Finite Group Databasebeta

← #25410

Subgroups — #25410

Order 256.

552 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#492C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#465C211yesmaximal, characteristic, complemented
4C2×C4×SD16C211yesmaximal, characteristic, complemented
5#2429C211yesmaximal, characteristic, complemented
6#2428C211yesmaximal, characteristic
7#2043C211yesmaximal, characteristic, complemented
8#2041C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1968C211yesmaximal, characteristic, complemented
12#1941C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1877C211yesmaximal, characteristic, complemented
15C8×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×#127C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121—21no—
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30#492—24no—
31#465—24no—
32#465—24no—
33C4×SD16—24no—
34C2×#106C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46C2×C4×C8C2211yescharacteristic
47#398—21no—
48#396—21no—
49#394—21no—
50#392—21no—
51#391C2211yescharacteristic
52#390C2211yescharacteristic
53#389—21no—
54#389C2211yescharacteristic
55#389C2211yescharacteristic
56#388—21no—
57#386—21no—
58C4×C22⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4—21no—
67C22×D4—21no—
68C22×D4—21no—
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—22no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—41no—
83C2×SD16—41no—
84C2×SD16D414yescomplemented
85C22×C8—22no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C4⋊Q8—22no—
93C4⋊Q8—41no—
94#127—22no—
95#127—41no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8—41no—
99C22⋊Q8—41no—
100#121—22no—
101#121—22no—
102#121—41no—
103#121—41no—
104#121—44no—
105C4×Q8—22no—
106C4×Q8—41no—
107C4×D4—22no—
108C4×D4—41no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4D411yescharacteristic
128C2×C22⋊C4D411yescharacteristic
129C2×C42—21no—
130C2×C42C2311yescharacteristic
131C2×C42C2311yescharacteristic
132#106—22no—
133#106D414yescomplemented
134#103—22no—
135#103—22no—
136#103—22no—
137#103—41no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—41no—
142C22⋊C8—22no—
143C4×C8—22no—
144C4×C8—22no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95D411yescharacteristic
150#95D411yescharacteristic
151C24—21no—
152C24—21no—
153C24C2×D411yescharacteristic
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—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8C2×D412yes—
166C2×D4—21no—
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—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—81no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4C2×D412yes—
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—21no—
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—81no—
205C22×C4C2×D411yescharacteristic
206C22×C4C2×D411yescharacteristic
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2411yescharacteristic
211SD16—28no—
212SD16—44no—
213SD16—44no—
214SD16—44no—
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×C8—44no—
224C2×C8C2×D412yes—
225C2×C8C2×D412yes—
226C2×C8#4112yes—
227C2×C8#4112yes—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4C2×D412yes—
242C4⋊C4C2×D412yes—
243C22⋊C4—22no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—81no—
251C22⋊C4SD1614yescomplemented
252C42—21no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—42no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×SD1612yes—
278C23C22×D411yescharacteristic
279Q8—22no—
280Q8—22no—
281Q8—41no—
282Q8—41no—
283Q8—41no—
284Q8—41no—
285Q8—41no—
286Q8—41no—
287D4—22no—
288D4—22no—
289D4—41no—
290D4—41no—
291D4—41no—
292D4—41no—
293D4—41no—
294D4—41no—
295D4—84no—
296D4—84no—
297D4—84no—
298D4—84no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
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×C4—41no—
333C2×C4—84no—
334C2×C4C2×SD1612yes—
335C2×C4C2×SD1612yes—
336C2×C4#13512yes—
337C2×C4C22×D411yescharacteristic
338C2×C4C22×D411yescharacteristic
339C2×C4C2×#4111yescharacteristic
340C2×C4C2×#4111yescharacteristic
341C2×C4H(5,𝔽2)11yescharacteristic
342C2×C4H(5,𝔽2)11yescharacteristic
343C8—22no—
344C8—41no—
345C8—41no—
346C8#12114yescomplemented
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—42no—
355C22—42no—
356C22—42no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22#53211yescharacteristic
369C22D4211yescharacteristic
370C22D4211yescharacteristic
371C22C22×SD1611yescharacteristic
372C22C2×#13511yescharacteristic
373C22#57711yescharacteristic
374C22#57711yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—42no—
380C4—42no—
381C4—41no—
382C4—42no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4C2×#12111yescharacteristic
387C4#54011yescharacteristic
388C4#54611yescharacteristic
389C4#55011yescharacteristic
390C2—42no—
391C2—82no—
392C2—161no—
393C2#242811yescharacteristic
394C2#316711yescharacteristic
395C2#317411yescharacteristic
396C2D4×SD1611yescharacteristic
397C2D4×SD1611yescharacteristic
398C2#328211yescharacteristic
399C2#328411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2428C2
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×#492C2semidirect
2. C2×#465C2semidirect
3. C2×#465C2semidirect
4. C2×C4×SD16C2semidirect
5. #2429C2semidirect
7. #2043C2semidirect
8. #2041C2semidirect
9. #2000C2semidirect
10. #2000C2semidirect
11. #1968C2semidirect
12. #1941C2semidirect
13. #1880C2semidirect
14. #1877C2semidirect
15. C8×C22⋊C4C2semidirect
84. C2×SD16D4semidirect
133. #106D4semidirect
251. C22⋊C4SD16semidirect
346. C8#121semidirect

Maximal subgroups

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

subgroupindex
C2×#4922
C2×#4652
C2×C4×SD162
#24292
#24282
#20432
#20412
#20002
#19682
#19412
#18802
#18772
C8×C22⋊C42