Finite Group Databasebeta

← #25552

Subgroups — #25552

Order 256.

583 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#468C211yesmaximal, characteristic, complemented
4#2417C211yesmaximal, characteristic, complemented
5#2057C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#2024C211yesmaximal, characteristic, complemented
8#2000C211yesmaximal, characteristic, complemented
9#1877C211yesmaximal, characteristic, complemented
10#1874C211yesmaximal, characteristic, complemented
11#1873C211yesmaximal, characteristic, complemented
12#1870C211yesmaximal, characteristic, complemented
13#1864C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18#555—24no—
19#555—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#124—21no—
22C2×#124—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121—21no—
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×#117—21no—
31C2×#117C2211yescharacteristic
32C2×#117C2211yescharacteristic
33C22×C4⋊C4C2211yescharacteristic
34#485—24no—
35#468—24no—
36C2×#107C2211yescharacteristic
37C2×#107C2211yescharacteristic
38C2×#106C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
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×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—41no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C4⋊Q8—22no—
84C4⋊Q8—22no—
85#124—28no—
86#124—28no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—41no—
91#121—28no—
92#121—22no—
93#121—22no—
94#121—22no—
95#121D414yes—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—22no—
100#117—28no—
101#117—22no—
102#117—22no—
103#117—22no—
104#117D414yes—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4D411yescharacteristic
128C2×C22⋊C4D411yescharacteristic
129C2×C42—21no—
130C2×C42—21no—
131C2×C42C2311yescharacteristic
132#107—22no—
133#107—22no—
134#106—22no—
135#103—22no—
136#103—22no—
137#103—22no—
138#102—22no—
139#102—22no—
140#102—41no—
141C22⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—41no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24—21no—
149C24C2×D411yescharacteristic
150C24C2×D411yescharacteristic
151C2×Q8—24no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—81no—
159C2×Q8—81no—
160C2×D4—22no—
161C2×D4—24no—
162C2×D4—24no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—41no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4C2×D412yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—24no—
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—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—81no—
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D412yes—
202C22×C4C2×D412yes—
203C22×C4C2×D411yescharacteristic
204C22×C4C2411yescharacteristic
205SD16—44no—
206M16—44no—
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×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8#4112yes—
219C4⋊C4—24no—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—81no—
233C4⋊C4C2×D412yes—
234C4⋊C4C2×D412yes—
235C4⋊C4C2×D412yes—
236C4⋊C4#4112yes—
237C22⋊C4—24no—
238C22⋊C4—24no—
239C22⋊C4—24no—
240C22⋊C4—24no—
241C22⋊C4—22no—
242C22⋊C4—24no—
243C22⋊C4—22no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—81no—
247C42—24no—
248C42—24no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—22no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23#12012yes—
272C23C22×D411yescharacteristic
273Q8—41no—
274Q8—41no—
275Q8—41no—
276Q8—41no—
277Q8—84no—
278D4—22no—
279D4—22no—
280D4—44no—
281D4—41no—
282D4—41no—
283D4—84no—
284D4—84no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—48no—
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—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—84no—
326C2×C4#12012yes—
327C2×C4#13712yes—
328C2×C4C22×D411yescharacteristic
329C2×C4C22×D411yescharacteristic
330C2×C4C2×#4111yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4H(5,𝔽2)11yescharacteristic
333C2×C42−1+411yescharacteristic
334C8—22no—
335C8—41no—
336C8—41no—
337C8—41no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—42no—
350C22—42no—
351C22—41no—
352C22—41no—
353C22—42no—
354C22—41no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22C2×#12011yescharacteristic
360C22#54611yescharacteristic
361C22#54711yescharacteristic
362C22C2×#13711yescharacteristic
363C22#57511yescharacteristic
364C22#57611yescharacteristic
365C22#57711yescharacteristic
366C4—22no—
367C4—22no—
368C4—41no—
369C4—42no—
370C4—41no—
371C4—42no—
372C4—42no—
373C4—42no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4#53211yescharacteristic
378C4#53711yescharacteristic
379C4#54611yescharacteristic
380C4#54711yescharacteristic
381C2—42no—
382C2—81no—
383C2—82no—
384C2#243011yescharacteristic
385C2#300711yescharacteristic
386C2#301211yescharacteristic
387C2#331911yescharacteristic
388C2#332911yescharacteristic
389C2#334811yescharacteristic
390C2#335711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#555C2
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×#485C2semidirect
3. C2×#468C2semidirect
4. #2417C2semidirect
5. #2057C2semidirect
6. #2033C2semidirect
7. #2024C2semidirect
8. #2000C2semidirect
9. #1877C2semidirect
10. #1874C2semidirect
11. #1873C2semidirect
12. #1870C2semidirect
13. #1864C2semidirect
14. #1854C2semidirect
15. #1736C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#4852
C2×#4682
#24172
#20572
#20332
#20242
#20002
#18772
#18742
#18732
#18702
#18642
#18542
#17362