Finite Group Databasebeta

← #28477

Subgroups — #28477

Order 256.

534 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#497C211yesmaximal, characteristic
2C2×#497C211yesmaximal, characteristic
3C2×#494C211yesmaximal, characteristic, complemented
4C2×#464C211yesmaximal, characteristic
5C2×#462C211yesmaximal, characteristic, complemented
6C2×#441C211yesmaximal, characteristic, complemented
7#2546C211yesmaximal, characteristic, complemented
8#2545C211yesmaximal, characteristic
9#2016C211yesmaximal, characteristic, complemented
10#2016C211yesmaximal, characteristic, complemented
11#2015C211yesmaximal, characteristic
12#2013C211yesmaximal, characteristic
13#1971C211yesmaximal, characteristic, complemented
14#1968C211yesmaximal, characteristic
15#1771C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×Q16C2211yescharacteristic
21C22×Q16C2211yescharacteristic
22C22×SD16C2211yescharacteristic
23C22×SD16C2211yescharacteristic
24C22×SD16C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26C2×C4⋊Q8C2211yescharacteristic
27C2×C4⋊Q8C2211yescharacteristic
28C2×C4⋊Q8C2211yescharacteristic
29C2×#124C2211yescharacteristic
30C2×#124C2211yescharacteristic
31C2×C4×Q8C2211yescharacteristic
32C2×C4×Q8C2211yescharacteristic
33#497—24no—
34#497—24no—
35#494—24no—
36#464—24no—
37#462—24no—
38#441—24no—
39C2×#106C2211yescharacteristic
40C2×C4⋊C8C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#97C2211yescharacteristic
48C2×#97C2211yescharacteristic
49C2×C4×C8C2211yescharacteristic
50#398—22no—
51#396—22no—
52#391C2211yescharacteristic
53#391C2211yescharacteristic
54#391C2211yescharacteristic
55#390C2211yescharacteristic
56#389—22no—
57#388—22no—
58C4×C4⋊C4C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C2×Q16—22no—
68C2×Q16—22no—
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—41no—
80C2×Q16—41no—
81C2×Q16—41no—
82C2×Q16—41no—
83C2×Q16—41no—
84C2×Q16D414yes—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—22no—
88C2×SD16—22no—
89C2×SD16—22no—
90C2×SD16—22no—
91C2×SD16—41no—
92C2×SD16—41no—
93C2×SD16—41no—
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C4⋊Q8—22no—
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—22no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105C4⋊Q8—41no—
106#124—22no—
107#124—22no—
108#124—22no—
109#124—41no—
110C4×Q8—22no—
111C4×Q8—22no—
112C4×Q8—22no—
113C4×Q8—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4D412yes—
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C22⋊C4—21no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—22no—
128C2×C42—21no—
129C2×C42—21no—
130C2×C42C2311yescharacteristic
131C2×C42C2311yescharacteristic
132C2×C42C2311yescharacteristic
133#106—22no—
134C4⋊C8D414yes—
135#103—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#102—22no—
141#97—22no—
142#97D414yes—
143C4×C8—22no—
144#95—22no—
145#95—22no—
146#95—22no—
147#95D412yes—
148C24—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8C2×D412yes—
178C2×Q8C2×D412yes—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4C2×D412yes—
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D412yes—
196C22×C4C2411yescharacteristic
197Q16—28no—
198Q16—44no—
199Q16—44no—
200Q16—44no—
201Q16—44no—
202Q16—44no—
203SD16—44no—
204SD16—44no—
205SD16—44no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8C2×D412yes—
213C2×C8C2×D412yes—
214C2×C8C2×D412yes—
215C2×C8C2×D412yes—
216C2×C8C2×D412yes—
217C2×C8#4112yes—
218C4⋊C4—24no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C22⋊C4—44no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—44no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C42—42no—
240C42C2×D412yes—
241C42C2×D412yes—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—42no—
246C23—42no—
247C23C22×D411yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—22no—
251Q8—22no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258Q8—41no—
259Q8—41no—
260Q8—41no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
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—41no—
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#12712yes—
299C2×C4#12712yes—
300C2×C4#13714yes—
301C2×C4C22×D411yescharacteristic
302C2×C4C22×D411yescharacteristic
303C2×C4C22×D411yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C8—22no—
308C8—22no—
309C8—22no—
310C8#12114yes—
311C22—82no—
312C22—81no—
313C22—81no—
314C22—82no—
315C22—81no—
316C22C2×#12711yescharacteristic
317C22D4211yescharacteristic
318C22#55011yescharacteristic
319C22C2×#13712yes—
320C22#57812yes—
321C4—22no—
322C4—22no—
323C4—24no—
324C4—42no—
325C4—42no—
326C4—42no—
327C4—81no—
328C4—81no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4C2×#12111yescharacteristic
333C4#53511yescharacteristic
334C4D4211yescharacteristic
335C4#55011yescharacteristic
336C2—161no—
337C2#254111yescharacteristic
338C2#315211yescharacteristic
339C2#315611yescharacteristic
340C2#327012yes—
341C2#339812yes—
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4C23
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
3. C2×#494C2semidirect
5. C2×#462C2semidirect
6. C2×#441C2semidirect
7. #2546C2semidirect
9. #2016C2semidirect
10. #2016C2semidirect
13. #1971C2semidirect
15. #1771C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4942
C2×#4642
C2×#4622
C2×#4412
#25462
#25452
#20162
#20152
#20132
#19712
#19682
#17712