276C2×#41C2×#12012yes—
277C2×#41C2×#12012yes—
278C2×#41C2×#12011yescharacteristic
279C2×#41C22×D811yescharacteristic
280C2×#41C2×AGL(1,ℤ/8)11yescharacteristic
281C2×#41C2×AGL(1,ℤ/8)12yes—
282C2×#41C23×D411yescharacteristic
283C22×Q8#44711yescharacteristic
284C22×Q8#44712yes—
285C22×Q8C22×D811yescharacteristic
286C22×D4#44711yescharacteristic
287C22×D4#44712yes—
288C22×D4C2×AGL(1,ℤ/8)11yescharacteristic
289C22×D4C2×AGL(1,ℤ/8)12yes—
290C23×C4C2×#12011yescharacteristic
291C23×C4C2×#12012yes—
292C2×Q16C2×#12012yes—
293C2×D8C2×#12014yes—
294C2×D8C2×#12014yes—
295C2×D8C2×#12012yes—
296C2×M16C2×#12012yes—
297C2×M16C2×#12014yes—
298C2×M16C2×#12011yescharacteristic
299C2×M16C2×#12012yes—
300C2×M16C2×#12012yes—
301C2×M16C2×#12012yes—
302C2×M16C2×#12012yes—
303C2×M16C23×D411yescharacteristic
304C2×M16C23×D412yes—
305C22×C8C2×#12012yes—
306C22×C8C2×#12011yescharacteristic
307C22×C8C2×#12012yes—
308C22×C8C2×#12012yes—
309C22×C8C2×#12011yescharacteristic
310C22×C8C2×#12011yescharacteristic
311C22×C8C2×#12012yes—
312C22×C8C2×#12011yescharacteristic
313C22×C8C2×#12012yes—
314C22×C8C2×#12014yes—
315C22×C8C2×#12011yescharacteristic
316C22×C8C23×D412yes—
317C22×C8C23×D411yescharacteristic
318C4⋊Q8#44712yes—
319C4⋊Q8#44714yes—
320C4⋊Q8C22×D812yes—
321C4⋊Q8C22×D814yes—
322#127#44712yes—
323#127#44714yes—
324#127C2×AGL(1,ℤ/8)12yes—
325#127C2×AGL(1,ℤ/8)14yes—
326#124#44714yes—
327#124#44712yes—
328#124#44714yes—
329#124#44712yes—
330#124C22×D814yes—
331#124C22×D812yes—
332#124C2×AGL(1,ℤ/8)14yes—
333#124C2×AGL(1,ℤ/8)12yes—
334C2×C42#44711yescharacteristic
335C2×C42#44712yes—
336C2×C42#44712yes—
337C2×C42#44711yescharacteristic
338C2×C42#44714yes—
339C2×C42#44712yes—
340C2×C42C2×#12014yes—
341C2×C42C2×#12012yes—
342#106C2×#12012yes—
343#106C2×#12014yes—
344#106C2×#12014yes—
345#106C2×#12014yes—
346#106C2×#12012yes—
347#106C2×#12014yes—
348C24#300312yes—
349C24#300311yescharacteristic
350C2×Q8C2×#44712yes—
351C2×Q8C2×#44714yes—
352C2×Q8C2×#44711yescharacteristic
353C2×Q8C2×#44712yes—
354C2×Q8C23×D811yescharacteristic
355C2×D4C2×#44712yes—
356C2×D4C2×#44714yes—
357C2×D4C2×#44711yescharacteristic
358C2×D4C2×#44712yes—
359C2×D4#300312yes—
360C2×D4#300312yes—
361C2×D4#300312yes—
362C2×D4C22×AGL(1,ℤ/8)11yescharacteristic
363C2×D4C22×AGL(1,ℤ/8)12yes—
364C22×C4#284611yescharacteristic
365C22×C4#284612yes—
366C22×C4#284611yescharacteristic
367C22×C4C2×#44712yes—
368C22×C4C2×#44711yescharacteristic
369C22×C4C2×#44711yescharacteristic
370C22×C4C2×#44712yes—
371C22×C4#300311yescharacteristic
372C22×C4#300312yes—
373C22×C4C22×#12011yescharacteristic
374C22×C4C22×#12012yes—
375C2×C8#284611yescharacteristic
376C2×C8#284612yes—
377C2×C8#284611yescharacteristic
378C2×C8#302511yescharacteristic
379C2×C8#302511yescharacteristic
380C2×C8#302711yescharacteristic
381C2×C8#302711yescharacteristic
382C2×C8C22×#12011yescharacteristic
383C2×C8C22×#12012yes—
384C2×C8C22×#12011yescharacteristic
