Finite Group Databasebeta

← #21420

Subgroups — #21420

Order 256.

839 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#525C211yesmaximal, characteristic
2#2280C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C4C211yesmaximal, characteristic, complemented
4#1996C212yesmaximal, complemented
5#1964C212yesmaximal, complemented
6#1898C214yesmaximal, complemented
7C4×#99C214yesmaximal, 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×C22⋊C4C2212yes—
29C22×C42—21no—
30C2×#99C2218yes—
31#381—21no—
32#380—21no—
33#380C2212yes—
34#379C2212yes—
35C4×C22⋊C4—21no—
36C4×C22⋊C4—21no—
37C4×C22⋊C4—21no—
38C4×C22⋊C4C2212yes—
39C4×C22⋊C4C2212yes—
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⋊C4C2×C411yescharacteristic, complemented
110C2×C4⋊C4C2×C411yescharacteristic, complemented
111C2×C4⋊C4C2×C412yescomplemented
112C2×C4⋊C4C2×C412yescomplemented
113C2×C22⋊C4—22no—
114C2×C22⋊C4—28no—
115C2×C22⋊C4—22no—
116C2×C22⋊C4—22no—
117C2×C22⋊C4C2×C412yes—
118C2×C22⋊C4C2×C412yes—
119C2×C22⋊C4C2×C412yescomplemented
120C2×C22⋊C4C2×C412yescomplemented
121C2×C22⋊C4C2314yes—
122C2×C22⋊C4C2312yes—
123C2×C22⋊C4C2314yes—
124C2×C42—22no—
125C2×C42—22no—
126C2×C42—22no—
127C2×C42—44no—
128C2×C42C2×C412yescomplemented
129#99—216no—
130#95—22no—
131#95—22no—
132#95—22no—
133#95—22no—
134C24—24no—
135C24—42no—
136C24—42no—
137C24C22×C411yescharacteristic
138C24C22×C411yescharacteristic
139C24C22×C412yes—
140C24C2×D411yescharacteristic
141C24C2×D411yescharacteristic
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144C24C2411yescharacteristic
145#41—44no—
146#41—44no—
147#41—44no—
148#41—44no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—42no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—22no—
157C2×D4—22no—
158C2×D4—28no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—42no—
162C2×D4—44no—
163C2×D4—44no—
164C22×C4—24no—
165C22×C4—24no—
166C22×C4—24no—
167C22×C4—28no—
168C22×C4—24no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—24no—
172C22×C4—42no—
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—44no—
184C22×C4—44no—
185C22×C4—44no—
186C22×C4C22⋊C411yescharacteristic
187C22×C4C22⋊C411yescharacteristic
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×C412yes—
203C22×C4C22×C411yescharacteristic
204C22×C4C22×C411yescharacteristic
205C22×C4C22×C412yes—
206C22×C4C22×C412yes—
207C22×C4C22×C411yescharacteristic
208C22×C4C22×C411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—44no—
226C22⋊C4—28no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—24no—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—28no—
239C22⋊C4—44no—
240C22⋊C4—44no—
241C22⋊C4#4118yescomplemented
242C42—22no—
243C42—22no—
244C42—44no—
245C42—44no—
246C42—44no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—24no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23—416no—
270C23—42no—
271C23—42no—
272C23C2×C22⋊C411yescharacteristic
273C23C2×C22⋊C411yescharacteristic
274C23C2×C22⋊C411yescharacteristic
275C23C2×C22⋊C411yescharacteristic
276C23C23×C411yescharacteristic
277C23C22×D411yescharacteristic
278C23C22×D411yescharacteristic
279C23C2×#4114yes—
280Q8—42no—
281Q8—42no—
282D4—42no—
283D4—44no—
284D4—44no—
285D4—44no—
286D4—44no—
287D4—44no—
288D4—44no—
289D4—42no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
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—24no—
309C2×C4—28no—
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—44no—
323C2×C4—44no—
324C2×C4—416no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4#9914yescomplemented
333C2×C4C2×C22⋊C411yescharacteristic
334C2×C4C2×C22⋊C411yescharacteristic
335C2×C4C2×C22⋊C411yescharacteristic
336C2×C4C2×C22⋊C411yescharacteristic
337C2×C4C2×C22⋊C411yescharacteristic
338C2×C4C2×C22⋊C411yescharacteristic
339C2×C4C2×C22⋊C411yescharacteristic
340C2×C4C2×C22⋊C411yescharacteristic
341C22—28no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—44no—
362C22—44no—
363C22—44no—
364C22—42no—
365C22—42no—
366C22—42no—
367C22—84no—
368C22C2×#9912yes—
369C22#41012yes—
370C22C22×C22⋊C411yescharacteristic
371C22C4×#4112yes—
372C22#52512yes—
373C22#53212yes—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—48no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—84no—
385C4C2×#9912yes—
386C4C2×#9912yes—
387C2—22no—
388C2—22no—
389C2—22no—
390C2—22no—
391C2—22no—
392C2—41no—
393C2—41no—
394C2—82no—
395C2#229711yescharacteristic
396C2C22×#9911yescharacteristic
397C2C2×#41011yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊C4
commutatorC23C23×C4
Frattini, cosocleC24C24
socleC22C22×C22⋊C4
prime radicalC2×#525C2
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ 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×C4×C22⋊C4C2semidirect
4. #1996C2semidirect
5. #1964C2semidirect
6. #1898C2semidirect
7. C4×#99C2semidirect
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
109. C2×C4⋊C4C2×C4semidirect
110. C2×C4⋊C4C2×C4semidirect
111. C2×C4⋊C4C2×C4semidirect
112. C2×C4⋊C4C2×C4semidirect
119. C2×C22⋊C4C2×C4semidirect
120. C2×C22⋊C4C2×C4semidirect
128. C2×C42C2×C4semidirect
241. C22⋊C4#41semidirect
332. 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×C4×C22⋊C42
#19962
#19642
#18982
C4×#992