Finite Group Databasebeta

← #25110

Subgroups — #25110

Order 256.

561 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#467C211yesmaximal, characteristic, complemented
2C2×#451C211yesmaximal, characteristic, complemented
3C2×C8×D4C211yesmaximal, characteristic
4C22×#103C211yesmaximal, characteristic, complemented
5#2417C211yesmaximal, characteristic
6#2409C211yesmaximal, characteristic
7#2068C211yesmaximal, characteristic, complemented
8#2056C211yesmaximal, characteristic, complemented
9#2036C211yesmaximal, characteristic, complemented
10#2001C211yesmaximal, characteristic, complemented
11#1969C211yesmaximal, characteristic, complemented
12#1933C211yesmaximal, characteristic, complemented
13#1892C211yesmaximal, characteristic, complemented
14#1892C211yesmaximal, characteristic, complemented
15#1852C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#467—24no—
28#451—24no—
29C8×D4—24no—
30C2×#107C2211yescharacteristic
31C2×#107C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33C2×#103—24no—
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#398—21no—
46#396—21no—
47#395—21no—
48#393—21no—
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#389C2211yescharacteristic
53#386—21no—
54#386—21no—
55#385—21no—
56#380—21no—
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×Q8C2311yescharacteristic
73C22×D4D411yescharacteristic
74C23×C4—21no—
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C23×C4C2311yescharacteristic
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—41no—
83C2×Q16—41no—
84C22×C8—22no—
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—41no—
89C22×C8—41no—
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22⋊Q8—22no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8—22no—
99C22⋊Q8—41no—
100C22⋊Q8—41no—
101C22⋊Q8—41no—
102C4×D4—22no—
103C4×D4—22no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
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×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4D411yescharacteristic
125C2×C22⋊C4D411yescharacteristic
126C2×C42C2311yescharacteristic
127#107—22no—
128#107—41no—
129C4⋊C8—22no—
130#103—22no—
131#103—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#103—41no—
137#103D414yescomplemented
138C22⋊C8—22no—
139C22⋊C8D414yes—
140C4×C8—22no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150C24C2×D411yescharacteristic
151C24C2×D411yescharacteristic
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—28no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—42no—
170C2×Q8—42no—
171C2×Q8—44no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—42no—
175C2×Q8C2×D412yes—
176C2×Q8#4112yes—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4Q1614yescomplemented
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—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—81no—
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D412yes—
208C22×C4C2×D411yescharacteristic
209C22×C4#4112yes—
210C22×C4C2411yescharacteristic
211Q16—44no—
212Q16—44no—
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—44no—
223C2×C8—44no—
224C2×C8C2×D412yes—
225C2×C8C2×D412yes—
226C2×C8C2×D412yes—
227C4⋊C4—22no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—81no—
243C4⋊C4C2×D412yes—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—81no—
252C42—21no—
253C42—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23#12112yes—
265C23C2×Q1612yes—
266C23C2×Q1612yes—
267C23#13512yes—
268C23C22×D411yescharacteristic
269Q8—22no—
270Q8—22no—
271Q8—22no—
272Q8—22no—
273Q8—48no—
274Q8—41no—
275Q8—41no—
276Q8—41no—
277Q8—41no—
278Q8—41no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282D4—28no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—48no—
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—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—84no—
327C2×C4#12112yes—
328C2×C4C2×Q1612yes—
329C2×C4C22×D411yescharacteristic
330C2×C4C22×D411yescharacteristic
331C2×C4C2×#4111yescharacteristic
332C2×C4C2×#4111yescharacteristic
333C2×C4H(5,𝔽2)11yescharacteristic
334C2×C4H(5,𝔽2)11yescharacteristic
335C8—22no—
336C8—22no—
337C8—22no—
338C8—41no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#46714yescomplemented
353C22C2×#12111yescharacteristic
354C22D4211yescharacteristic
355C22#54611yescharacteristic
356C22C22×Q1611yescharacteristic
357C22C2×#13511yescharacteristic
358C22#57711yescharacteristic
359C22#57811yescharacteristic
360C4—22no—
361C4—22no—
362C4—41no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4#52511yescharacteristic
373C4#53811yescharacteristic
374C4D4211yescharacteristic
375C4#54611yescharacteristic
376C2—24no—
377C2—42no—
378C2#242411yescharacteristic
379C2C2×#46711yescharacteristic
380C2#306311yescharacteristic
381C2#328211yescharacteristic
382C2D4×Q1611yescharacteristic
383C2#329911yescharacteristic
384C2#330011yescharacteristic
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×#467C2semidirect
2. C2×#451C2semidirect
4. C22×#103C2semidirect
7. #2068C2semidirect
8. #2056C2semidirect
9. #2036C2semidirect
10. #2001C2semidirect
11. #1969C2semidirect
12. #1933C2semidirect
13. #1892C2semidirect
14. #1892C2semidirect
15. #1852C2semidirect
137. #103D4semidirect
179. C2×D4Q16semidirect
352. C22#467semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×#4512
C2×C8×D42
C22×#1032
#24172
#24092
#20682
#20562
#20362
#20012
#19692
#19332
#18922
#18522