Finite Group Databasebeta

← C2×#3198 #60742

Subgroups — C2×#3198 #60742

Order 256.

1189 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#537C211yesmaximal, characteristic, complemented
2C2×#535C211yesmaximal, characteristic, complemented
3#3198C2116yesmaximal, direct product factor
4C2×#481C211yesmaximal, characteristic, complemented
5C2×#485C211yesmaximal, characteristic, complemented
6C2×#482C211yesmaximal, characteristic, complemented
7C2×#480C211yesmaximal, characteristic, complemented
8C2×#469C211yesmaximal, characteristic, complemented
9C2×#468C211yesmaximal, characteristic, complemented
10C2×#466C211yesmaximal, characteristic, complemented
11C2×#465C211yesmaximal, characteristic, complemented
12C2×#449C211yesmaximal, characteristic, complemented
13C2×#449C211yesmaximal, characteristic, complemented
14C2×#450C211yesmaximal, characteristic, complemented
15C2×#447C211yesmaximal, characteristic, complemented
16C2×#408C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C22×#41C2211yescharacteristic, complemented
19C23×D4C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22C22×D8C2211yescharacteristic
23C22×D8C2211yescharacteristic
24C22×M16C2211yescharacteristic, complemented
25C23×C8C2211yescharacteristic, complemented
26#537—24no—
27#537C2218yescomplemented
28#535—24no—
29#535C2218yescomplemented
30C2×#127—21no—
31C2×#124—21no—
32C2×C22⋊Q8—21no—
33C2×C22⋊Q8C2211yescharacteristic
34C2×C22⋊Q8C2211yescharacteristic, complemented
35C2×#121—21no—
36C2×#121—21no—
37C2×#121—21no—
38C2×#121C2211yescharacteristic
39C2×#121C2211yescharacteristic
40C2×#121C2211yescharacteristic
41C2×#121C2211yescharacteristic
42C2×#121C2211yescharacteristic
43C2×#121C2211yescharacteristic, complemented
44C2×#120—21no—
45C2×#120—21no—
46C2×#117C2211yescharacteristic
47C22×C4⋊C4C2211yescharacteristic, complemented
48#485C2218yescomplemented
49#482C2218yescomplemented
50#481C2218yescomplemented
51#480C2218yescomplemented
52#469C2218yescomplemented
53#468C2218yescomplemented
54#466C2218yescomplemented
55#465C2218yescomplemented
56#450C2218yescomplemented
57#449C2218yescomplemented
58#449C2218yescomplemented
59#447C2218yescomplemented
60C2×#107C2211yescharacteristic, complemented
61C2×#107C2211yescharacteristic, complemented
62C2×#106C2211yescharacteristic, complemented
63C2×#106C2211yescharacteristic, complemented
64C2×#103C2211yescharacteristic, complemented
65C2×#103C2211yescharacteristic, complemented
66C2×#102C2211yescharacteristic
67C2×#102C2211yescharacteristic
68C2×#102C2211yescharacteristic
69C2×#102C2211yescharacteristic
70C2×#102C2211yescharacteristic, complemented
71C2×#102C2211yescharacteristic, complemented
72#408C2218yescomplemented
73C2×C22⋊C8C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×C22⋊C8C2211yescharacteristic, complemented
76C2×C22⋊C8C2211yescharacteristic, complemented
77C25—21no—
78C2×#41—22no—
79C2×#41—42no—
80C2×#41—42no—
81C2×#41—42no—
82C2×#41—42no—
83C2×#41D414yes—
84C2×#41C2314yes—
85C2×#41C2314yescomplemented
86C22×Q8D411yescharacteristic
87C22×Q8C2311yescharacteristic
88C22×D4—21no—
89C22×D4—21no—
90C22×D4—21no—
91C22×D4—21no—
92C22×D4—21no—
93C22×D4—21no—
94C22×D4—21no—
95C22×D4—21no—
96C22×D4—22no—
97C22×D4—42no—
98C22×D4—42no—
99C22×D4D411yescharacteristic
