Finite Group Databasebeta

← C2×#3225 #60796

Subgroups — C2×#3225 #60796

Order 256.

1043 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic, complemented
2C2×#536C211yesmaximal, characteristic, complemented
3#3225C2116yesmaximal, direct product factor
4C2×#497C211yesmaximal, characteristic, complemented
5C2×#495C211yesmaximal, characteristic, complemented
6C2×#489C211yesmaximal, characteristic, complemented
7C2×#487C211yesmaximal, characteristic, complemented
8C2×#464C211yesmaximal, characteristic, complemented
9C2×#462C211yesmaximal, characteristic, complemented
10C2×#463C211yesmaximal, characteristic, complemented
11C2×#461C211yesmaximal, characteristic, complemented
12C2×#452C211yesmaximal, characteristic, complemented
13C2×#449C211yesmaximal, characteristic
14C2×#451C211yesmaximal, characteristic, complemented
15C2×#448C211yesmaximal, characteristic, complemented
16C2×#433C211yesmaximal, characteristic, complemented
17C22×#41C2211yescharacteristic
18C23×Q8C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×Q16C2211yescharacteristic
21C22×Q16C2211yescharacteristic
22C22×SD16C2211yescharacteristic
23C22×SD16C2211yescharacteristic
24C22×SD16C2211yescharacteristic
25C22×SD16C2211yescharacteristic
26C22×D8C2211yescharacteristic
27#542—24no—
28#542C2218yescomplemented
29#536—24no—
30#536C2218yescomplemented
31C2×C4⋊Q8C2211yescharacteristic
32C2×C4⋊Q8C2211yescharacteristic
33C2×C4⋊Q8C2211yescharacteristic
34C2×#126—21no—
35C2×#124—21no—
36C2×#124C2211yescharacteristic
37C2×#124C2211yescharacteristic
38C2×#124C2211yescharacteristic
39C2×#123—21no—
40C2×#123—21no—
41C2×#123—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8C2211yescharacteristic
46C2×#121C2211yescharacteristic
47C2×C4×Q8C2211yescharacteristic
48C2×C4×D4C2211yescharacteristic
49C2×#117C2211yescharacteristic
50#497C2218yescomplemented
51#495C2218yescomplemented
52#489C2218yescomplemented
53#487C2218yescomplemented
54#464C2218yescomplemented
55#463C2218yescomplemented
56#462C2218yescomplemented
57#461C2218yescomplemented
58#452C2218yescomplemented
59#451C2218yescomplemented
60#449C2218yes—
61#448C2218yescomplemented
62#433C2218yescomplemented
63C2×C4⋊C8C2211yescharacteristic
64C2×C4⋊C8C2211yescharacteristic
65C2×#103C2211yescharacteristic
66C2×#103C2211yescharacteristic
67C2×#103C2211yescharacteristic
68C2×#103C2211yescharacteristic
69C2×#103C2211yescharacteristic
70C2×#103C2211yescharacteristic
71C2×#102C2211yescharacteristic
72C2×#102C2211yescharacteristic
73C2×C22⋊C8C2211yescharacteristic
74C2×C22⋊C8C2211yescharacteristic
75C2×#97C2211yescharacteristic
76C2×C4×C8C2211yescharacteristic
77C2×#41—22no—
78C2×#41—42no—
79C2×#41—42no—
80C2×#41C2314yes—
81C22×Q8—21no—
82C22×Q8—22no—
83C22×Q8—21no—
84C22×Q8—42no—
85C22×Q8—42no—
86C22×Q8C2311yescharacteristic
87C22×Q8C2311yescharacteristic
88C22×Q8C2311yescharacteristic
89C22×Q8C2311yescharacteristic
90C22×Q8C2314yes—
91C22×D4—21no—
92C22×D4—21no—
93C22×D4C2311yescharacteristic
94C22×D4C2311yescharacteristic
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4C2311yescharacteristic
99C2×Q16—42no—
100C2×Q16—42no—
101C2×Q16—42no—
102C2×Q16C2314yes—
103C2×Q16C2314yes—
104C2×Q16C2314yes—
105C2×SD16—42no—
106C2×SD16—42no—
107C2×SD16—42no—
108C2×SD16—42no—
109C2×SD16C2314yes—
110C2×SD16C2314yes—
111C2×SD16C2314yes—
112C2×SD16C2314yes—
113C2×D8—42no—
114C2×D8C2314yes—
115C22×C8C2311yescharacteristic
116C22×C8C2311yescharacteristic
117C22×C8C2311yescharacteristic
118C22×C8C2311yescharacteristic
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8C2314yes—
123C4⋊Q8C2314yes—
124C4⋊Q8C2314yes—
125#126—28no—
126#124—28no—
127#124—22no—
128#124—22no—
129#124—22no—
130#124C2314yes—
131#124C2314yes—
132#124C2314yes—
133#123—28no—
134#123—28no—
135#123—28no—
136C22⋊Q8—22no—
137C22⋊Q8—28no—
138C22⋊Q8—28no—
139C22⋊Q8—28no—
140C22⋊Q8C2314yes—
141#121—22no—
142#121C2314yes—
143C4×Q8—22no—
144C4×Q8C2314yes—
145C4×D4—22no—
146C4×D4C2314yes—
147#117D414yes—
148#117C2314yes—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4D411yescharacteristic
157C2×C4⋊C4D411yescharacteristic
158C2×C4⋊C4C2311yescharacteristic
159C2×C4⋊C4C2311yescharacteristic
160C2×C22⋊C4—21no—
