Finite Group Databasebeta

← #32968

Subgroups — #32968

Order 256.

637 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#520C211yesmaximal, characteristic
2C2×#479C211yesmaximal, characteristic, complemented
3C2×#478C211yesmaximal, characteristic, complemented
4#2629C211yesmaximal, characteristic, complemented
5#1865C211yesmaximal, characteristic, complemented
6#1864C211yesmaximal, characteristic, complemented
7#1863C211yesmaximal, characteristic, complemented
8#1862C211yesmaximal, characteristic, complemented
9#1853C211yesmaximal, characteristic, complemented
10#1852C211yesmaximal, characteristic, complemented
11#1803C211yesmaximal, characteristic, complemented
12#1802C211yesmaximal, characteristic, complemented
13#1798C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic
15#1788C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C2×#125C2211yescharacteristic
18#520—24no—
19#520—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×C4⋊C4—21no—
29C22×C4⋊C4C2211yescharacteristic
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#398—21no—
49#398—21no—
50#396—21no—
51#396—21no—
52#389C2211yescharacteristic
53#389C2211yescharacteristic
54#384C2211yescharacteristic
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×C4—21no—
76C23×C4C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81#125—22no—
82#125—22no—
83C4×Q8—22no—
84C4×Q8—22no—
85C4×Q8—22no—
86C4×Q8D414yes—
87C4×D4—28no—
88C4×D4—28no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4D414yes—
93#117—28no—
94#117—22no—
95#117—22no—
96C2×C4⋊C4—28no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4D411yescharacteristic
121C2×C22⋊C4D411yescharacteristic
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126#107—22no—
127#107—22no—
128#106—22no—
129#106—22no—
130C4⋊C8—22no—
131C4⋊C8—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#102—22no—
137#102—22no—
138#102—22no—
139#102—22no—
140C22⋊C8—22no—
141C22⋊C8—22no—
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—21no—
176C22×C4—21no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×Q812yes—
196C22×C4#4112yes—
197C22×C4C2411yescharacteristic
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—21no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
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—21no—
224C4⋊C4—21no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—81no—
228C4⋊C4—81no—
229C4⋊C4—81no—
230C4⋊C4—81no—
231C4⋊C4C2×D412yes—
232C4⋊C4C2×D412yes—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C42—24no—
240C42—24no—
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—22no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
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—84no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4#12012yes—
311C2×C4#12012yes—
312C2×C4C22⋊Q812yes—
313C2×C4C22×D411yescharacteristic
314C2×C4C22×D411yescharacteristic
315C2×C4C22×Q811yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C8—41no—
320C8—41no—
321C8—41no—
322C8—41no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—24no—
327C22—42no—
328C22—42no—
329C22—41no—
330C22—42no—
331C22—41no—
332C22—42no—
333C22—41no—
334C22—42no—
335C22—41no—
336C22—42no—
337C22C2×#12011yescharacteristic
338C22C2×C22⋊Q811yescharacteristic
339C22#52511yescharacteristic
340C22#57611yescharacteristic
341C22#57711yescharacteristic
342C22#57711yescharacteristic
343C22#57811yescharacteristic
344C4—22no—
345C4—24no—
346C4—22no—
347C4—22no—
348C4—41no—
349C4—42no—
350C4—42no—
351C4—41no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4C2×C22⋊Q811yescharacteristic
357C4C2×C22⋊Q811yescharacteristic
358C4#52511yescharacteristic
359C4#52511yescharacteristic
360C2—42no—
361C2—44no—
362C2#284811yescharacteristic
363C2#301211yescharacteristic
364C2#301311yescharacteristic
365C2#308311yescharacteristic
366C2#308411yescharacteristic
367C2#311911yescharacteristic
368C2#312011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ 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. #2629C2semidirect
5. #1865C2semidirect
6. #1864C2semidirect
7. #1863C2semidirect
8. #1862C2semidirect
9. #1853C2semidirect
10. #1852C2semidirect
11. #1803C2semidirect
12. #1802C2semidirect
13. #1798C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5202
C2×#4792
C2×#4782
#26292
#18652
#18642
#18632
#18622
#18532
#18522
#18032
#18022
#17982
#17882