Finite Group Databasebeta

← #24735

Subgroups — #24735

Order 256.

532 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#463C211yesmaximal, characteristic, complemented
2C2×#463C211yesmaximal, characteristic, complemented
3C2×#459C211yesmaximal, characteristic
4C2×C4×D8C211yesmaximal, characteristic, complemented
5#2544C211yesmaximal, characteristic, complemented
6#2349C211yesmaximal, characteristic, complemented
7#2068C211yesmaximal, characteristic, complemented
8#2067C211yesmaximal, characteristic, complemented
9#2042C211yesmaximal, characteristic, complemented
10#2041C211yesmaximal, characteristic, complemented
11#1958C211yesmaximal, characteristic, complemented
12#1939C211yesmaximal, characteristic, complemented
13#1879C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#127C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×C4×D4C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#463—24no—
27#463—24no—
28#459—24no—
29C4×D8—24no—
30C2×#107C2211yescharacteristic
31C2×C4⋊C8C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#394—21no—
46#394—21no—
47#391C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic
50#389C2211yescharacteristic
51#386—21no—
52#386—21no—
53#385—21no—
54#385—21no—
55#382C2211yescharacteristic
56#380—21no—
57C4×C4⋊C4C2211yescharacteristic
58C2×#95—21no—
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C23×C4—21no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×D8—22no—
77C2×D8—22no—
78C2×D8—22no—
79C2×D8—22no—
80C2×D8—22no—
81C2×D8D414yes—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86#127—22no—
87#127—41no—
88#124—22no—
89#124—22no—
90#124—22no—
91#124—41no—
92C4×D4—22no—
93C4×D4—22no—
94C4×D4—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4D414yescomplemented
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4D411yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C42C2311yescharacteristic
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—22no—
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—22no—
125#103—22no—
126#103—22no—
127#103—41no—
128#102—22no—
129#102—22no—
130#102—22no—
131#102—22no—
132#102—22no—
133#102—22no—
134#102—41no—
135C4×C8—22no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24—21no—
146C24—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—42no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4C2×D412yes—
174C2×D4C2×D412yes—
175C2×D4C2×D412yes—
176C2×D4#4112yes—
177C22×C4—21no—
178C22×C4—24no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2411yescharacteristic
201SD16—44no—
202D8—28no—
203D8—44no—
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—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8C2×D412yes—
216C4⋊C4—22no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—41no—
225C4⋊C4—81no—
226C4⋊C4—81no—
227C4⋊C4C2×D412yes—
228C22⋊C4—24no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42C2×D412yes—
243C42#4112yes—
244C23—22no—
245C23—22no—
246C23—22no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23C22×D411yescharacteristic
266Q8—41no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270D4—22no—
271D4—22no—
272D4—22no—
273D4—22no—
274D4—22no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—84no—
282D4—84no—
283D4#12114yes—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
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—84no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4#12112yes—
319C2×C4#12112yes—
320C2×C4#13512yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C42−1+411yescharacteristic
327C8—22no—
328C8—41no—
329C8—41no—
330C8—41no—
331C22—24no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22C2×#12111yescharacteristic
353C22D4211yescharacteristic
354C22#54811yescharacteristic
355C22C2×#13511yescharacteristic
356C22#57511yescharacteristic
357C22#57611yescharacteristic
358C22#57611yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—41no—
363C4—41no—
364C4—41no—
365C4—42no—
366C4—81no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4C2×#12111yescharacteristic
371C4#53711yescharacteristic
372C4#54711yescharacteristic
373C4#55311yescharacteristic
374C2—44no—
375C2—82no—
376C2—161no—
377C2#241611yescharacteristic
378C2#303511yescharacteristic
379C2#304111yescharacteristic
380C2#327311yescharacteristic
381C2#328011yescharacteristic
382C2#336111yescharacteristic
383C2#337011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
LCSG ▷ C22×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
1. C2×#463C2semidirect
2. C2×#463C2semidirect
4. C2×C4×D8C2semidirect
5. #2544C2semidirect
6. #2349C2semidirect
7. #2068C2semidirect
8. #2067C2semidirect
9. #2042C2semidirect
10. #2041C2semidirect
11. #1958C2semidirect
12. #1939C2semidirect
13. #1879C2semidirect
14. #1877C2semidirect
15. #1804C2semidirect
99. C4×D4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×#4592
C2×C4×D82
#25442
#23492
#20682
#20672
#20422
#20412
#19582
#19392
#18792
#18772
#18042