Finite Group Databasebeta

← #31019

Subgroups — #31019

Order 256.

488 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×#484C211yesmaximal, characteristic, complemented
2C2×#451C211yesmaximal, characteristic, complemented
3#2417C211yesmaximal, characteristic
4#2409C211yesmaximal, characteristic
5#2076C211yesmaximal, characteristic, complemented
6#2068C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2036C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2001C211yesmaximal, characteristic, complemented
11#1969C211yesmaximal, characteristic, complemented
12#1954C211yesmaximal, characteristic
13#1933C211yesmaximal, characteristic, complemented
14#1872C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C22×C4⋊C4C2211yescharacteristic
25#484—24no—
26#451—24no—
27C2×#107C2211yescharacteristic
28C2×#107C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C4×C8C2211yescharacteristic
43#398—21no—
44#396—21no—
45#395—21no—
46#393—21no—
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#391C2211yescharacteristic
50#389C2211yescharacteristic
51#386—21no—
52#386—21no—
53#385—21no—
54#380—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×Q8C2311yescharacteristic
71C22×D4D411yescharacteristic
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C23×C4C2311yescharacteristic
76C2×Q16—22no—
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—41no—
80C2×Q16—41no—
81C2×Q16—41no—
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22⋊Q8—22no—
87C22⋊Q8—22no—
88C22⋊Q8—22no—
89C22⋊Q8—41no—
90C22⋊Q8—41no—
91C22⋊Q8—41no—
92C22⋊Q8—41no—
93C22⋊Q8—41no—
94C4×D4—41no—
95C4×D4—41no—
96C2×C4⋊C4—22no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—41no—
104C2×C4⋊C4D411yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
117C2×C22⋊C4D411yescharacteristic
118C2×C42C2311yescharacteristic
119#107—22no—
120#107—22no—
121C4⋊C8—41no—
122#103—22no—
123#103—22no—
124#103—41no—
125#103—41no—
126#103—41no—
127#103—41no—
128#103—41no—
129#103D414yescomplemented
130C22⋊C8—22no—
131C22⋊C8—22no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C4×C8—41no—
135#95—21no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24C2×D411yescharacteristic
145C24C2×D411yescharacteristic
146C2×Q8—21no—
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—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8C2×D412yes—
170C2×Q8#4112yes—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4Q1614yescomplemented
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
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—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—81no—
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2411yescharacteristic
205Q16—44no—
206Q16—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×C8C2×D412yes—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
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—81no—
244C42—21no—
245C42—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23C2×Q1612yes—
259C23C2×Q1612yes—
260C23C22×D411yescharacteristic
261Q8—22no—
262Q8—22no—
263Q8—22no—
264Q8—22no—
265Q8—48no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274D4—44no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—48no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
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—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—84no—
320C2×C4C2×Q1612yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C2×#4111yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—22no—
328C8—41no—
329C8—41no—
330C8—41no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—22no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22#53411yescharacteristic
346C22D4211yescharacteristic
347C22#54611yescharacteristic
348C22C22×Q1611yescharacteristic
349C22#57511yescharacteristic
350C22#57711yescharacteristic
351C22#57811yescharacteristic
352C4—22no—
353C4—41no—
354C4—41no—
355C4—42no—
356C4—42no—
357C4—41no—
358C4—42no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4#53511yescharacteristic
365C4#53811yescharacteristic
366C4#54611yescharacteristic
367C4#56011yescharacteristic
368C2—42no—
369C2—42no—
370C2#267811yescharacteristic
371C2#318911yescharacteristic
372C2#320111yescharacteristic
373C2#327511yescharacteristic
374C2D4×Q1611yescharacteristic
375C2#329911yescharacteristic
376C2#332711yescharacteristic
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
1. C2×#484C2semidirect
2. C2×#451C2semidirect
5. #2076C2semidirect
6. #2068C2semidirect
7. #2056C2semidirect
8. #2036C2semidirect
9. #2001C2semidirect
10. #2001C2semidirect
11. #1969C2semidirect
13. #1933C2semidirect
14. #1872C2semidirect
15. #1787C2semidirect
129. #103D4semidirect
173. C2×D4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4842
C2×#4512
#24172
#24092
#20762
#20682
#20562
#20362
#20012
#19692
#19542
#19332
#18722
#17872