Finite Group Databasebeta

← #28325

Subgroups — #28325

Order 256.

573 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#481C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic, complemented
4C2×#449C211yesmaximal, characteristic, complemented
5#2619C211yesmaximal, characteristic, complemented
6#2311C211yesmaximal, characteristic, complemented
7#2097C211yesmaximal, characteristic, complemented
8#2092C211yesmaximal, characteristic, complemented
9#2014C211yesmaximal, characteristic, complemented
10#2011C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1972C211yesmaximal, characteristic, complemented
13#1970C211yesmaximal, characteristic, complemented
14#1968C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C2×#127C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#481—24no—
27#480—24no—
28#449—24no—
29#449—24no—
30C2×#107C2211yescharacteristic
31C2×#107C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#394—22no—
47#392—22no—
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#390C2211yescharacteristic
54#386—22no—
55#382C2211yescharacteristic
56#380—22no—
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59C2×#41—22no—
60C2×#41—22no—
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—41no—
64C2×#41—41no—
65C22×Q8C2311yescharacteristic
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4—22no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×D8—22no—
83C2×D8—22no—
84C2×D8—22no—
85C2×D8—22no—
86C2×D8—41no—
87C2×D8—41no—
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#127—41no—
93#127—41no—
94#121—22no—
95#121—22no—
96#121—22no—
97#121—22no—
98#117—41no—
99#117—41no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4D412yes—
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—22no—
111C2×C22⋊C4—22no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4—22no—
114C2×C22⋊C4D412yes—
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#107—22no—
118#107—22no—
119#106—22no—
120#106—22no—
121#103—22no—
122#103—22no—
123#102—22no—
124#102—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129C22⋊C8D414yescomplemented
130C22⋊C8D414yescomplemented
131#97—41no—
132C4×C8—41no—
133#95—22no—
134#95—22no—
135#95—22no—
136C24—21no—
137C24—21no—
138C24—21no—
139C24C2×D411yescharacteristic
140#41—44no—
141#41—44no—
142#41—44no—
143#41—44no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8#4112yes—
150C2×D4—24no—
151C2×D4—21no—
152C2×D4—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—21no—
168C2×D4—44no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4—42no—
172C2×D4—42no—
173C2×D4—82no—
174C2×D4—82no—
175C2×D4—82no—
176C2×D4—82no—
177C2×D4#4112yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—24no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—42no—
191C22×C4—82no—
192C22×C4C2×D412yes—
193C22×C4C2×D412yes—
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D412yes—
196C22×C4C2×D412yes—
197C22×C4C2411yescharacteristic
198SD16—44no—
199SD16—44no—
200D8—44no—
201D8—44no—
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×C8C2×D412yes—
211C2×C8C2×D412yes—
212C2×C8C2×D412yes—
213C2×C8C2×D412yes—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—82no—
228C22⋊C4—82no—
229C22⋊C4—82no—
230C22⋊C4—82no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—21no—
235C23—21no—
236C23—21no—
237C23—22no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23—41no—
242C23—42no—
243C23—42no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—42no—
251C23—42no—
252C23—42no—
253C23—42no—
254C23#12712yes—
255C23C22×D411yescharacteristic
256Q8—22no—
257Q8—22no—
258Q8—41no—
259Q8—41no—
260D4—22no—
261D4—22no—
262D4—48no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—88no—
274D4—84no—
275D4—84no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—48no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—84no—
301C2×C4#12712yes—
302C2×C4C22×D411yescharacteristic
303C2×C4C22×D411yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C4H(5,𝔽2)11yescharacteristic
308C8—22no—
309C8—22no—
310C8—22no—
311C8—22no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—42no—
316C22—42no—
317C22—44no—
318C22—42no—
319C22—81no—
320C22—81no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—82no—
329C22—82no—
330C22—81no—
331C22—81no—
332C22#52511yescharacteristic
333C22C2×#12711yescharacteristic
334C22#53411yescharacteristic
335C22#57612yes—
336C22#57712yes—
337C4—22no—
338C4—42no—
339C4—42no—
340C4—42no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4#54611yescharacteristic
346C4#54611yescharacteristic
347C4#54611yescharacteristic
348C4#54611yescharacteristic
349C2—42no—
350C2—82no—
351C2—161no—
352C2—161no—
353C2#253911yescharacteristic
354C2#311912yes—
355C2#315411yescharacteristic
356C2#315511yescharacteristic
357C2#319812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#481C2semidirect
2. C2×#480C2semidirect
3. C2×#449C2semidirect
4. C2×#449C2semidirect
5. #2619C2semidirect
6. #2311C2semidirect
7. #2097C2semidirect
8. #2092C2semidirect
9. #2014C2semidirect
10. #2011C2semidirect
11. #1973C2semidirect
12. #1972C2semidirect
13. #1970C2semidirect
14. #1968C2semidirect
15. #1750C2semidirect
129. C22⋊C8D4semidirect
130. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4812
C2×#4802
C2×#4492
#26192
#23112
#20972
#20922
#20142
#20112
#19732
#19722
#19702
#19682
#17502