Finite Group Databasebeta

← #24040

Subgroups — #24040

Order 256.

661 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#548C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#416C211yesmaximal, characteristic, complemented
4#2485C211yesmaximal, characteristic, complemented
5#2042C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1892C211yesmaximal, characteristic, complemented
10#1869C211yesmaximal, characteristic, complemented
11#1865C211yesmaximal, characteristic, complemented
12#1863C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic, complemented
14#1787C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C23×C8C2211yescharacteristic
19#548—24no—
20#548—24no—
21C2×#124—21no—
22C2×#124C2211yescharacteristic
23C2×C22⋊Q8—21no—
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121—21no—
26C2×#121C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4C2211yescharacteristic
30C2×#117C2211yescharacteristic
31#485—24no—
32C2×#107C2211yescharacteristic
33C2×#106C2211yescharacteristic
34#416—24no—
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50#397—21no—
51#394—21no—
52#389C2211yescharacteristic
53#389C2211yescharacteristic
54#388—21no—
55#387—21no—
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—41no—
67C2×#41—41no—
68C2×#41D414yes—
69C22×Q8—22no—
70C22×Q8—22no—
71C22×Q8—21no—
72C22×Q8—22no—
73C22×Q8—22no—
74C22×Q8—41no—
75C22×Q8—41no—
76C22×Q8C2311yescharacteristic
77C22×Q8C2311yescharacteristic
78C22×D4—21no—
79C22×D4—21no—
80C22×D4D411yescharacteristic
81C22×D4C2311yescharacteristic
82C23×C4—21no—
83C23×C4—21no—
84C23×C4C2311yescharacteristic
85C23×C4C2311yescharacteristic
86C22×C8—22no—
87C22×C8—41no—
88C22×C8—41no—
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93#124—28no—
94#124—22no—
95#124—22no—
96C22⋊Q8—28no—
97C22⋊Q8—22no—
98C22⋊Q8D414yes—
99#121—28no—
100#121—22no—
101#121—22no—
102C4×Q8—22no—
103C4×Q8—22no—
104C4×D4—28no—
105C4×D4—22no—
106C4×D4—22no—
107#117—22no—
108#117—22no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C22⋊C4—21no—
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×C42—21no—
131C2×C42C2311yescharacteristic
132#107—22no—
133#106—22no—
134#103—22no—
135#103—22no—
136#103—22no—
137#103—22no—
138#103—22no—
139#103—22no—
140#103—41no—
141#103—41no—
142#102—22no—
143#102—22no—
144#102—41no—
145#102—41no—
146C22⋊C8—22no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154C24—21no—
155C24C2×D411yescharacteristic
156C24C2×D411yescharacteristic
157#41—28no—
158#41—44no—
159#41—44no—
160#41—44no—
161C2×Q8—24no—
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—42no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×Q8—44no—
175C2×Q8—44no—
176C2×Q8C2×D412yes—
177C2×Q8C2×D412yes—
178C2×Q8#4112yes—
179C2×Q8#4112yes—
180C2×D4—24no—
181C2×D4—22no—
182C2×D4—24no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4C2×D412yes—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D412yes—
223C22×C4C2×D412yes—
224C22×C4C2411yescharacteristic
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×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—44no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—42no—
251C4⋊C4—81no—
252C4⋊C4C2×D412yes—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—81no—
262C22⋊C4—81no—
263C42—24no—
264C42—21no—
265C42—21no—
266C42—42no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23#12012yes—
287C23#13512yes—
288C23C22×D411yescharacteristic
289Q8—22no—
290Q8—22no—
291Q8—22no—
292Q8—22no—
293Q8—22no—
294Q8—48no—
295Q8—41no—
296Q8—41no—
297Q8#12314yes—
298D4—22no—
299D4—22no—
300D4—48no—
301D4—44no—
302D4—48no—
303D4—41no—
304D4—41no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—48no—
326C2×C4—48no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
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#12012yes—
348C2×C4C22×D411yescharacteristic
349C2×C4C22×D411yescharacteristic
350C2×C4C2×#4111yescharacteristic
351C2×C4C2×#4111yescharacteristic
352C2×C4H(5,𝔽2)11yescharacteristic
353C2×C4H(5,𝔽2)11yescharacteristic
354C8—41no—
355C8—41no—
356C8—41no—
357C8—41no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C2×#12011yescharacteristic
380C22#54611yescharacteristic
381C22#54611yescharacteristic
382C22C2×#13511yescharacteristic
383C22#57511yescharacteristic
384C22#57711yescharacteristic
385C22#57811yescharacteristic
386C4—22no—
387C4—24no—
388C4—41no—
389C4—42no—
390C4—41no—
391C4—42no—
392C4—41no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4C2×#12311yescharacteristic
398C4#53411yescharacteristic
399C4#54611yescharacteristic
400C4#54611yescharacteristic
401C2—42no—
402C2—42no—
403C2—82no—
404C2#240511yescharacteristic
405C2#300211yescharacteristic
406C2#301311yescharacteristic
407C2#330011yescharacteristic
408C2#330111yescharacteristic
409C2#331311yescharacteristic
410C2#332111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#548C2
LCSG ▷ C22×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×#485C2semidirect
3. C2×#416C2semidirect
4. #2485C2semidirect
5. #2042C2semidirect
6. #2042C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1892C2semidirect
10. #1869C2semidirect
11. #1865C2semidirect
12. #1863C2semidirect
13. #1788C2semidirect
14. #1787C2semidirect
15. #1787C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5482
C2×#4852
C2×#4162
#24852
#20422
#20342
#20332
#18922
#18692
#18652
#18632
#17882
#17872