Finite Group Databasebeta

← #22470

Subgroups — #22470

Order 256.

601 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#529C211yesmaximal, characteristic
2C2×#435C211yesmaximal, characteristic, complemented
3C2×C8×D4C211yesmaximal, characteristic, complemented
4#1985C211yesmaximal, characteristic, complemented
5#1984C211yesmaximal, characteristic, complemented
6#1870C211yesmaximal, characteristic, complemented
7#1869C211yesmaximal, characteristic, complemented
8#1869C211yesmaximal, characteristic, complemented
9#1869C211yesmaximal, characteristic, complemented
10#1818C211yesmaximal, characteristic, complemented
11#1817C211yesmaximal, characteristic, complemented
12#1798C211yesmaximal, characteristic, complemented
13#1750C211yesmaximal, characteristic, complemented
14#1729C211yesmaximal, characteristic, complemented
15C2×#336C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#529—24no—
21#529—24no—
22C2×#126C411yescharacteristic
23C2×#126C411yescharacteristic
24C2×#125C411yescharacteristic
25C2×#124C411yescharacteristic
26C2×#123C411yescharacteristic
27C2×#123C411yescharacteristic
28C2×C22⋊Q8C411yescharacteristic
29C2×#121C411yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C2×#117C2211yescharacteristic
35C2×#117C2211yescharacteristic
36C22×C42C2211yescharacteristic
37#435—24no—
38C8×D4—24no—
39C2×C4⋊C8C2211yescharacteristic
40C2×C4⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×#97C2211yescharacteristic
50C2×C4×C8C2211yescharacteristic
51#336C2211yescharacteristic
52#336C2211yescharacteristic
53#336C2211yescharacteristic
54#336C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2×C411yescharacteristic
64C22×D4C2×C411yescharacteristic
65C22×D4C2×C411yescharacteristic
66C22×D4C2×C411yescharacteristic
67C23×C4C2×C411yescharacteristic
68C23×C4C2×C411yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—41no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—22no—
80C22×C8—22no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#126—24no—
93#126—24no—
94#125—24no—
95#124—24no—
96#123—24no—
97#123—24no—
98C22⋊Q8—24no—
99#121—24no—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4D414yes—
107C4×D4D414yes—
108#117—22no—
109#117—22no—
110#117—22no—
111#117—22no—
112C2×C4⋊C4C2×C411yescharacteristic
113C2×C4⋊C4C2×C411yescharacteristic
114C2×C4⋊C4C2×C411yescharacteristic
115C2×C4⋊C4C2×C411yescharacteristic
116C2×C4⋊C4C2×C411yescharacteristic
117C2×C4⋊C4C2×C411yescharacteristic
118C2×C4⋊C4C2×C411yescharacteristic
119C2×C4⋊C4C2×C411yescharacteristic
120C2×C4⋊C4C2×C411yescharacteristic
121C2×C4⋊C4C2×C411yescharacteristic
122C2×C22⋊C4C2×C411yescharacteristic
123C2×C22⋊C4C2×C411yescharacteristic
124C2×C22⋊C4C2×C411yescharacteristic
125C2×C22⋊C4C2×C411yescharacteristic
126C2×C22⋊C4C2×C411yescharacteristic
127C2×C22⋊C4C2×C411yescharacteristic
128C2×C22⋊C4C2×C411yescharacteristic
129C2×C22⋊C4C2×C411yescharacteristic
130C2×C22⋊C4C2×C411yescharacteristic
131C2×C22⋊C4C2×C411yescharacteristic
132C2×C42—22no—
133C2×C42—22no—
134C2×C42C2×C411yescharacteristic
135C2×C42C2×C411yescharacteristic
136C2×C42C2311yescharacteristic
137C2×C42C2311yescharacteristic
138C2×C42C2311yescharacteristic
139C2×C42C2311yescharacteristic
140C4⋊C8—22no—
141C4⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150#97—22no—
151C4×C8—22no—
152C24C22×C411yescharacteristic
153C24C22×C411yescharacteristic
154C24C22×C411yescharacteristic
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—41no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—22no—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—41no—
166C2×D4C22⋊C414yes—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4C22×C411yescharacteristic
188C22×C4C22×C411yescharacteristic
189C22×C4C22×C411yescharacteristic
190C22×C4C22×C411yescharacteristic
191C22×C4C22×C411yescharacteristic
192C22×C4C22×C411yescharacteristic
193C22×C4C22×C411yescharacteristic
194C22×C4C22×C411yescharacteristic
195C22×C4C22×C411yescharacteristic
196C22×C4C22×C411yescharacteristic
197C22×C4C22×C411yescharacteristic
198C22×C4C2×D412yes—
199C22×C4C2×D412yes—
200C22×C4C2×D412yes—
201C22×C4C2×D412yes—
202C22×C4C2411yescharacteristic
203M16—44no—
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—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—44no—
227C2×C8—44no—
228C2×C8—44no—
229C2×C8#4112yes—
230C2×C8#4112yes—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4C22⋊C414yes—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4C22⋊C414yes—
250C22⋊C4C22⋊C414yes—
251C42—22no—
252C42—22no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42C2×D412yes—
260C42C2×D412yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23C2×C22⋊C412yes—
279C23C2×C22⋊C412yes—
280C23#13112yes—
281C23#13112yes—
282C23C23×C411yescharacteristic
283Q8—44no—
284D4—28no—
285D4—44no—
286D4—44no—
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—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
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—44no—
324C2×C4—44no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4C2×C22⋊C412yes—
330C2×C4C2×C22⋊C412yes—
331C2×C4C2×C22⋊C412yes—
332C2×C4C2×C22⋊C412yes—
333C2×C4#13112yes—
334C2×C4C22×D411yescharacteristic
335C2×C4C22×D411yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4C2×#4111yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C42−1+411yescharacteristic
340C8—22no—
341C8—22no—
342C8—41no—
343C8—41no—
344C8—41no—
345C8—41no—
346C8—41no—
347C8—41no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22#40814yes—
369C22C22×C22⋊C411yescharacteristic
370C22C4×#4111yescharacteristic
371C22#52011yescharacteristic
372C22C2×#13111yescharacteristic
373C22C2×#13111yescharacteristic
374C22C2×#13111yescharacteristic
375C22#56811yescharacteristic
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4#54611yescharacteristic
388C4#54611yescharacteristic
389C4#54711yescharacteristic
390C4#54711yescharacteristic
391C2—24no—
392C2—42no—
393C2—81no—
394C2#233911yescharacteristic
395C2C2×#40811yescharacteristic
396C2#288011yescharacteristic
397C2#296611yescharacteristic
398C2#296811yescharacteristic
399C2#299211yescharacteristic
400C2#299411yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radicalC2×#121C4
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
2. C2×#435C2semidirect
3. C2×C8×D4C2semidirect
4. #1985C2semidirect
5. #1984C2semidirect
6. #1870C2semidirect
7. #1869C2semidirect
8. #1869C2semidirect
9. #1869C2semidirect
10. #1818C2semidirect
11. #1817C2semidirect
12. #1798C2semidirect
13. #1750C2semidirect
14. #1729C2semidirect
15. C2×#336C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4352
C2×C8×D42
#19852
#19842
#18702
#18692
#18182
#18172
#17982
#17502
#17292
C2×#3362