Finite Group Databasebeta

← #24100

Subgroups — #24100

Order 256.

650 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#547C211yesmaximal, characteristic
2C2×#482C211yesmaximal, characteristic, complemented
3C2×#428C211yesmaximal, characteristic, complemented
4#2485C211yesmaximal, characteristic, complemented
5#2058C211yesmaximal, characteristic, complemented
6#2057C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1895C211yesmaximal, characteristic, complemented
10#1870C211yesmaximal, characteristic, complemented
11#1864C211yesmaximal, characteristic, complemented
12#1862C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1824C211yesmaximal, characteristic, complemented
16C22×#41—21no—
17C22×M16C2211yescharacteristic
18#547—24no—
19#547—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#123—21no—
22C2×#123—21no—
23C2×C22⋊Q8—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×#117C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31#482—24no—
32#428—24no—
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#107C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46#397—21no—
47#394—21no—
48#389C2211yescharacteristic
49#389C2211yescharacteristic
50#388—21no—
51#387—21no—
52#380C2211yescharacteristic
53#380C2211yescharacteristic
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—28no—
64C2×#41—44no—
65C2×#41—44no—
66C22×Q8—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4D411yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—22no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85#123—28no—
86#123—28no—
87C22⋊Q8—28no—
88#121—22no—
89#121—22no—
90#121—22no—
91#121D414yes—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96#117—22no—
97#117—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4D414yes—
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4D411yescharacteristic
123C2×C22⋊C4D411yescharacteristic
124C2×C42—21no—
125C2×C42C2311yescharacteristic
126#107—22no—
127#107—22no—
128#107—22no—
129#106—22no—
130#106—22no—
131#106—22no—
132#102—22no—
133#102—22no—
134#102—22no—
135#102—22no—
136C22⋊C8—22no—
137C22⋊C8—22no—
138C22⋊C8—22no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24C2×D411yescharacteristic
146C24C2×D411yescharacteristic
147#41—432no—
148C2×Q8—24no—
149C2×Q8—42no—
150C2×Q8—42no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4C2×D412yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—24no—
171C22×C4—24no—
172C22×C4—24no—
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—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—42no—
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D412yes—
196C22×C4C2×D412yes—
197C22×C4C2×D411yescharacteristic
198C22×C4C2411yescharacteristic
199M16—28no—
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—
210C2×C8#4112yescomplemented
211C2×C8#4112yescomplemented
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—24no—
215C4⋊C4—22no—
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—21no—
226C4⋊C4—21no—
227C4⋊C4—42no—
228C4⋊C4—81no—
229C4⋊C4C2×D412yes—
230C4⋊C4C2×D412yes—
231C4⋊C4C2×D412yes—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—22no—
235C22⋊C4—24no—
236C22⋊C4—22no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C42—21no—
242C42—21no—
243C42—42no—
244C23—21no—
245C23—21no—
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—
263C23#12012yes—
264C23AGL(1,ℤ/8)12yes—
265C23C22×D411yescharacteristic
266Q8—48no—
267D4—28no—
268D4—22no—
269D4—22no—
270D4—48no—
271D4—48no—
272D4—41no—
273D4—41no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—48no—
292C2×C4—48no—
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—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4#12012yes—
317C2×C4AGL(1,ℤ/8)12yes—
318C2×C4C22×D411yescharacteristic
319C2×C4C22×D411yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4C2×#4111yescharacteristic
322C2×C42−1+411yescharacteristic
323C2×C42−1+411yescharacteristic
324C8—22no—
325C8—22no—
326C8—41no—
327C8—41no—
328C22—22no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—42no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22C2×#12011yescharacteristic
350C22#54711yescharacteristic
351C22#54711yescharacteristic
352C22C2×AGL(1,ℤ/8)11yescharacteristic
353C22C2×AGL(1,ℤ/8)11yescharacteristic
354C22#57711yescharacteristic
355C22#57811yescharacteristic
356C4—22no—
357C4—22no—
358C4—41no—
359C4—42no—
360C4—42no—
361C4—42no—
362C4—41no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4#52511yescharacteristic
368C4#53611yescharacteristic
369C4#54711yescharacteristic
370C4#54711yescharacteristic
371C2—42no—
372C2—42no—
373C2—82no—
374C2#240611yescharacteristic
375C2#300311yescharacteristic
376C2#301311yescharacteristic
377C2#334311yescharacteristic
378C2#334511yescharacteristic
379C2#335311yescharacteristic
380C2#335411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#547C2
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×#482C2semidirect
3. C2×#428C2semidirect
4. #2485C2semidirect
5. #2058C2semidirect
6. #2057C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1895C2semidirect
10. #1870C2semidirect
11. #1864C2semidirect
12. #1862C2semidirect
13. #1853C2semidirect
14. #1852C2semidirect
15. #1824C2semidirect
210. C2×C8#41semidirect
211. C2×C8#41semidirect

Maximal subgroups

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

subgroupindex
C2×#5472
C2×#4822
C2×#4282
#24852
#20582
#20572
#20342
#20332
#18952
#18702
#18642
#18622
#18532
#18522
#18242