Finite Group Databasebeta

← #22676

Subgroups — #22676

Order 256.

543 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#442C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic
3C2×C4×D8C211yesmaximal, characteristic
4#2349C211yesmaximal, characteristic, complemented
5D4×C4⋊C4C211yesmaximal, characteristic, complemented
6#1944C211yesmaximal, characteristic, complemented
7#1941C211yesmaximal, characteristic, complemented
8#1864C211yesmaximal, characteristic, complemented
9#1864C211yesmaximal, characteristic, complemented
10#1862C211yesmaximal, characteristic, complemented
11#1862C211yesmaximal, characteristic, complemented
12#1847C211yesmaximal, characteristic, complemented
13#1804C211yesmaximal, characteristic, complemented
14#1802C211yesmaximal, characteristic, complemented
15#1771C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×C4×D4C2211yescharacteristic
18C2×C4×D4C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C22×C4⋊C4—21no—
24#442—24no—
25#442C418yescomplemented
26C4×D8—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×#106C2211yescharacteristic
30C2×#106C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×#97C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#389C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#389C2211yescharacteristic
46#385—21no—
47#385—21no—
48#385—21no—
49#385—21no—
50#384C2211yescharacteristic
51#382C2211yescharacteristic
52#380—21no—
53#380—21no—
54C4×C4⋊C4C2211yescharacteristic
55C2×#95—21no—
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C22×D4C2311yescharacteristic
63C22×D4C2311yescharacteristic
64C23×C4—21no—
65C23×C4—21no—
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C2×D8C2×C414yes—
71C2×D8D414yes—
72C2×D8Q814yes—
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C4×D4—22no—
78C4×D4—22no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83C4×D4—41no—
84C4×D4—41no—
85C4×D4—41no—
86C4×D4—41no—
87C4×D4C2×C414yescomplemented
88C4×D4C2×C414yescomplemented
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—44no—
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4D411yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C42C2311yescharacteristic
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113#107—22no—
114#107—41no—
115#106—22no—
116#106—41no—
117#106C2×C414yescomplemented
118C4⋊C8—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102C2×C414yescomplemented
124#102C2×C414yescomplemented
125#97—22no—
126#97C2×C414yescomplemented
127C4×C8—22no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95D411yescharacteristic
135#95D411yescharacteristic
136C24—21no—
137C24—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4C22×C412yes—
145C2×D4C22×C412yes—
146C2×D4C2×D412yes—
147C2×D4C2×D412yes—
148C2×D4C2×Q812yes—
149C2×D4C2×Q812yes—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—24no—
153C22×C4—24no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—42no—
159C22×C4—42no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—42no—
163C22×C4—42no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—42no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4—81no—
172C22×C4—81no—
173C22×C4—81no—
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2×D411yescharacteristic
177C22×C4C2×D411yescharacteristic
178C22×C4C2×D411yescharacteristic
179C22×C4C2×D411yescharacteristic
180C22×C4C2411yescharacteristic
181D8C4⋊C4116yescomplemented
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8C22×C412yes—
191C2×C8C22×C412yes—
192C2×C8C2×D412yes—
193C2×C8C2×Q812yes—
194C4⋊C4—22no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—41no—
206C4⋊C4—42no—
207C4⋊C4—42no—
208C4⋊C4—42no—
209C4⋊C4—81no—
210C4⋊C4—81no—
211C4⋊C4—81no—
212C4⋊C4C22×C412yes—
213C4⋊C4C22×C412yes—
214C22⋊C4—24no—
215C22⋊C4—24no—
216C22⋊C4—42no—
217C22⋊C4—42no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42C22×C412yes—
225C42#4112yes—
226C23—22no—
227C23—22no—
228C23—22no—
229C23—22no—
230C23—22no—
231C23—22no—
232C23—41no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23C22×D411yescharacteristic
241D4—22no—
242D4—22no—
243D4—22no—
244D4—22no—
245D4—22no—
246D4—22no—
247D4C2×C4⋊C414yes—
248D4C2×C4⋊C414yescomplemented
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—48no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—84no—
284C2×C4—84no—
285C2×C4—84no—
286C2×C4—84no—
287C2×C4—84no—
288C2×C4C4×D412yes—
289C2×C4C4×D412yes—
290C2×C4AGL(1,ℤ/8)12yes—
291C2×C4C23×C411yescharacteristic
292C2×C4C22×D411yescharacteristic
293C2×C4C22×Q811yescharacteristic
294C2×C4C2×#4111yescharacteristic
295C2×C4H(5,𝔽2)11yescharacteristic
296C2×C42−1+411yescharacteristic
297C8—22no—
298C8—41no—
299C8—41no—
300C8C2×C4⋊C414yescomplemented
301C22—24no—
302C22—24no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—42no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22C2×C4×D411yescharacteristic
316C22D4211yescharacteristic
317C22D4×Q811yescharacteristic
318C22C2×AGL(1,ℤ/8)11yescharacteristic
319C22#57511yescharacteristic
320C22#57611yescharacteristic
321C22#57711yescharacteristic
322C4—22no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—42no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4C22×C4⋊C411yescharacteristic
335C4#52011yescharacteristic
336C4#54711yescharacteristic
337C4#55411yescharacteristic
338C2—44no—
339C2—44no—
340C2D4×C4⋊C411yescharacteristic
341C2C4×AGL(1,ℤ/8)11yescharacteristic
342C2#294611yescharacteristic
343C2#326411yescharacteristic
344C2#326711yescharacteristic
345C2#338411yescharacteristic
346C2#338911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #2349C2semidirect
5. D4×C4⋊C4C2semidirect
6. #1944C2semidirect
7. #1941C2semidirect
8. #1864C2semidirect
9. #1864C2semidirect
10. #1862C2semidirect
11. #1862C2semidirect
12. #1847C2semidirect
13. #1804C2semidirect
14. #1802C2semidirect
15. #1771C2semidirect
25. #442C4semidirect
87. C4×D4C2×C4semidirect
88. C4×D4C2×C4semidirect
117. #106C2×C4semidirect
123. #102C2×C4semidirect
124. #102C2×C4semidirect
126. #97C2×C4semidirect
181. D8C4⋊C4semidirect
248. D4C2×C4⋊C4semidirect
300. C8C2×C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4422
C2×C4×D82
#23492
D4×C4⋊C42
#19442
#19412
#18642
#18622
#18472
#18042
#18022
#17712