Finite Group Databasebeta

← #22495

Subgroups — #22495

Order 256.

695 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547C211yesmaximal, characteristic
2C2×#435C211yesmaximal, characteristic, complemented
3#2352C211yesmaximal, characteristic, complemented
4#1993C211yesmaximal, characteristic, complemented
5#1987C211yesmaximal, characteristic, complemented
6#1895C211yesmaximal, characteristic, complemented
7#1893C211yesmaximal, characteristic, complemented
8#1875C211yesmaximal, characteristic, complemented
9#1874C211yesmaximal, characteristic, complemented
10#1824C211yesmaximal, characteristic, complemented
11#1823C211yesmaximal, characteristic, complemented
12#1804C211yesmaximal, characteristic, complemented
13#1762C211yesmaximal, characteristic, complemented
14#1736C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C22×#41—21no—
17C22×M16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19#547—24no—
20#547C418yescomplemented
21C2×C4⋊Q8C2211yescharacteristic
22C2×#123—21no—
23C2×#123—21no—
24C2×C22⋊Q8—21no—
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33#435—24no—
34C2×C4⋊C8C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×#97C2211yescharacteristic
42#388—21no—
43#387—21no—
44#386—21no—
45#382C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C2×#41—28no—
63C2×#41—44no—
64C2×#41—44no—
65C22×Q8—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4D411yescharacteristic
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
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
86C4⋊Q8—22no—
87C4⋊Q8C2×C414yescomplemented
88#123—28no—
89#123—28no—
90C22⋊Q8—28no—
91#121—22no—
92#121—22no—
93#121C2×C414yes—
94#121C2×C414yescomplemented
95C4×D4—22no—
96C4×D4C2×C414yes—
97C4×D4C2×C414yescomplemented
98C4×D4D414yes—
99#117—22no—
100#117—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C2×C414yes—
110C2×C4⋊C4C2×C414yescomplemented
111C2×C4⋊C4D411yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4D411yescharacteristic
126C2×C22⋊C4D411yescharacteristic
127C2×C42—21no—
128C2×C42—21no—
129C2×C42C2311yescharacteristic
130C4⋊C8—22no—
131#102—22no—
132#102—22no—
133#102—41no—
134#102—41no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137#97—22no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143C24—21no—
144C24C2×D411yescharacteristic
145C24C2×D411yescharacteristic
146#41—432no—
147C2×Q8—24no—
148C2×Q8—42no—
149C2×Q8—42no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—24no—
153C2×D4—24no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—42no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4C22⋊C414yes—
164C2×D4C22×C412yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—24no—
174C22×C4—24no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—24no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—81no—
192C22×C4C22×C412yes—
193C22×C4C22×C412yes—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D412yes—
199C22×C4C2×D412yes—
200C22×C4C2411yescharacteristic
201M16—44no—
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—44no—
214C2×C8#4112yescomplemented
215C4⋊C4—24no—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—42no—
229C4⋊C4C22⋊C414yes—
230C4⋊C4C22×C412yes—
231C4⋊C4C22×C412yes—
232C4⋊C4C22×C412yes—
233C4⋊C4#4112yescomplemented
234C22⋊C4—24no—
235C22⋊C4—24no—
236C22⋊C4—24no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—81no—
240C22⋊C4C22⋊C414yes—
241C22⋊C4C22⋊C414yes—
242C42—42no—
243C42—81no—
244C42C22×C412yes—
245C42C2×D412yes—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—22no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23C2×C22⋊C412yes—
264C23C2×C22⋊C412yes—
265C23AGL(1,ℤ/8)12yes—
266C23AGL(1,ℤ/8)12yes—
267C23C22×D411yescharacteristic
268Q8—48no—
269D4—28no—
270D4—22no—
271D4—22no—
272D4—48no—
273D4—48no—
274D4—41no—
275D4—41no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—48no—
297C2×C4—48no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—48no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—84no—
317C2×C4C2×C22⋊C412yes—
318C2×C4C2×C22⋊C412yes—
319C2×C4C2×C22⋊C412yes—
320C2×C4C2×C22⋊C412yes—
321C2×C4C23×C411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C42−1+411yescharacteristic
327C8—22no—
328C8—41no—
329C8—41no—
330C8—41no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—42no—
345C22—42no—
346C22—42no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22#41814yes—
352C22C22×C22⋊C411yescharacteristic
353C22#54611yescharacteristic
354C22#54711yescharacteristic
355C22C2×AGL(1,ℤ/8)11yescharacteristic
356C22C2×AGL(1,ℤ/8)11yescharacteristic
357C22#57711yescharacteristic
358C22#57811yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—42no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4C4×#4111yescharacteristic
371C4#52011yescharacteristic
372C4#54611yescharacteristic
373C4#54711yescharacteristic
374C2—24no—
375C2—42no—
376C2—82no—
377C2#233911yescharacteristic
378C2C2×#41811yescharacteristic
379C2#289811yescharacteristic
380C2#330611yescharacteristic
381C2#330811yescharacteristic
382C2#334311yescharacteristic
383C2#334511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
LCSG ▷ C2×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×#435C2semidirect
3. #2352C2semidirect
4. #1993C2semidirect
5. #1987C2semidirect
6. #1895C2semidirect
7. #1893C2semidirect
8. #1875C2semidirect
9. #1874C2semidirect
10. #1824C2semidirect
11. #1823C2semidirect
12. #1804C2semidirect
13. #1762C2semidirect
14. #1736C2semidirect
15. C2×#340C2semidirect
20. #547C4semidirect
87. C4⋊Q8C2×C4semidirect
94. #121C2×C4semidirect
97. C4×D4C2×C4semidirect
110. C2×C4⋊C4C2×C4semidirect
214. C2×C8#41semidirect
233. 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×#4352
#23522
#19932
#19872
#18952
#18932
#18752
#18742
#18242
#18232
#18042
#17622
#17362
C2×#3402