385C42C2×#44712yes—
386C42C2×#44712yes—
387C42C2×#44714yes—
388C42C2×#44714yes—
389C23#3287811yescharacteristic
390C23#3287812yes—
391C23#3287811yescharacteristic
392C23C2×#300312yes—
393C23C2×#300311yescharacteristic
394C2×C4#3287812yes—
395C2×C4#3287811yescharacteristic
396C2×C4#3287811yescharacteristic
397C2×C4#3289511yescharacteristic
398C2×C4#3289511yescharacteristic
399C2×C4#3289911yescharacteristic
400C2×C4#3289911yescharacteristic
401C2×C4C2×#284611yescharacteristic
402C2×C4C22×#44712yes—
403C2×C4C22×#44711yescharacteristic
404C8#3289911yescharacteristic
405C8#3289911yescharacteristic
406C8C2×#302511yescharacteristic
407C8C2×#302711yescharacteristic
408C22#34652211yescharacteristic
409C22#34652211yescharacteristic
410C22C2×#3287811yescharacteristic
411C4#34652211yescharacteristic
412C4#34652211yescharacteristic
413C4C2×#3289511yescharacteristic
414C4C2×#3289911yescharacteristic
415C2#7223171212yesdirect product factor
416C2C2×#34652211yescharacteristic
subgroupG/H
center, socleC22#346522
commutator, Frattini, cosocleC2×#131C25
LCSG ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×#131 ◁ G
LECSG ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#486342C2semidirect
2. C2×#500005C2semidirect
3. C2×#130847C2semidirect
4. C2×#10584532C2semidirect
5. C2×#511937C2semidirect
6. C2×#129596C2semidirect
7. C2×#130234C2semidirect
8. C2×#486345C2semidirect
9. C2×#130296C2semidirect
10. C2×#130852C2semidirect
11. C2×#500001C2semidirect
12. #72231712C2direct (N × H)
13. #10584532C22semidirect
19. C2×#33533C22semidirect
20. C2×#33531C22semidirect
21. C2×#33388C22semidirect
22. C2×#24288C22semidirect
23. C2×#24287C22semidirect
24. #511937C22semidirect
25. #500005C22semidirect
26. #500001C22semidirect
27. C2×#13079C22semidirect
28. C2×#13076C22semidirect
29. C2×#12988C22semidirect
30. C2×#12982C22semidirect
31. C2×#12874C22semidirect
32. C2×#12874C22semidirect
33. C2×#12810C22semidirect
34. C2×#12809C22semidirect
35. #486345C22semidirect
36. #486342C22semidirect
37. C2×#11872C22semidirect
38. C2×#11870C22semidirect
39. C2×#11870C22semidirect
40. C2×#11486C22semidirect
41. C2×#11485C22semidirect
42. #130852C22semidirect
43. #130847C22semidirect
44. #130296C22semidirect
45. #130234C22semidirect
46. #129596C22semidirect
82. #33533C23semidirect
83. #33531C23semidirect
84. #33388C23semidirect
87. #24288C23semidirect
88. #24287C23semidirect
91. #13079C23semidirect
92. #13076C23semidirect
93. #12988C23semidirect
94. #12982C23semidirect
95. #12874C23semidirect
96. #12874C23semidirect
97. #12810C23semidirect
98. #12809C23semidirect
99. #11872C23semidirect
100. #11870C23semidirect
101. #11870C23semidirect
102. #11486C23semidirect
103. #11485C23semidirect
415. C2#72231712direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#4863422
C2×#5000052
C2×#1308472
C2×#105845322
C2×#5119372
C2×#1295962
C2×#1302342
C2×#4863452
C2×#1302962
C2×#1308522
C2×#5000012
#722317122