100C22×D4D411yescharacteristic
101C22×D4D411yescharacteristic
102C22×D4C2311yescharacteristic
103C22×D4C2311yescharacteristic
104C22×D4C2311yescharacteristic
105C22×D4C2314yes—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C2311yescharacteristic
109C23×C4C2311yescharacteristic
110C23×C4C2311yescharacteristic
111C2×SD16—42no—
112C2×SD16—42no—
113C2×SD16C2314yes—
114C2×SD16C2314yes—
115C2×D8—42no—
116C2×D8—42no—
117C2×D8C2314yes—
118C2×D8C2314yes—
119C2×M16—42no—
120C2×M16C2314yescomplemented
121C22×C8—42no—
122C22×C8C2311yescharacteristic
123C22×C8C2311yescharacteristic
124C22×C8C2311yescharacteristic
125C22×C8C2311yescharacteristic
126C22×C8C2314yescomplemented
127#127—28no—
128#124—28no—
129C22⋊Q8—28no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8C2314yes—
133C22⋊Q8C2314yescomplemented
134#121—28no—
135#121—28no—
136#121—28no—
137#121—22no—
138#121—22no—
139#121—22no—
140#121—22no—
141#121—22no—
142#121—22no—
143#121C2314yes—
144#121C2314yes—
145#121C2314yes—
146#121C2314yes—
147#121C2314yes—
148#121C2314yescomplemented
149#120—28no—
150#120—28no—
151#117—22no—
152#117C2314yes—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4C2311yescharacteristic
158C2×C4⋊C4C2311yescharacteristic
159C2×C4⋊C4C2311yescharacteristic
160C2×C4⋊C4C2311yescharacteristic
161C2×C4⋊C4C2314yescomplemented
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—21no—
171C2×C42—21no—
172#107C2314yescomplemented
173#107C2314yescomplemented
174#106C2314yescomplemented
175#106C2314yescomplemented
176#103C2314yescomplemented
177#103C2314yescomplemented
178#102C2314yes—
179#102C2314yes—
180#102C2314yes—
181#102C2314yes—
182#102C2314yescomplemented
183#102C2314yescomplemented
184C22⋊C8C2314yes—
185C22⋊C8C2314yes—
186C22⋊C8C2314yescomplemented
187C22⋊C8C2314yescomplemented
188C24—21no—
189C24—21no—
190C24—21no—
191C24—24no—
192C24—44no—
193C24—82no—
194C24—82no—
195C24C2×D411yescharacteristic
196C24C2×D411yescharacteristic
197C24C2×D411yescharacteristic
198#41—48no—
199#41—44no—
200#41—44no—
201#41—44no—
202#41—44no—
203#41—48no—
204C2×Q8—21no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—42no—
208C2×Q8C2×D414yes—
209C2×Q8C2412yes—
210C2×D4—24no—
211C2×D4—24no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—44no—
228C2×D4—44no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—44no—
233C2×D4—48no—
234C2×D4—44no—
235C2×D4—82no—
236C2×D4—82no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4—82no—
240C2×D4—82no—
241C2×D4C2×D414yes—
242C2×D4C2×D414yes—
243C2×D4C2×D414yes—
244C2×D4C2412yes—
245C2×D4C2412yes—
246C2×D4C2412yes—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—44no—
261C22×C4—82no—
262C22×C4C2×D412yes—
263C22×C4C2×D412yes—
264C22×C4C2×D412yes—
265C22×C4C2×D411yescharacteristic
266C22×C4C2×D411yescharacteristic
267C22×C4C2×D411yescharacteristic
268C22×C4C2412yes—
269C22×C4C2412yes—
270C22×C4C2412yes—
271C22×C4C2411yescharacteristic
272SD16—44no—
273SD16—44no—
274D8—44no—
275D8—44no—
276M16—44no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—44no—
282C2×C8C2412yes—
283C2×C8C2412yes—
284C2×C8C2412yes—
285C2×C8C2412yes—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4C2412yes—
294C4⋊C4C2412yes—
295C4⋊C4C2412yes—
296C4⋊C4C2412yes—
