Finite Group Databasebeta

← #30449

Subgroups — #30449

Order 256.

529 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#494C211yesmaximal, characteristic, complemented
2C2×#462C211yesmaximal, characteristic, complemented
3C2×#451C211yesmaximal, characteristic
4C2×#451C211yesmaximal, characteristic
5#2434C211yesmaximal, characteristic
6#2429C211yesmaximal, characteristic
7#2056C211yesmaximal, characteristic, complemented
8#2043C211yesmaximal, characteristic, complemented
9#2016C211yesmaximal, characteristic, complemented
10#2016C211yesmaximal, characteristic, complemented
11#2001C211yesmaximal, characteristic
12#2001C211yesmaximal, characteristic
13#1933C211yesmaximal, characteristic, complemented
14#1923C211yesmaximal, characteristic
15#1878C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×Q16C2211yescharacteristic
21C22×Q16C2211yescharacteristic
22C22×Q16C2211yescharacteristic
23C2×C4⋊Q8C2211yescharacteristic
24C2×C4⋊Q8C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30#494—24no—
31#462—24no—
32#451—24no—
33#451—24no—
34C2×C4⋊C8C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#398—21no—
47#396—21no—
48#395—21no—
49#395—21no—
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#391C2211yescharacteristic
53#389—21no—
54#389C2211yescharacteristic
55#388—21no—
56#385—21no—
57#385—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×Q8C2311yescharacteristic
73C22×Q8C2311yescharacteristic
74C22×Q8C2311yescharacteristic
75C23×C4—21no—
76C23×C4C2311yescharacteristic
77C2×Q16—22no—
78C2×Q16—22no—
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—22no—
83C2×Q16—22no—
84C2×Q16—22no—
85C2×Q16—41no—
86C2×Q16—41no—
87C2×Q16—41no—
88C2×Q16—41no—
89C2×Q16—41no—
90C2×Q16—41no—
91C2×Q16—41no—
92C2×Q16—41no—
93C2×Q16—41no—
94C2×Q16D414yes—
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C22×C8C2311yescharacteristic
99C4⋊Q8—22no—
100C4⋊Q8—22no—
101C4⋊Q8—22no—
102C4⋊Q8—41no—
103C4⋊Q8—41no—
104C4⋊Q8—41no—
105C22⋊Q8—22no—
106C22⋊Q8—22no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110C22⋊Q8—41no—
111C4×Q8—22no—
112C4×Q8—41no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4C2311yescharacteristic
125C2×C4⋊C4C2311yescharacteristic
126C2×C4⋊C4C2311yescharacteristic
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4D411yescharacteristic
131C2×C22⋊C4D411yescharacteristic
132C2×C42—21no—
133C2×C42C2311yescharacteristic
134C2×C42C2311yescharacteristic
135C4⋊C8—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#103—41no—
141#103—41no—
142#103—41no—
143#103D414yes—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C4×C8—22no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154#95D411yescharacteristic
155#95D411yescharacteristic
156C24C2×D411yescharacteristic
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—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—44no—
190C2×Q8—44no—
191C2×Q8C2×D412yes—
192C2×Q8C2×D412yes—
193C2×Q8C2×D412yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2411yescharacteristic
217Q16—44no—
218Q16—44no—
219Q16—44no—
220Q16—44no—
221Q16—44no—
222Q16—44no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8C2×D412yes—
234C2×C8C2×D412yes—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4C2×D412yes—
252C22⋊C4—22no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4Q1614yescomplemented
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—42no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23C2×Q1612yes—
269C23C22×D411yescharacteristic
270Q8—22no—
271Q8—22no—
272Q8—22no—
273Q8—22no—
274Q8—22no—
275Q8—22no—
276Q8—48no—
277Q8—41no—
278Q8—41no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282Q8—41no—
283Q8—41no—
284Q8—41no—
285Q8—41no—
286Q8—41no—
287Q8—41no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—48no—
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—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4C2×Q1612yes—
333C2×C4C2×Q1612yes—
334C2×C4#13712yes—
335C2×C4C22×D411yescharacteristic
336C2×C4C22×D411yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C4H(5,𝔽2)11yescharacteristic
340C2×C4H(5,𝔽2)11yescharacteristic
341C8—22no—
342C8—22no—
343C8—41no—
344C8—41no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22#53511yescharacteristic
353C22D4211yescharacteristic
354C22D4211yescharacteristic
355C22C22×Q1611yescharacteristic
356C22C2×#13711yescharacteristic
357C22#57811yescharacteristic
358C22#57811yescharacteristic
359C4—22no—
360C4—22no—
361C4—22no—
362C4—41no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—41no—
367C4—42no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4#53411yescharacteristic
373C4#53511yescharacteristic
374C4D4211yescharacteristic
375C4#56011yescharacteristic
376C2—42no—
377C2#266911yescharacteristic
378C2#320911yescharacteristic
379C2#322311yescharacteristic
380C2#327011yescharacteristic
381C2#327611yescharacteristic
382C2D4×Q1611yescharacteristic
383C2D4×Q1611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC24C2×D4
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×#494C2semidirect
2. C2×#462C2semidirect
7. #2056C2semidirect
8. #2043C2semidirect
9. #2016C2semidirect
10. #2016C2semidirect
13. #1933C2semidirect
15. #1878C2semidirect
256. C22⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#4942
C2×#4622
C2×#4512
#24342
#24292
#20562
#20432
#20162
#20012
#19332
#19232
#18782