Finite Group Databasebeta

← #21874

Subgroups — #21874

Order 256.

799 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2#2280C211yesmaximal, characteristic, complemented
3C2×#380C211yesmaximal, characteristic, complemented
4#1996C212yesmaximal, complemented
5#1964C212yesmaximal, complemented
6#1898C214yesmaximal, complemented
7#1799C214yesmaximal, complemented
8C22×#41C411yescharacteristic, complemented
9C24×C4C2211yescharacteristic
10#525—21no—
11#525—21no—
12#525—21no—
13#525—21no—
14#525—21no—
15#525—21no—
16#525—21no—
17#525—21no—
18C2×#123C412yescomplemented
19C2×C22⋊Q8C411yescharacteristic, complemented
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C411yescharacteristic, complemented
22C2×#121C2211yescharacteristic
23C2×#120C2212yes—
24C2×C4×D4C412yescomplemented
25C2×C4×D4C2212yes—
26C2×#117—21no—
27C2×#117C411yescharacteristic, complemented
28C22×C4⋊C4—21no—
29C22×C22⋊C4C2212yes—
30C2×#99C2218yes—
31#381—21no—
32#380—21no—
33#380—21no—
34#380—21no—
35#380C2212yes—
36#380C2212yes—
37#380C2212yes—
38#379C2212yes—
39C4×C22⋊C4—21no—
40C4×C22⋊C4C2212yes—
41C2×#95—21no—
42#342C2218yes—
43C25C2311yescharacteristic
44C2×#41—21no—
45C2×#41—21no—
46C2×#41—21no—
47C2×#41—21no—
48C2×#41—44no—
49C22×Q8C2×C411yescharacteristic, complemented
50C22×D4C2×C412yes—
51C22×D4C2×C412yes—
52C22×D4C2×C411yescharacteristic, complemented
53C22×D4C2312yes—
54C23×C4—22no—
55C23×C4—44no—
56C23×C4C2×C411yescharacteristic
57C23×C4C2×C411yescharacteristic
58C23×C4C2×C411yescharacteristic
59C23×C4C2×C411yescharacteristic
60C23×C4C2×C412yescomplemented
61C23×C4D411yescharacteristic
62C23×C4D411yescharacteristic
63C23×C4D411yescharacteristic
64C23×C4D411yescharacteristic
65C23×C4D411yescharacteristic
66C23×C4D411yescharacteristic
67C23×C4D411yescharacteristic
68C23×C4D411yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71#123—22no—
72#123—22no—
73#123—22no—
74#123—22no—
75C22⋊Q8—21no—
76C22⋊Q8—21no—
77C22⋊Q8—21no—
78C22⋊Q8—21no—
79C22⋊Q8—21no—
80C22⋊Q8—21no—
81C22⋊Q8—21no—
82C22⋊Q8—21no—
83#121—21no—
84#121—21no—
85#121—21no—
86#121—21no—
87#121—21no—
88#121—21no—
89#121—21no—
90#121—21no—
91#120—22no—
92#120—22no—
93#120—22no—
94#120—22no—
95C4×D4—22no—
96C4×D4—22no—
97C4×D4—22no—
98C4×D4—22no—
99C4×D4—22no—
100C4×D4—22no—
101C4×D4—22no—
102C4×D4—22no—
103#117—21no—
104#117—21no—
105#117—21no—
106#117—21no—
107#117—44no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—44no—
112C2×C4⋊C4C2×C411yescharacteristic, complemented
113C2×C4⋊C4C2×C411yescharacteristic, complemented
114C2×C4⋊C4C2×C412yescomplemented
115C2×C4⋊C4C2×C412yescomplemented
116C2×C22⋊C4—22no—
117C2×C22⋊C4—22no—
118C2×C22⋊C4—22no—
119C2×C22⋊C4—44no—
120C2×C22⋊C4C2×C412yes—
121C2×C22⋊C4C2×C412yes—
122C2×C22⋊C4C2×C412yescomplemented
123C2×C22⋊C4C2×C412yescomplemented
124C2×C22⋊C4C2314yes—
125C2×C22⋊C4C2314yes—
126C2×C22⋊C4C2312yes—
127C2×C42—22no—
128C2×C42C2×C412yescomplemented
129#99—48no—
130#95—22no—
131#95—22no—
132#95—22no—
133#95—22no—
134C24—22no—
135C24—22no—
136C24—42no—
137C24—42no—
138C24C22×C411yescharacteristic
139C24C22×C411yescharacteristic
140C24C22×C412yes—
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144C24C2×D411yescharacteristic
145C24C2411yescharacteristic
146#41—44no—
147#41—44no—
148#41—44no—
149#41—44no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—42no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—22no—
159C2×D4—24no—
160C2×D4—24no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—42no—
164C2×D4—44no—
165C2×D4—44no—
166C22×C4—24no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—24no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4—44no—
187C22×C4—44no—
