Finite Group Databasebeta

← #25545

Subgroups — #25545

Order 256.

579 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#554C211yesmaximal, characteristic
2C2×#484C211yesmaximal, characteristic, complemented
3C2×#467C211yesmaximal, characteristic, complemented
4#2417C211yesmaximal, characteristic
5#2057C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2001C211yesmaximal, characteristic, complemented
9#1878C211yesmaximal, characteristic, complemented
10#1872C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1869C211yesmaximal, characteristic
13#1863C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15C2×#340C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C23×C8C2211yescharacteristic
18#554—24no—
19#554—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#125—21no—
22C2×C22⋊Q8—21no—
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34#484—24no—
35#467—24no—
36C2×#107C2211yescharacteristic
37C2×#107C2211yescharacteristic
38C2×#107C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#398—21no—
49#396—21no—
50#391C2211yescharacteristic
51#389C2211yescharacteristic
52#386—21no—
53#380—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4D411yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×Q16—22no—
73C2×Q16—22no—
74C2×Q16—41no—
75C22×C8—22no—
76C22×C8—22no—
77C22×C8—41no—
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C4⋊Q8—22no—
83C4⋊Q8—22no—
84#125—28no—
85C22⋊Q8—28no—
86C22⋊Q8—28no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92C22⋊Q8—22no—
93C22⋊Q8—41no—
94C22⋊Q8D414yes—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×D4—28no—
98C4×D4—22no—
99C4×D4—22no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4D411yescharacteristic
116C2×C4⋊C4D414yes—
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4D411yescharacteristic
129C2×C22⋊C4D411yescharacteristic
130C2×C42—21no—
131C2×C42C2311yescharacteristic
132#107—22no—
133#107—22no—
134#107—22no—
135#103—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#103—41no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24C2×D411yescharacteristic
149C24C2×D411yescharacteristic
150C2×Q8—24no—
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—81no—
163C2×Q8—81no—
164C2×Q8C2×D412yes—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4Q1614yescomplemented
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—24no—
178C22×C4—24no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4—81no—
191C22×C4—81no—
192C22×C4—81no—
193C22×C4—81no—
194C22×C4—81no—
195C22×C4C2×D411yescharacteristic
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D412yes—
199C22×C4C2×D412yes—
200C22×C4C2×D411yescharacteristic
201C22×C4C2411yescharacteristic
202Q16—44no—
203C2×C8—21no—
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—44no—
215C2×C8#4112yes—
216C4⋊C4—24no—
217C4⋊C4—24no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—24no—
222C4⋊C4—22no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—81no—
235C4⋊C4C2×D412yes—
236C4⋊C4C2×D412yes—
237C4⋊C4C2×D412yes—
238C4⋊C4#4112yes—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—81no—
246C42—24no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23#12012yes—
260C23C2×Q1612yes—
261C23C2×Q1612yes—
262C23#13712yes—
263C23C22×D411yescharacteristic
264Q8—22no—
265Q8—22no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—84no—
273D4—28no—
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—22no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—41no—
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—84no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4#12012yes—
320C2×C4C2×Q1612yes—
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—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22#45114yes—
345C22C2×#12011yescharacteristic
346C22#54611yescharacteristic
347C22#54711yescharacteristic
348C22C22×Q1611yescharacteristic
349C22C2×#13711yescharacteristic
350C22#57611yescharacteristic
351C22#57711yescharacteristic
352C4—22no—
353C4—22no—
354C4—41no—
355C4—42no—
356C4—41no—
357C4—42no—
358C4—42no—
359C4—42no—
360C4—42no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4#53211yescharacteristic
365C4#53711yescharacteristic
366C4#54611yescharacteristic
367C4#54711yescharacteristic
368C2—24no—
369C2—42no—
370C2#243011yescharacteristic
371C2C2×#45111yescharacteristic
372C2#301211yescharacteristic
373C2#329911yescharacteristic
374C2#331411yescharacteristic
375C2#333811yescharacteristic
376C2#334411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
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×#484C2semidirect
3. C2×#467C2semidirect
5. #2057C2semidirect
6. #2036C2semidirect
7. #2024C2semidirect
8. #2001C2semidirect
9. #1878C2semidirect
10. #1872C2semidirect
11. #1872C2semidirect
13. #1863C2semidirect
14. #1852C2semidirect
15. C2×#340C2semidirect
167. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#5542
C2×#4842
C2×#4672
#24172
#20572
#20362
#20242
#20012
#18782
#18722
#18692
#18632
#18522
C2×#3402