161C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
170C2×C22⋊C4D411yescharacteristic
171C2×C42—21no—
172C2×C42C2311yescharacteristic
173C2×C42C2311yescharacteristic
174C4⋊C8C2314yes—
175C4⋊C8C2314yes—
176#103C2314yes—
177#103C2314yes—
178#103C2314yes—
179#103C2314yes—
180#103C2314yes—
181#103C2314yes—
182#102C2314yes—
183#102C2314yes—
184C22⋊C8C2314yes—
185C22⋊C8C2314yes—
186#97C2314yes—
187C4×C8C2314yes—
188C24—21no—
189C24—21no—
190C24C2×D411yescharacteristic
191#41—44no—
192#41—44no—
193#41—48no—
194C2×Q8—21no—
195C2×Q8—24no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—24no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—44no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—48no—
212C2×Q8—82no—
213C2×Q8C2412yes—
214C2×Q8C2412yes—
215C2×Q8C2412yes—
216C2×Q8C2412yes—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—44no—
226C2×D4—82no—
227C2×D4C2412yes—
228C2×D4C2412yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—82no—
243C22×C4C2×D412yes—
244C22×C4C2×D411yescharacteristic
245C22×C4C2×D411yescharacteristic
246C22×C4C2×D411yescharacteristic
247C22×C4C2×D411yescharacteristic
248C22×C4C2×D411yescharacteristic
249C22×C4C2412yes—
250C22×C4C2411yescharacteristic
251Q16—44no—
252Q16—44no—
253Q16—44no—
254SD16—44no—
255SD16—44no—
256SD16—44no—
257SD16—44no—
258D8—44no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8C2412yes—
264C2×C8C2412yes—
265C2×C8C2412yes—
266C2×C8C2412yes—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4C2×D414yes—
277C4⋊C4C2×D414yes—
278C4⋊C4C2412yes—
279C4⋊C4C2412yes—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—24no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4C2×D414yes—
290C22⋊C4C2×D414yes—
291C42—24no—
292C42C2×D412yes—
293C42C2×D412yes—
294C42C2412yes—
295C42C2412yes—
296C23—22no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—42no—
302C23—42no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23C22×D412yes—
308C23C22×D411yescharacteristic
309Q8—22no—
310Q8—22no—
311Q8—22no—
312Q8—22no—
313Q8—22no—
314Q8—22no—
315Q8—22no—
316Q8—22no—
317Q8—48no—
318Q8—42no—
319Q8—42no—
320Q8—42no—
321Q8—42no—
322Q8—84no—
323D4—22no—
324D4—22no—
325D4—22no—
326D4—22no—
327D4—48no—
328D4—42no—
329D4—42no—
330D4—84no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—84no—
354C2×C4C22×D412yes—
355C2×C4C22×D412yes—
356C2×C4C22×D412yes—
357C2×C4C22×D412yes—
358C2×C4C22×D412yes—
359C2×C4C22×D411yescharacteristic
360C2×C4H(5,𝔽2)12yes—
361C2×C4H(5,𝔽2)12yes—
362C2×C4C2511yescharacteristic
363C8—22no—
364C8—22no—
365C8—22no—
366C8—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—44no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—84no—
378C22—84no—
379C22#53512yes—
380C22#57712yes—
381C22#57812yes—
382C22C23×D411yescharacteristic
383C4—22no—
384C4—22no—
385C4—22no—
386C4—22no—
387C4—22no—
388C4—42no—
389C4—42no—
390C4—42no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4#53512yes—
395C4C2×H(5,𝔽2)11yescharacteristic
396C4C2×H(5,𝔽2)11yescharacteristic
397C2—42no—
398C2—82no—
399C2—82no—
400C2#322514yesdirect product factor
401C2C2×#53511yescharacteristic
402C2C2×#57711yescharacteristic
403C2C2×#57811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC2×C4C25
prime radicalC2×#449C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#542C2semidirect
2. C2×#536C2semidirect
3. #3225C2direct (N × H)
4. C2×#497C2semidirect
5. C2×#495C2semidirect
6. C2×#489C2semidirect
7. C2×#487C2semidirect
8. C2×#464C2semidirect
9. C2×#462C2semidirect
10. C2×#463C2semidirect
11. C2×#461C2semidirect
12. C2×#452C2semidirect
14. C2×#451C2semidirect
15. C2×#448C2semidirect
16. C2×#433C2semidirect
28. #542C22semidirect
30. #536C22semidirect
50. #497C22semidirect
51. #495C22semidirect
52. #489C22semidirect
53. #487C22semidirect
54. #464C22semidirect
55. #463C22semidirect
56. #462C22semidirect
57. #461C22semidirect
58. #452C22semidirect
59. #451C22semidirect
61. #448C22semidirect
62. #433C22semidirect
400. C2#3225direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#5362
#32252
C2×#4972
C2×#4952
C2×#4892
C2×#4872
C2×#4642
C2×#4622
C2×#4632
C2×#4612
C2×#4522
C2×#4492
C2×#4512
C2×#4482
C2×#4332