188C22×C4C22⋊C411yescharacteristic
189C22×C4C22⋊C411yescharacteristic
190C22×C4C22⋊C411yescharacteristic
191C22×C4C22⋊C411yescharacteristic
192C22×C4C22⋊C411yescharacteristic
193C22×C4C22⋊C411yescharacteristic
194C22×C4C22⋊C411yescharacteristic
195C22×C4C22⋊C411yescharacteristic
196C22×C4C22⋊C411yescharacteristic
197C22×C4C22⋊C411yescharacteristic
198C22×C4C22⋊C411yescharacteristic
199C22×C4C22⋊C411yescharacteristic
200C22×C4C22⋊C411yescharacteristic
201C22×C4C22⋊C411yescharacteristic
202C22×C4C22⋊C411yescharacteristic
203C22×C4C22⋊C411yescharacteristic
204C22×C4C22×C412yes—
205C22×C4C22×C411yescharacteristic
206C22×C4C22×C411yescharacteristic
207C22×C4C22×C412yes—
208C22×C4C22×C412yes—
209C22×C4C22×C411yescharacteristic
210C22×C4C22×C411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—22no—
226C4⋊C4—22no—
227C4⋊C4—44no—
228C4⋊C4—44no—
229C4⋊C4—44no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—24no—
241C22⋊C4—28no—
242C22⋊C4—24no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—44no—
246C42—22no—
247C42—22no—
248C42—44no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—22no—
262C23—22no—
263C23—22no—
264C23—24no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—48no—
274C23—48no—
275C23—42no—
276C23—42no—
277C23C2×C22⋊C411yescharacteristic
278C23C2×C22⋊C411yescharacteristic
279C23C2×C22⋊C411yescharacteristic
280C23C2×C22⋊C411yescharacteristic
281C23C23×C411yescharacteristic
282C23C22×D411yescharacteristic
283C23C22×D411yescharacteristic
284C23H(5,𝔽2)12yes—
285C232−1+412yes—
286Q8—42no—
287Q8—42no—
288D4—42no—
289D4—44no—
290D4—44no—
291D4—42no—
292D4—42no—
293D4—44no—
294D4—44no—
295D4—44no—
296D4—42no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—24no—
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—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—88no—
339C2×C4#9914yescomplemented
340C2×C4C2×C22⋊C411yescharacteristic
341C2×C4C2×C22⋊C411yescharacteristic
342C2×C4C2×C22⋊C411yescharacteristic
343C2×C4C2×C22⋊C411yescharacteristic
344C2×C4C2×C22⋊C411yescharacteristic
345C2×C4C2×C22⋊C411yescharacteristic
346C2×C4C2×C22⋊C411yescharacteristic
347C2×C4C2×C22⋊C411yescharacteristic
348C22—24no—
349C22—24no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—42no—
367C22—42no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—84no—
378C22C2×#9912yes—
379C22#41012yes—
380C22C22×C22⋊C411yescharacteristic
381C22#52012yes—
382C22#53511yescharacteristic
383C22#53611yescharacteristic
384C22#53712yes—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—84no—
395C4—84no—
396C4C2×#9912yes—
397C4C2×#9912yes—
398C2—21no—
399C2—22no—
400C2—21no—
401C2—22no—
402C2—22no—
403C2—22no—
404C2—41no—
405C2—41no—
406C2—82no—
407C2#231111yescharacteristic
408C2C22×#9911yescharacteristic
409C2C2×#41011yescharacteristic
subgroupG/H
center, socleC22C22×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC24C24
prime radicalC2×#525C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2280C2semidirect
3. C2×#380C2semidirect
4. #1996C2semidirect
5. #1964C2semidirect
6. #1898C2semidirect
7. #1799C2semidirect
8. C22×#41C4semidirect
18. C2×#123C4semidirect
19. C2×C22⋊Q8C4semidirect
21. C2×#121C4semidirect
24. C2×C4×D4C4semidirect
27. C2×#117C4semidirect
49. C22×Q8C2×C4semidirect
52. C22×D4C2×C4semidirect
60. C23×C4C2×C4semidirect
112. C2×C4⋊C4C2×C4semidirect
113. C2×C4⋊C4C2×C4semidirect
114. C2×C4⋊C4C2×C4semidirect
115. C2×C4⋊C4C2×C4semidirect
122. C2×C22⋊C4C2×C4semidirect
123. C2×C22⋊C4C2×C4semidirect
128. C2×C42C2×C4semidirect
339. C2×C4#99semidirect

Maximal subgroups

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

subgroupindex
C2×#5252
#22802
C2×#3802
#19962
#19642
#18982
#17992