Finite Group Databasebeta

← #28327

Subgroups — #28327

Order 256.

544 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#484C211yesmaximal, characteristic, complemented
2C2×#480C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4C2×#449C211yesmaximal, characteristic
5#2620C211yesmaximal, characteristic, complemented
6#2311C211yesmaximal, characteristic, complemented
7#2095C211yesmaximal, characteristic, complemented
8#2093C211yesmaximal, characteristic, complemented
9#2015C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1973C211yesmaximal, characteristic, complemented
13#1970C211yesmaximal, characteristic, complemented
14#1969C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×D8C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C22×C4⋊C4C2211yescharacteristic
26#484—24no—
27#480—24no—
28#452—24no—
29#449—24no—
30C2×#107C2211yescharacteristic
31C2×#107C2211yescharacteristic
32C2×#107C2211yescharacteristic
33C2×#107C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#396—22no—
47#394—22no—
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#390C2211yescharacteristic
54#386—22no—
55#384C2211yescharacteristic
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×Q8C2311yescharacteristic
67C22×D4—21no—
68C22×D4—21no—
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—41no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—41no—
82C2×SD16—41no—
83C2×D8—22no—
84C2×D8—22no—
85C2×D8—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#124—41no—
91#124—41no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94#121—22no—
95#121—22no—
96#117—41no—
97#117—41no—
98C2×C4⋊C4—22no—
99C2×C4⋊C4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
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#107—22no—
120#107—22no—
121#103—22no—
122#103—22no—
123#103—22no—
124#103—22no—
125#102—22no—
126#102—22no—
127#102—22no—
128#102—22no—
129C22⋊C8D414yes—
130C22⋊C8D414yes—
131#97—41no—
132C4×C8—41no—
133#95—22no—
134#95—22no—
135#95—22no—
136C24—21no—
137C24—21no—
138C24C2×D411yescharacteristic
139#41—44no—
140#41—44no—
141#41—44no—
142#41—44no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8#4112yes—
155C2×D4—24no—
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—82no—
171C2×D4#4112yes—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—24no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—82no—
187C22×C4C2×D412yes—
188C22×C4C2×D412yes—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D412yes—
191C22×C4C2×D412yes—
192C22×C4C2411yescharacteristic
193Q16—44no—
194SD16—44no—
195SD16—44no—
196D8—44no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8C2×D412yes—
206C2×C8C2×D412yes—
207C2×C8C2×D412yes—
208C2×C8C2×D412yes—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—82no—
221C22⋊C4—42no—
222C22⋊C4—44no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—82no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—41no—
238C23—41no—
239C23—42no—
240C23—42no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—42no—
245C23—42no—
246C23#12712yes—
247C23C22×D411yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256D4—22no—
257D4—22no—
258D4—48no—
259D4—41no—
260D4—41no—
261D4—41no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—84no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—48no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—84no—
294C2×C4#12712yes—
295C2×C4C22×D411yescharacteristic
296C2×C4C22×D411yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4C2×#4111yescharacteristic
299C2×C4H(5,𝔽2)11yescharacteristic
300C2×C4H(5,𝔽2)11yescharacteristic
301C8—22no—
302C8—22no—
303C8—22no—
304C8—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—42no—
309C22—42no—
310C22—44no—
311C22—42no—
312C22—81no—
313C22—81no—
314C22—82no—
315C22—82no—
316C22—81no—
317C22—81no—
318C22—82no—
319C22—81no—
320C22#52511yescharacteristic
321C22C2×#12711yescharacteristic
322C22#53411yescharacteristic
323C22#57612yes—
324C22#57812yes—
325C4—22no—
326C4—42no—
327C4—42no—
328C4—42no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4#54611yescharacteristic
335C4#54611yescharacteristic
336C4#54611yescharacteristic
337C4#54611yescharacteristic
338C2—42no—
339C2—82no—
340C2—161no—
341C2#253911yescharacteristic
342C2#312412yes—
343C2#315511yescharacteristic
344C2#315611yescharacteristic
345C2#320312yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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×#484C2semidirect
2. C2×#480C2semidirect
3. C2×#452C2semidirect
5. #2620C2semidirect
6. #2311C2semidirect
7. #2095C2semidirect
8. #2093C2semidirect
9. #2015C2semidirect
10. #2012C2semidirect
11. #1973C2semidirect
12. #1973C2semidirect
13. #1970C2semidirect
14. #1969C2semidirect
15. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4802
C2×#4522
C2×#4492
#26202
#23112
#20952
#20932
#20152
#20122
#19732
#19702
#19692
#17502