Finite Group Databasebeta

← #22556

Subgroups — #22556

Order 256.

509 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#431C211yesmaximal, characteristic, complemented
2C2×#408C211yesmaximal, characteristic
3#2297C211yesmaximal, characteristic
4#1924C211yesmaximal, characteristic, complemented
5#1923C211yesmaximal, characteristic, complemented
6#1923C211yesmaximal, characteristic, complemented
7#1923C211yesmaximal, characteristic, complemented
8#1851C211yesmaximal, characteristic, complemented
9#1850C211yesmaximal, characteristic, complemented
10#1817C211yesmaximal, characteristic, complemented
11#1798C211yesmaximal, characteristic, complemented
12#1780C211yesmaximal, characteristic
13#1753C211yesmaximal, characteristic, complemented
14C8×C22⋊C4C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×#117C2211yescharacteristic
20C22×C42C2211yescharacteristic
21#431—24no—
22C2×C4⋊C8C2211yescharacteristic
23C2×C4⋊C8C2211yescharacteristic
24C2×C4⋊C8C2211yescharacteristic
25C2×C4⋊C8C2211yescharacteristic
26#408—24no—
27C2×C22⋊C8C2211yescharacteristic
28C2×C22⋊C8C2211yescharacteristic
29C2×C22⋊C8C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×#97C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#391C411yescharacteristic
42#391C411yescharacteristic
43#390C411yescharacteristic
44#390C411yescharacteristic
45#386C411yescharacteristic
46#386C411yescharacteristic
47#386C411yescharacteristic
48#386C411yescharacteristic
49C4×C22⋊C4C2211yescharacteristic
50C4×C22⋊C4C2211yescharacteristic
51C4×C22⋊C4C2211yescharacteristic
52C4×C22⋊C4C2211yescharacteristic
53#336C2211yescharacteristic
54#336C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8C2×C411yescharacteristic
68C22×D4C2×C411yescharacteristic
69C22×D4C2×C411yescharacteristic
70C22×D4C2×C411yescharacteristic
71C23×C4C2×C411yescharacteristic
72C23×C4C2×C411yescharacteristic
73C23×C4C2311yescharacteristic
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—41no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—41no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#117—41no—
91#117—41no—
92C2×C4⋊C4C2×C411yescharacteristic
93C2×C4⋊C4C2×C411yescharacteristic
94C2×C22⋊C4C2×C411yescharacteristic
95C2×C22⋊C4C2×C411yescharacteristic
96C2×C22⋊C4C2×C411yescharacteristic
97C2×C22⋊C4C2×C411yescharacteristic
98C2×C22⋊C4C2×C411yescharacteristic
99C2×C22⋊C4C2×C411yescharacteristic
100C2×C22⋊C4C2×C411yescharacteristic
101C2×C22⋊C4C2×C411yescharacteristic
102C2×C22⋊C4C2×C411yescharacteristic
103C2×C22⋊C4C2×C411yescharacteristic
104C2×C42—22no—
105C2×C42—22no—
106C2×C42C2×C411yescharacteristic
107C2×C42C2×C411yescharacteristic
108C2×C42C2311yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112C4⋊C8—22no—
113C4⋊C8—22no—
114C4⋊C8—41no—
115C4⋊C8—41no—
116C22⋊C8—22no—
117C22⋊C8—22no—
118C22⋊C8—22no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—22no—
122C22⋊C8—22no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125C22⋊C8D414yes—
126#97—41no—
127C4×C8—22no—
128C4×C8—22no—
129C4×C8—41no—
130#95C2×C411yescharacteristic
131#95C2×C411yescharacteristic
132#95C2×C411yescharacteristic
133#95C2×C411yescharacteristic
134#95C2×C411yescharacteristic
135#95C2×C411yescharacteristic
136#95C2×C411yescharacteristic
137#95C2×C411yescharacteristic
138C24C22×C411yescharacteristic
139C24C22×C411yescharacteristic
140C24C22×C411yescharacteristic
141#41—44no—
142#41—44no—
143#41—44no—
144#41—44no—
145C2×Q8—22no—
146C2×Q8—22no—
147C2×Q8—41no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—41no—
177C22×C4C22×C411yescharacteristic
178C22×C4C22×C411yescharacteristic
179C22×C4C22×C411yescharacteristic
180C22×C4C22×C411yescharacteristic
181C22×C4C22×C411yescharacteristic
182C22×C4C22×C411yescharacteristic
183C22×C4C22×C411yescharacteristic
184C22×C4C22×C411yescharacteristic
185C22×C4C22×C411yescharacteristic
186C22×C4C22×C411yescharacteristic
187C22×C4C22×C411yescharacteristic
188C22×C4C2×D412yes—
189C22×C4#4112yes—
190C22×C4#4112yes—
191C22×C4#4112yes—
192C22×C4C2411yescharacteristic
193M16—44no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—44no—
217C2×C8C2×D412yes—
218C2×C8C2×D412yes—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C22⋊C4—22no—
222C22⋊C4—22no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—22no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—41no—
230C22⋊C4—41no—
231C42—22no—
232C42—22no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23#11712yes—
261C23C4×D412yes—
262C23C23×C411yescharacteristic
263Q8—44no—
264D4—44no—
265D4—44no—
266D4—44no—
267C2×C4—28no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—44no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4#11712yes—
310C2×C4C4×D412yes—
311C2×C4#13112yes—
312C2×C4#13112yes—
313C2×C4#13112yes—
314C2×C4C22×D411yescharacteristic
315C2×C4C2×#4111yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4H(5,𝔽2)11yescharacteristic
319C2×C4H(5,𝔽2)11yescharacteristic
320C8—22no—
321C8—22no—
322C8—41no—
323C8—41no—
324C8—41no—
325C8—41no—
326C8—41no—
327C8—41no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22C2×#11711yescharacteristic
350C22C2×C4×D411yescharacteristic
351C22#51811yescharacteristic
352C22C2×#13111yescharacteristic
353C22C2×#13111yescharacteristic
354C22C2×#13111yescharacteristic
355C22#56811yescharacteristic
356C4—22no—
357C4—22no—
358C4—22no—
359C4—22no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4#54611yescharacteristic
368C4#54611yescharacteristic
369C4#55111yescharacteristic
370C4#55111yescharacteristic
371C2—42no—
372C2—42no—
373C2—81no—
374C2#234211yescharacteristic
375C2#292211yescharacteristic
376C2#292511yescharacteristic
377C2#292911yescharacteristic
378C2#293311yescharacteristic
379C2#297911yescharacteristic
380C2#298111yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radicalC22×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#431C2semidirect
4. #1924C2semidirect
5. #1923C2semidirect
6. #1923C2semidirect
7. #1923C2semidirect
8. #1851C2semidirect
9. #1850C2semidirect
10. #1817C2semidirect
11. #1798C2semidirect
13. #1753C2semidirect
14. C8×C22⋊C4C2semidirect
15. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4312
C2×#4082
#22972
#19242
#19232
#18512
#18502
#18172
#17982
#17802
#17532
C8×C22⋊C42
#17502