Finite Group Databasebeta

← #31010

Subgroups — #31010

Order 256.

517 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#463C211yesmaximal, characteristic, complemented
2C2×#448C211yesmaximal, characteristic, complemented
3#2436C211yesmaximal, characteristic, complemented
4#2422C211yesmaximal, characteristic, complemented
5#2072C211yesmaximal, characteristic, complemented
6#2068C211yesmaximal, characteristic, complemented
7#2042C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#2012C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic
11#1973C211yesmaximal, characteristic, complemented
12#1959C211yesmaximal, characteristic, complemented
13#1924C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×D8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×#124C2211yescharacteristic
22C2×#123—21no—
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26#463—24no—
27#448—24no—
28C2×#107C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×#97C2211yescharacteristic
43#397—21no—
44#394—21no—
45#393—21no—
46#393—21no—
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#390C2211yescharacteristic
50#389C2211yescharacteristic
51#386—21no—
52#386—21no—
53#385—21no—
54#384C2211yescharacteristic
55#380C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
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×D4C2311yescharacteristic
73C22×D4C2311yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—41no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×SD16D414yescomplemented
83C2×D8—22no—
84C2×D8—41no—
85C2×D8—41no—
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90#124—22no—
91#124—22no—
92#124—41no—
93#124—41no—
94#123—44no—
95#121—22no—
96#121—41no—
97#121—41no—
98#121—41no—
99C4×D4—22no—
100C4×D4—41no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
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⋊C4—21no—
118C2×C22⋊C4D411yescharacteristic
119C2×C22⋊C4D411yescharacteristic
120C2×C42C2311yescharacteristic
121C2×C42C2311yescharacteristic
122#107—41no—
123C4⋊C8—22no—
124#103—22no—
125#103—22no—
126#103—22no—
127#103—41no—
128#103—41no—
129#103—41no—
130#102—22no—
131#102—41no—
132#102—41no—
133#102—41no—
134C22⋊C8—22no—
135C22⋊C8—41no—
136#97—41no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95D411yescharacteristic
144#95D411yescharacteristic
145C24—21no—
146C24—21no—
147C24C2×D411yescharacteristic
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—21no—
158C2×Q8—42no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—44no—
162C2×Q8—44no—
163C2×Q8—44no—
164C2×Q8—44no—
165C2×Q8C2×D412yes—
166C2×Q8#4112yes—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—81no—
184C2×D4C2×D412yes—
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—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—81no—
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2411yescharacteristic
209SD16—44no—
210SD16—44no—
211D8—44no—
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×C8—21no—
222C2×C8—21no—
223C2×C8C2×D412yes—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—81no—
235C22⋊C4—22no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C22⋊C4—42no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23AGL(1,ℤ/8)12yes—
272C23C22×D411yescharacteristic
273Q8—22no—
274Q8—22no—
275Q8—22no—
276Q8—22no—
277Q8—48no—
278Q8—41no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282D4—22no—
283D4—22no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—41no—
290D4—84no—
291D4—84no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—48no—
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—41no—
330C2×C4—84no—
331C2×C4C22×D411yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C2×C4H(5,𝔽2)11yescharacteristic
336C2×C4H(5,𝔽2)11yescharacteristic
337C8—22no—
338C8—41no—
339C8—41no—
340C8—41no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—42no—
349C22—42no—
350C22—42no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22#53511yescharacteristic
363C22D4211yescharacteristic
364C22#54611yescharacteristic
365C22C2×AGL(1,ℤ/8)11yescharacteristic
366C22#57511yescharacteristic
367C22#57711yescharacteristic
368C22#57811yescharacteristic
369C4—41no—
370C4—42no—
371C4—42no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4#53411yescharacteristic
381C4#53811yescharacteristic
382C4#54611yescharacteristic
383C4#56011yescharacteristic
384C2—42no—
385C2—82no—
386C2—161no—
387C2#267811yescharacteristic
388C2#321311yescharacteristic
389C2#322511yescharacteristic
390C2#326511yescharacteristic
391C2#326911yescharacteristic
392C2#330811yescharacteristic
393C2#332611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#463C2semidirect
2. C2×#448C2semidirect
3. #2436C2semidirect
4. #2422C2semidirect
5. #2072C2semidirect
6. #2068C2semidirect
7. #2042C2semidirect
8. #2033C2semidirect
9. #2012C2semidirect
11. #1973C2semidirect
12. #1959C2semidirect
13. #1924C2semidirect
14. #1877C2semidirect
15. #1787C2semidirect
82. C2×SD16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4632
C2×#4482
#24362
#24222
#20722
#20682
#20422
#20332
#20122
#19992
#19732
#19592
#19242
#18772
#17872