Finite Group Databasebeta

← #31031

Subgroups — #31031

Order 256.

502 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#485C211yesmaximal, characteristic, complemented
2C2×#448C211yesmaximal, characteristic, complemented
3#2414C211yesmaximal, characteristic
4#2409C211yesmaximal, characteristic, complemented
5#2077C211yesmaximal, characteristic, complemented
6#2068C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1968C211yesmaximal, characteristic, complemented
12#1954C211yesmaximal, characteristic, complemented
13#1933C211yesmaximal, characteristic, complemented
14#1873C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25#485—24no—
26#448—24no—
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C4×C8C2211yescharacteristic
43#395—21no—
44#394—21no—
45#393—21no—
46#392—21no—
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#390—21no—
50#390C2211yescharacteristic
51#389C2211yescharacteristic
52#388—21no—
53#386—21no—
54#385—21no—
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×D4—21no—
71C22×D4—21no—
72C22×D4D411yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C23×C4C2311yescharacteristic
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×SD16—41no—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91#121—22no—
92#121—41no—
93#121—41no—
94#121—41no—
95C4×D4—41no—
96C4×D4—41no—
97#117—22no—
98#117—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4D411yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42C2311yescharacteristic
121#107—22no—
122#106—22no—
123C4⋊C8—41no—
124#103—22no—
125#103—22no—
126#103—41no—
127#103—41no—
128#103—41no—
129#103D414yescomplemented
130#102—41no—
131#102—41no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C4×C8—41no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24C2×D411yescharacteristic
146C24C2×D411yescharacteristic
147C2×Q8—21no—
148C2×Q8—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—42no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—44no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—44no—
164C2×Q8C2×D412yes—
165C2×Q8#4112yes—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—41no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—81no—
179C2×D4—81no—
180C2×D4—81no—
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—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2411yescharacteristic
208SD16—44no—
209SD16—44no—
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—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8C2×D412yes—
222C4⋊C4—21no—
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—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—81no—
235C4⋊C4C2×D412yes—
236C22⋊C4—41no—
237C22⋊C4—41no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—81no—
246C22⋊C4—81no—
247C42—21no—
248C42—21no—
249C42—42no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23#13512yes—
271C23C22×D411yescharacteristic
272Q8—22no—
273Q8—22no—
274Q8—22no—
275Q8—22no—
276Q8—48no—
277Q8—41no—
278Q8—41no—
279Q8—41no—
280Q8—41no—
281D4—44no—
282D4—41no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—84no—
287D4—84no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—48no—
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—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—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4#13712yes—
330C2×C4C22×D411yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4H(5,𝔽2)11yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C2×C4H(5,𝔽2)11yescharacteristic
336C8—22no—
337C8—41no—
338C8—41no—
339C8—41no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#53411yescharacteristic
362C22D4211yescharacteristic
363C22#54611yescharacteristic
364C22C2×#13511yescharacteristic
365C22C2×#13711yescharacteristic
366C22#57711yescharacteristic
367C22#57711yescharacteristic
368C4—22no—
369C4—41no—
370C4—41no—
371C4—42no—
372C4—42no—
373C4—41no—
374C4—42no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4#53511yescharacteristic
380C4#53811yescharacteristic
381C4#54611yescharacteristic
382C4#56011yescharacteristic
383C2—42no—
384C2—81no—
385C2—161no—
386C2#267811yescharacteristic
387C2#319011yescharacteristic
388C2#320011yescharacteristic
389C2#327811yescharacteristic
390C2#328211yescharacteristic
391C2#330111yescharacteristic
392C2#332911yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2414C2
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
1. C2×#485C2semidirect
2. C2×#448C2semidirect
4. #2409C2semidirect
5. #2077C2semidirect
6. #2068C2semidirect
7. #2055C2semidirect
8. #2033C2semidirect
9. #2000C2semidirect
10. #2000C2semidirect
11. #1968C2semidirect
12. #1954C2semidirect
13. #1933C2semidirect
14. #1873C2semidirect
15. #1787C2semidirect
129. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4852
C2×#4482
#24142
#24092
#20772
#20682
#20552
#20332
#20002
#19682
#19542
#19332
#18732
#17872