297C22⋊C4—24no—
298C22⋊C4—24no—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C42—24no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—48no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—84no—
330C23—84no—
331C23—816no—
332C23C22×D412yes—
333C23C22×D412yes—
334C23C22×D412yes—
335C23C22×D411yescharacteristic
336Q8—22no—
337Q8—22no—
338Q8—44no—
339Q8—42no—
340D4—22no—
341D4—22no—
342D4—22no—
343D4—22no—
344D4—22no—
345D4—22no—
346D4—44no—
347D4—44no—
348D4—44no—
349D4—48no—
350D4—48no—
351D4—42no—
352D4—42no—
353D4—42no—
354D4—84no—
355D4—84no—
356D4—84no—
357D4—84no—
358D4—84no—
359D4—84no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—48no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—84no—
384C2×C4C22×D412yes—
385C2×C4C22×D412yes—
386C2×C4C22×D412yes—
387C2×C4C22×D411yescharacteristic
388C2×C4H(5,𝔽2)12yes—
389C2×C4H(5,𝔽2)12yes—
390C2×C4C2511yescharacteristic
391C8—22no—
392C8—22no—
393C8—22no—
394C8—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—42no—
416C22—816no—
417C22—84no—
418C22—84no—
419C22—84no—
420C22#53412yes—
421C22#57612yes—
422C22#57712yes—
423C22C23×D411yescharacteristic
424C4—22no—
425C4—22no—
426C4—22no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4#53412yes—
433C4C2×H(5,𝔽2)11yescharacteristic
434C4C2×H(5,𝔽2)11yescharacteristic
435C2—42no—
436C2—42no—
437C2—42no—
438C2—82no—
439C2—82no—
440C2—82no—
441C2#319814yesdirect product factor
442C2C2×#53411yescharacteristic
443C2C2×#57611yescharacteristic
444C2C2×#57711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#537C2semidirect
2. C2×#535C2semidirect
3. #3198C2direct (N × H)
4. C2×#481C2semidirect
5. C2×#485C2semidirect
6. C2×#482C2semidirect
7. C2×#480C2semidirect
8. C2×#469C2semidirect
9. C2×#468C2semidirect
10. C2×#466C2semidirect
11. C2×#465C2semidirect
12. C2×#449C2semidirect
13. C2×#449C2semidirect
14. C2×#450C2semidirect
15. C2×#447C2semidirect
16. C2×#408C2semidirect
18. C22×#41C22semidirect
24. C22×M16C22semidirect
25. C23×C8C22semidirect
27. #537C22semidirect
29. #535C22semidirect
34. C2×C22⋊Q8C22semidirect
43. C2×#121C22semidirect
47. C22×C4⋊C4C22semidirect
48. #485C22semidirect
49. #482C22semidirect
50. #481C22semidirect
51. #480C22semidirect
52. #469C22semidirect
53. #468C22semidirect
54. #466C22semidirect
55. #465C22semidirect
56. #450C22semidirect
57. #449C22semidirect
58. #449C22semidirect
59. #447C22semidirect
60. C2×#107C22semidirect
61. C2×#107C22semidirect
62. C2×#106C22semidirect
63. C2×#106C22semidirect
64. C2×#103C22semidirect
65. C2×#103C22semidirect
70. C2×#102C22semidirect
71. C2×#102C22semidirect
72. #408C22semidirect
75. C2×C22⋊C8C22semidirect
76. C2×C22⋊C8C22semidirect
85. C2×#41C23semidirect
120. C2×M16C23semidirect
126. C22×C8C23semidirect
133. C22⋊Q8C23semidirect
148. #121C23semidirect
161. C2×C4⋊C4C23semidirect
172. #107C23semidirect
173. #107C23semidirect
174. #106C23semidirect
175. #106C23semidirect
176. #103C23semidirect
177. #103C23semidirect
182. #102C23semidirect
183. #102C23semidirect
186. C22⋊C8C23semidirect
187. C22⋊C8C23semidirect
441. C2#3198direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5372
C2×#5352
#31982
C2×#4812
C2×#4852
C2×#4822
C2×#4802
C2×#4692
C2×#4682
C2×#4662
C2×#4652
C2×#4492
C2×#4502
C2×#4472
C2×#4082