Finite Group Databasebeta

← #25547

Subgroups — #25547

Order 256.

670 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic, complemented
3C2×#468C211yesmaximal, characteristic, complemented
4#2415C211yesmaximal, characteristic, complemented
5#2057C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#1877C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic, complemented
13#1862C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×#41—21no—
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19#547—24no—
20#547—24no—
21C2×C4⋊Q8C2211yescharacteristic
22C2×#123—21no—
23C2×#123—21no—
24C2×C22⋊Q8—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34#484—24no—
35#468—24no—
36C2×#107C2211yescharacteristic
37C2×#107C2211yescharacteristic
38C2×#107C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#397—21no—
49#394—21no—
50#391C2211yescharacteristic
51#389—21no—
52#389C2211yescharacteristic
53#388—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—28no—
64C2×#41—44no—
65C2×#41—44no—
66C22×Q8—21no—
67C22×Q8C2311yescharacteristic
68C22×D4—21no—
69C22×D4—21no—
70C22×D4D411yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—41no—
80C2×M16—22no—
81C2×M16—22no—
82C2×M16—41no—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C4⋊Q8—22no—
88C4⋊Q8—22no—
89#123—28no—
90#123—28no—
91C22⋊Q8—28no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—22no—
95C22⋊Q8—41no—
96#121—22no—
97#121—22no—
98#121—22no—
99#121D414yes—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103C4×D4—22no—
104#117—22no—
105#117—41no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4D411yescharacteristic
116C2×C4⋊C4D414yes—
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4D411yescharacteristic
131C2×C22⋊C4D411yescharacteristic
132C2×C42—21no—
133C2×C42C2311yescharacteristic
134#107—22no—
135#107—22no—
136#107—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#102—22no—
141#102—22no—
142#102—41no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—41no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150C24—21no—
151C24C2×D411yescharacteristic
152C24C2×D411yescharacteristic
153#41—432no—
154C2×Q8—24no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—42no—
162C2×Q8—42no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—24no—
166C2×D4—24no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4C2×D412yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—24no—
183C22×C4—24no—
184C22×C4—24no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D412yes—
208C22×C4C2×D412yes—
209C22×C4C2×D411yescharacteristic
210C22×C4C2411yescharacteristic
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×C8#4112yescomplemented
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—24no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4C2×D412yes—
241C4⋊C4C2×D412yes—
242C4⋊C4C2×D412yes—
243C4⋊C4#4112yescomplemented
244C22⋊C4—24no—
245C22⋊C4—24no—
246C22⋊C4—22no—
247C22⋊C4—24no—
248C22⋊C4—22no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—21no—
254C42—21no—
255C42—42no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—22no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23#12012yes—
276C23AGL(1,ℤ/8)12yes—
277C23C22×D411yescharacteristic
278Q8—48no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282Q8—41no—
283D4—28no—
284D4—22no—
285D4—22no—
286D4—48no—
287D4—48no—
288D4—41no—
289D4—41no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—48no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
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#12012yes—
333C2×C4AGL(1,ℤ/8)12yes—
334C2×C4C22×D411yescharacteristic
335C2×C4C22×D411yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4C2×#4111yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C42−1+411yescharacteristic
340C8—22no—
341C8—41no—
342C8—41no—
343C8—41no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—42no—
359C22—42no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22C2×#12011yescharacteristic
366C22#54611yescharacteristic
367C22#54711yescharacteristic
368C22C2×AGL(1,ℤ/8)11yescharacteristic
369C22C2×AGL(1,ℤ/8)11yescharacteristic
370C22#57811yescharacteristic
371C22#57811yescharacteristic
372C4—22no—
373C4—22no—
374C4—42no—
375C4—41no—
376C4—42no—
377C4—42no—
378C4—42no—
379C4—41no—
380C4—81no—
381C4—81no—
382C4—81no—
383C4#53211yescharacteristic
384C4#53711yescharacteristic
385C4#54611yescharacteristic
386C4#54711yescharacteristic
387C2—42no—
388C2—42no—
389C2—82no—
390C2#243011yescharacteristic
391C2#300311yescharacteristic
392C2#301111yescharacteristic
393C2#330811yescharacteristic
394C2#333011yescharacteristic
395C2#334311yescharacteristic
396C2#335311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#484C2semidirect
3. C2×#468C2semidirect
4. #2415C2semidirect
5. #2057C2semidirect
6. #2033C2semidirect
7. #2024C2semidirect
8. #2000C2semidirect
9. #1877C2semidirect
10. #1875C2semidirect
11. #1872C2semidirect
12. #1870C2semidirect
13. #1862C2semidirect
14. #1852C2semidirect
15. #1736C2semidirect
224. C2×C8#41semidirect
243. C4⋊C4#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4842
C2×#4682
#24152
#20572
#20332
#20242
#20002
#18772
#18752
#18722
#18702
#18622
#18522
#17362