Finite Group Databasebeta

← #25105

Subgroups — #25105

Order 256.

596 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×#470C211yesmaximal, characteristic, complemented
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#450C211yesmaximal, characteristic
4C2×#418C211yesmaximal, characteristic
5#2435C211yesmaximal, characteristic
6#2362C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2042C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1926C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1873C211yesmaximal, characteristic, complemented
15#1851C211yesmaximal, characteristic, complemented
16C23×D4C2211yescharacteristic
17C22×Q16C2211yescharacteristic, complemented
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C4×Q8C2211yescharacteristic, complemented
26C2×#117C2211yescharacteristic
27#470—24no—
28#464—24no—
29#450—24no—
30C2×#106C2211yescharacteristic, complemented
31C2×C4⋊C8C2211yescharacteristic, complemented
32C2×C4⋊C8C2211yescharacteristic, complemented
33#418—24no—
34C2×#103C2211yescharacteristic, complemented
35C2×#103C2211yescharacteristic, complemented
36C2×#103C2211yescharacteristic, complemented
37C2×#103C2211yescharacteristic, complemented
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45#394—21no—
46#394—21no—
47#391C2211yescharacteristic, complemented
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#389C2211yescharacteristic, complemented
51#388—21no—
52#388—21no—
53#387—21no—
54#379—21no—
55#379—21no—
56C4×C4⋊C4—21no—
57C4×C4⋊C4C2211yescharacteristic, complemented
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61#336C2211yescharacteristic, complemented
62#336C2211yescharacteristic, complemented
63C25—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—22no—
69C22×D4—21no—
70C22×D4—22no—
71C22×D4—41no—
72C22×D4—41no—
73C22×D4C2311yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×Q16—22no—
77C2×Q16—22no—
78C2×Q16—41no—
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—41no—
84C2×SD16—41no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—41no—
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#124—22no—
93#124—22no—
94#124—41no—
95#124—41no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8—22no—
99C22⋊Q8D414yes—
100C4×Q8—22no—
101C4×Q8—41no—
102#117—22no—
103#117—41no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4D411yescharacteristic
124C2×C22⋊C4D411yescharacteristic
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42C2311yescharacteristic
129C2×C42C2311yescharacteristic
130#106—22no—
131C4⋊C8—22no—
132C4⋊C8—41no—
133#103—22no—
134#103—22no—
135#103—22no—
136#103—41no—
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102D414yes—
143C22⋊C8—22no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95D411yescharacteristic
149#95D411yescharacteristic
150C24—21no—
151C24—21no—
152C24—42no—
153C24—42no—
154C24—42no—
155C24—42no—
156C24—42no—
157C24—81no—
158C24—81no—
159C24C2×D411yescharacteristic
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8C2×D412yes—
172C2×D4—28no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—42no—
186C2×D4—42no—
187C2×D4—44no—
188C2×D4—42no—
189C2×D4C2×D412yes—
190C2×D4#4112yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D412yes—
207C22×C4#4112yes—
208C22×C4C2411yescharacteristic
209Q16—44no—
210SD16—44no—
211SD16—44no—
212M16—44no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8C2×D412yes—
225C4⋊C4—24no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4C2×D412yes—
236C4⋊C4C2×D412yes—
237C22⋊C4—24no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—81no—
247C22⋊C4—81no—
248C22⋊C4—81no—
249C22⋊C4—81no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—42no—
255C42—42no—
256C42—42no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—22no—
265C23—48no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—84no—
279C23—84no—
280C23—84no—
281C23—84no—
282C23—84no—
283C23—84no—
284C23#12112yes—
285C23C22×D411yescharacteristic
286Q8—22no—
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291Q8—41no—
292Q8—41no—
293Q8—41no—
294D4—22no—
295D4—22no—
296D4—22no—
297D4—22no—
298D4—48no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4#12112yes—
340C2×C4C22×D411yescharacteristic
341C2×C4C22×D411yescharacteristic
342C2×C4C2×#4111yescharacteristic
343C2×C4C2×#4111yescharacteristic
344C2×C4H(5,𝔽2)11yescharacteristic
345C2×C4H(5,𝔽2)11yescharacteristic
346C8—22no—
347C8—41no—
348C8—41no—
349C8—41no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—42no—
363C22—816no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22C2×#12111yescharacteristic
373C22D4211yescharacteristic
374C22#54611yescharacteristic
375C22#57511yescharacteristic
376C22#57511yescharacteristic
377C22#57711yescharacteristic
378C22#57711yescharacteristic
379C4—22no—
380C4—42no—
381C4—41no—
382C4—42no—
383C4—41no—
384C4—42no—
385C4—41no—
386C4—41no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4#52511yescharacteristic
391C4#53811yescharacteristic
392C4D4211yescharacteristic
393C4#54611yescharacteristic
394C2—42no—
395C2—82no—
396C2—82no—
397C2#242411yescharacteristic
398C2#304011yescharacteristic
399C2#304311yescharacteristic
400C2#326711yescharacteristic
401C2#327511yescharacteristic
402C2#331111yescharacteristic
403C2#332311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#470C2semidirect
2. C2×#464C2semidirect
6. #2362C2semidirect
7. #2065C2semidirect
8. #2055C2semidirect
9. #2042C2semidirect
10. #2015C2semidirect
11. #1983C2semidirect
12. #1926C2semidirect
13. #1880C2semidirect
14. #1873C2semidirect
15. #1851C2semidirect
17. C22×Q16C22semidirect
25. C2×C4×Q8C22semidirect
30. C2×#106C22semidirect
31. C2×C4⋊C8C22semidirect
32. C2×C4⋊C8C22semidirect
34. C2×#103C22semidirect
35. C2×#103C22semidirect
36. C2×#103C22semidirect
37. C2×#103C22semidirect
47. #391C22semidirect
50. #389C22semidirect
57. C4×C4⋊C4C22semidirect
59. #340C22semidirect
60. #340C22semidirect
61. #336C22semidirect
62. #336C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4642
C2×#4502
C2×#4182
#24352
#23622
#20652
#20552
#20422
#20152
#19832
#19262
#18802
#18732
#18512