Finite Group Databasebeta

← #32961

Subgroups — #32961

Order 256.

660 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#41C211yesmaximal, characteristic
2C2×#479C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic, complemented
4#2851C211yesmaximal, characteristic, complemented
5#1865C211yesmaximal, characteristic, complemented
6#1864C211yesmaximal, characteristic, complemented
7#1863C211yesmaximal, characteristic, complemented
8#1862C211yesmaximal, characteristic, complemented
9#1854C211yesmaximal, characteristic, complemented
10#1854C211yesmaximal, characteristic, complemented
11#1805C211yesmaximal, characteristic, complemented
12#1804C211yesmaximal, characteristic, complemented
13#1797C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic
15#1788C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C2×#125C2211yescharacteristic
18C4×#41—24no—
19C4×#41—24no—
20C2×C4×Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×D4—21no—
23C2×C4×D4—21no—
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×#117—21no—
27C2×#117C2211yescharacteristic
28C22×C42—21no—
29C22×C42C2211yescharacteristic
30#479—24no—
31#478—24no—
32C2×#107C2211yescharacteristic
33C2×#107C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×C4⋊C8C2211yescharacteristic
37C2×C4⋊C8C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#387—21no—
51#387—21no—
52#385—21no—
53#385—21no—
54#382C2211yescharacteristic
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41Q814yes—
69C22×Q8C2311yescharacteristic
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4—21no—
75C23×C4D411yescharacteristic
76C23×C4D411yescharacteristic
77C23×C4C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#125—22no—
83#125—22no—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8—22no—
87C4×Q8D414yes—
88C4×D4—28no—
89C4×D4—28no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4—22no—
93C4×D4D414yes—
94#117—28no—
95#117—22no—
96#117—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C42—28no—
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—21no—
118C2×C42—22no—
119C2×C42—22no—
120C2×C42D411yescharacteristic
121C2×C42D411yescharacteristic
122C2×C42C2311yescharacteristic
123C2×C42C2311yescharacteristic
124#107—22no—
125#107—22no—
126#106—22no—
127#106—22no—
128C4⋊C8—22no—
129C4⋊C8—22no—
130#103—22no—
131#103—22no—
132#103—22no—
133#103—22no—
134#102—22no—
135#102—22no—
136#102—22no—
137#102—22no—
138C22⋊C8—22no—
139C22⋊C8—22no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146C24—21no—
147C24C2×D411yescharacteristic
148#41—28no—
149#41—28no—
150#41—28no—
151#41—28no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8C2×D412yes—
156C2×Q8C2×Q812yes—
157C2×Q8#4112yes—
158C2×D4—24no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4C2×D412yes—
165C2×D4C2×Q812yes—
166C2×D4#4112yes—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—21no—
170C22×C4—24no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—24no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—24no—
181C22×C4—22no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×Q812yes—
199C22×C4#4112yes—
200C22×C4C2411yescharacteristic
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—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C4⋊C4—24no—
214C4⋊C4—24no—
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—21no—
224C4⋊C4—21no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4C2×D412yes—
230C4⋊C4C2×D412yes—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C42—24no—
237C42—24no—
238C42—22no—
239C42—24no—
240C42—22no—
241C42—21no—
242C42—21no—
243C42C2×D412yes—
244C42C2×D412yes—
245C23—21no—
246C23—22no—
247C23—22no—
248C23—21no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23C22⋊Q812yes—
259C23C22×D411yescharacteristic
260Q8—22no—
261Q8—22no—
262Q8—22no—
263Q8C22⋊Q814yes—
264D4—216no—
265D4—22no—
266D4—22no—
267D4—22no—
268D4C22⋊Q814yes—
269C2×C4—216no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—44no—
291C2×C4—48no—
292C2×C4—44no—
293C2×C4—48no—
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#12012yes—
311C2×C4#12012yes—
312C2×C4C22⋊Q812yes—
313C2×C4#13512yes—
314C2×C4C22×D411yescharacteristic
315C2×C4C22×D411yescharacteristic
316C2×C4C22×Q811yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C8—41no—
321C8—41no—
322C8—41no—
323C8—41no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—24no—
328C22—42no—
329C22—42no—
330C22—41no—
331C22—42no—
332C22—41no—
333C22—42no—
334C22—41no—
335C22—42no—
336C22—41no—
337C22—42no—
338C22C2×#12011yescharacteristic
339C22C2×C22⋊Q811yescharacteristic
340C22#52511yescharacteristic
341C22C2×#13511yescharacteristic
342C22#57511yescharacteristic
343C22#57511yescharacteristic
344C22#57511yescharacteristic
345C4—22no—
346C4—24no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—42no—
351C4—41no—
352C4—42no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4C2×C22⋊Q811yescharacteristic
358C4C2×C22⋊Q811yescharacteristic
359C4#52511yescharacteristic
360C4#52511yescharacteristic
361C2—42no—
362C2—44no—
363C2#284811yescharacteristic
364C2#300211yescharacteristic
365C2#300511yescharacteristic
366C2#307611yescharacteristic
367C2#308011yescharacteristic
368C2#310211yescharacteristic
369C2#311111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#479C2semidirect
3. C2×#478C2semidirect
4. #2851C2semidirect
5. #1865C2semidirect
6. #1864C2semidirect
7. #1863C2semidirect
8. #1862C2semidirect
9. #1854C2semidirect
10. #1854C2semidirect
11. #1805C2semidirect
12. #1804C2semidirect
13. #1797C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#412
C2×#4792
C2×#4782
#28512
#18652
#18642
#18632
#18622
#18542
#18052
#18042
#17972
#17882