Finite Group Databasebeta

← #22429

Subgroups — #22429

Order 256.

733 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#548C211yesmaximal, characteristic
2C2×#436C211yesmaximal, characteristic, complemented
3C2×#417C211yesmaximal, characteristic, complemented
4C2×#419C211yesmaximal, characteristic, complemented
5#2352C211yesmaximal, characteristic, complemented
6#1983C211yesmaximal, characteristic, complemented
7#1958C211yesmaximal, characteristic, complemented
8#1895C211yesmaximal, characteristic, complemented
9#1894C211yesmaximal, characteristic, complemented
10#1875C211yesmaximal, characteristic, complemented
11#1872C211yesmaximal, characteristic, complemented
12#1805C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic, complemented
14#1787C211yesmaximal, characteristic, complemented
15C4×#103C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20#548—24no—
21#548C418yescomplemented
22C2×#124—21no—
23C2×#124C2211yescharacteristic
24C2×C22⋊Q8—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×#121—21no—
27C2×#121C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4—21no—
30C2×C4×D4C2211yescharacteristic
31C2×#117C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33#436—24no—
34C2×C4⋊C8C2211yescharacteristic
35#419—24no—
36#417—24no—
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×#102C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×C4×C8C2211yescharacteristic
52#388—21no—
53#387—21no—
54#386—21no—
55#382C2211yescharacteristic
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59C4×C22⋊C4—21no—
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#340C2211yescharacteristic
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—41no—
68C2×#41—41no—
69C2×#41C2×C414yes—
70C22×Q8—21no—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C2×C414yescomplemented
77C22×Q8C2311yescharacteristic
78C22×Q8C2311yescharacteristic
79C22×D4—21no—
80C22×D4—21no—
81C22×D4D411yescharacteristic
82C22×D4C2311yescharacteristic
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C2×M16—22no—
89C2×M16—22no—
90C2×M16—22no—
91C2×M16—22no—
92C2×M16—41no—
93C2×M16—41no—
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98#124—28no—
99#124—22no—
100#124C2×C414yescomplemented
101C22⋊Q8—22no—
102C22⋊Q8—28no—
103C22⋊Q8C2×C414yes—
104#121—28no—
105#121—22no—
106#121C2×C414yescomplemented
107C4×Q8—22no—
108C4×Q8C2×C414yescomplemented
109C4×D4—28no—
110C4×D4C2×C414yes—
111C4×D4D414yes—
112#117—22no—
113#117—41no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4D411yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C22⋊C4—21no—
132C2×C22⋊C4D411yescharacteristic
133C2×C22⋊C4D411yescharacteristic
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42C2311yescharacteristic
138C4⋊C8—22no—
139#103—22no—
140#103—22no—
141#103—22no—
142#103—22no—
143#103—22no—
144#103—41no—
145#103—41no—
146#103D414yescomplemented
147#102—22no—
148#102—22no—
149#102—41no—
150#102—41no—
151C22⋊C8—22no—
152C22⋊C8—22no—
153C4×C8—22no—
154#95—21no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159C24—21no—
160C24C2×D411yescharacteristic
161C24C2×D411yescharacteristic
162#41—28no—
163#41—44no—
164#41—44no—
165#41—44no—
166C2×Q8—24no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—28no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—42no—
177C2×Q8—42no—
178C2×Q8—44no—
179C2×Q8—44no—
180C2×Q8—44no—
181C2×Q8C22×C412yes—
182C2×Q8C22×C412yes—
183C2×Q8C2×D412yes—
184C2×Q8#4112yes—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—21no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4C22⋊C414yes—
199C2×D4C22×C412yes—
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—21no—
209C22×C4—24no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—24no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—81no—
224C22×C4C22×C412yes—
225C22×C4C22×C412yes—
226C22×C4C2×D411yescharacteristic
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D412yes—
231C22×C4C2×D412yes—
232C22×C4C2411yescharacteristic
233M16—44no—
234M16—44no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8C2×D412yes—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—42no—
258C4⋊C4C22⋊C414yes—
259C4⋊C4C22×C412yes—
260C4⋊C4C2×D412yes—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—81no—
268C22⋊C4C22⋊C414yes—
269C22⋊C4C22⋊C414yes—
270C42—24no—
271C42—42no—
272C42—81no—
273C42C22×C412yes—
274C42C2×D412yes—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—22no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23C2×C22⋊C412yes—
295C23C2×C22⋊C412yes—
296C23C22×D411yescharacteristic
297Q8—22no—
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—22no—
302Q8—48no—
303Q8—41no—
304Q8—41no—
305Q8C4×D414yes—
306D4—22no—
307D4—22no—
308D4—44no—
309D4—48no—
310D4—48no—
311D4—41no—
312D4—41no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—84no—
353C2×C4C2×C22⋊C412yes—
354C2×C4C2×C22⋊C412yes—
355C2×C4C2×C22⋊C412yes—
356C2×C4C2×C22⋊C412yes—
357C2×C4C23×C411yescharacteristic
358C2×C4C22×D411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4C2×#4111yescharacteristic
361C2×C4H(5,𝔽2)11yescharacteristic
362C2×C4H(5,𝔽2)11yescharacteristic
363C8—22no—
364C8—41no—
365C8—41no—
366C8—41no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22C22×C22⋊C411yescharacteristic
389C22D4211yescharacteristic
390C22#54611yescharacteristic
391C22#57511yescharacteristic
392C22#57511yescharacteristic
393C22#57711yescharacteristic
394C22#57811yescharacteristic
395C4—22no—
396C4—22no—
397C4—24no—
398C4—22no—
399C4—22no—
400C4—42no—
401C4—42no—
402C4—42no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4C2×C4×D411yescharacteristic
407C4#51811yescharacteristic
408C4D4211yescharacteristic
409C4#54611yescharacteristic
410C2—42no—
411C2—42no—
412C2—82no—
413C2D4×C22⋊C411yescharacteristic
414C2#289611yescharacteristic
415C2#289811yescharacteristic
416C2#327511yescharacteristic
417C2#327711yescharacteristic
418C2#332611yescharacteristic
419C2#332711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#548C2
LCSG ▷ C2×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
2. C2×#436C2semidirect
3. C2×#417C2semidirect
4. C2×#419C2semidirect
5. #2352C2semidirect
6. #1983C2semidirect
7. #1958C2semidirect
8. #1895C2semidirect
9. #1894C2semidirect
10. #1875C2semidirect
11. #1872C2semidirect
12. #1805C2semidirect
13. #1788C2semidirect
14. #1787C2semidirect
15. C4×#103C2semidirect
21. #548C4semidirect
76. C22×Q8C2×C4semidirect
100. #124C2×C4semidirect
106. #121C2×C4semidirect
108. C4×Q8C2×C4semidirect
146. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#4362
C2×#4172
C2×#4192
#23522
#19832
#19582
#18952
#18942
#18752
#18722
#18052
#17882
#17872
C4×#1032