Finite Group Databasebeta

← #25558

Subgroups — #25558

Order 256.

551 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#556C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#470C211yesmaximal, characteristic, complemented
4#2417C211yesmaximal, characteristic
5#2058C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2001C211yesmaximal, characteristic, complemented
9#1878C211yesmaximal, characteristic, complemented
10#1873C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic
13#1863C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18#556—24no—
19#556—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#126—21no—
22C2×#126—21no—
23C2×#124—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C4×Q8—21no—
29C2×C4×Q8C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C2×#117C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34#485—24no—
35#470—24no—
36C2×#107C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#398—21no—
49#396—21no—
50#391C2211yescharacteristic
51#389C2211yescharacteristic
52#386—21no—
53#380—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4D411yescharacteristic
67C23×C4—21no—
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—41no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—41no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82#126—28no—
83#126—28no—
84#124—28no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8—41no—
92C22⋊Q8D414yes—
93C4×Q8—28no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×D4—22no—
97C4×D4—22no—
98#117—22no—
99#117—22no—
100#117—22no—
101#117D414yes—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4D411yescharacteristic
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⋊C4—21no—
126C2×C22⋊C4D411yescharacteristic
127C2×C22⋊C4D411yescharacteristic
128C2×C42—21no—
129C2×C42—21no—
130C2×C42—21no—
131C2×C42C2311yescharacteristic
132#107—22no—
133#106—22no—
134#106—22no—
135#103—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#103—41no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24C2×D411yescharacteristic
149C24C2×D411yescharacteristic
150C2×Q8—24no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8C2×D412yes—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—41no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D412yes—
193C22×C4C2×D412yes—
194C22×C4C2×D411yescharacteristic
195C22×C4C2411yescharacteristic
196Q16—44no—
197M16—44no—
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#4112yes—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—22no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—81no—
227C4⋊C4C2×D412yes—
228C4⋊C4C2×D412yes—
229C4⋊C4C2×D412yes—
230C4⋊C4#4112yes—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—81no—
240C42—24no—
241C42—24no—
242C42—24no—
243C42—21no—
244C42—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#12012yes—
259C23C22×D411yescharacteristic
260Q8—22no—
261Q8—22no—
262Q8—48no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269D4—44no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
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—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—84no—
313C2×C4#12012yes—
314C2×C4#13712yes—
315C2×C4C22×D411yescharacteristic
316C2×C4C22×D411yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4H(5,𝔽2)11yescharacteristic
320C2×C42−1+411yescharacteristic
321C8—22no—
322C8—41no—
323C8—41no—
324C8—41no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22C2×#12011yescharacteristic
340C22#54611yescharacteristic
341C22#54711yescharacteristic
342C22C2×#13711yescharacteristic
343C22#57511yescharacteristic
344C22#57711yescharacteristic
345C22#57811yescharacteristic
346C4—22no—
347C4—22no—
348C4—41no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—41no—
353C4—42no—
354C4—42no—
355C4—81no—
356C4—81no—
357C4—81no—
358C4#53211yescharacteristic
359C4#53711yescharacteristic
360C4#54611yescharacteristic
361C4#54711yescharacteristic
362C2—42no—
363C2—81no—
364C2#243011yescharacteristic
365C2#300711yescharacteristic
366C2#301311yescharacteristic
367C2#331111yescharacteristic
368C2#332811yescharacteristic
369C2#334911yescharacteristic
370C2#335611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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×#485C2semidirect
3. C2×#470C2semidirect
5. #2058C2semidirect
6. #2035C2semidirect
7. #2024C2semidirect
8. #2001C2semidirect
9. #1878C2semidirect
10. #1873C2semidirect
11. #1872C2semidirect
13. #1863C2semidirect
14. #1854C2semidirect
15. #1736C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5562
C2×#4852
C2×#4702
#24172
#20582
#20352
#20242
#20012
#18782
#18732
#18722
#18702
#18632
#18542
#17362