Finite Group Databasebeta

← #23965

Subgroups — #23965

Order 256.

666 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#555C211yesmaximal, characteristic, complemented
2C2×#449C211yesmaximal, characteristic, complemented
3C2×#416C211yesmaximal, characteristic, complemented
4#2407C211yesmaximal, characteristic, complemented
5#2024C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#2000C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1875C211yesmaximal, characteristic, complemented
13#1873C211yesmaximal, characteristic, complemented
14#1869C211yesmaximal, characteristic, complemented
15#1825C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#555—24no—
21#555—24no—
22C2×C4⋊Q8C2211yescharacteristic
23C2×#124—21no—
24C2×#124—21no—
25C2×C22⋊Q8C2211yescharacteristic
26C2×#121—21no—
27C2×#121C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×#121C2211yescharacteristic
30C2×C4×D4C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×#117—21no—
33C2×#117C2211yescharacteristic
34C2×#117C2211yescharacteristic
35#449—24no—
36#416—24no—
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48#394—21no—
49#392—21no—
50#391C2211yescharacteristic
51#390C2211yescharacteristic
52#386—21no—
53#385—21no—
54#380C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—41no—
68C2×#41—41no—
69C22×Q8—21no—
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—21no—
74C22×D4—21no—
75C22×D4D411yescharacteristic
76C22×D4C2311yescharacteristic
77C22×D4C2311yescharacteristic
78C23×C4—21no—
79C23×C4—21no—
80C23×C4C2311yescharacteristic
81C23×C4C2311yescharacteristic
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—41no—
85C2×D8—22no—
86C2×D8—22no—
87C2×D8—41no—
88C22×C8—22no—
89C22×C8—41no—
90C22×C8—41no—
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C4⋊Q8—22no—
96C4⋊Q8—22no—
97#124—28no—
98#124—28no—
99C22⋊Q8—22no—
100C22⋊Q8—41no—
101#121—28no—
102#121—22no—
103#121—22no—
104#121—22no—
105#121—22no—
106#121—41no—
107#121D414yescomplemented
108C4×D4—22no—
109C4×D4—22no—
110C4×D4—22no—
111C4×D4—22no—
112#117—28no—
113#117—22no—
114#117—22no—
115#117—22no—
116#117D414yescomplemented
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
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⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4D411yescharacteristic
136C2×C22⋊C4D411yescharacteristic
137C2×C42—21no—
138C2×C42—21no—
139C2×C42C2311yescharacteristic
140#103—22no—
141#103—22no—
142#103—22no—
143#102—22no—
144#102—22no—
145#102—22no—
146#102—41no—
147#102—41no—
148C22⋊C8—22no—
149C22⋊C8—41no—
150C22⋊C8—41no—
151#95—21no—
152#95—21no—
153#95—21no—
154C24—21no—
155C24—21no—
156C24C2×D411yescharacteristic
157C24C2×D411yescharacteristic
158#41—28no—
159#41—44no—
160#41—44no—
161#41—44no—
162C2×Q8—24no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—81no—
169C2×Q8—81no—
170C2×Q8#4112yes—
171C2×D4—22no—
172C2×D4—24no—
173C2×D4—24no—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—81no—
192C2×D4—81no—
193C2×D4—81no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4C2×D412yes—
197C2×D4#4112yes—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—24no—
203C22×C4—21no—
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—21no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2×D411yescharacteristic
223C22×C4C2×D412yes—
224C22×C4C2×D412yes—
225C22×C4C2×D411yescharacteristic
226C22×C4C2411yescharacteristic
227SD16—44no—
228D8—44no—
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—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—44no—
242C4⋊C4—24no—
243C4⋊C4—22no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—42no—
252C4⋊C4C2×D412yes—
253C4⋊C4C2×D412yes—
254C4⋊C4C2×D412yes—
255C22⋊C4—24no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—22no—
260C22⋊C4—24no—
261C22⋊C4—22no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—81no—
266C22⋊C4—81no—
267C42—24no—
268C42—24no—
269C42—21no—
270C42—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—22no—
279C23—21no—
280C23—22no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#12012yes—
299C23C22×D411yescharacteristic
300Q8—22no—
301Q8—22no—
302Q8—41no—
303Q8—41no—
304Q8—84no—
305D4—22no—
306D4—22no—
307D4—22no—
308D4—22no—
309D4—44no—
310D4—48no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—41no—
315D4—84no—
316D4—84no—
317D4—84no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—48no—
337C2×C4—48no—
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—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4#12012yes—
357C2×C4#13512yes—
358C2×C4C22×D411yescharacteristic
359C2×C4C22×D411yescharacteristic
360C2×C4C2×#4111yescharacteristic
361C2×C4C2×#4111yescharacteristic
362C2×C4H(5,𝔽2)11yescharacteristic
363C2×C4H(5,𝔽2)11yescharacteristic
364C8—41no—
365C8—41no—
366C8—41no—
367C8—41no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—42no—
376C22—41no—
377C22—42no—
378C22—42no—
379C22—41no—
380C22—41no—
381C22—42no—
382C22—41no—
383C22—42no—
384C22—41no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22C2×#12011yescharacteristic
397C22#54611yescharacteristic
398C22#54611yescharacteristic
399C22C2×#13511yescharacteristic
400C22#57511yescharacteristic
401C22#57611yescharacteristic
402C22#57711yescharacteristic
403C4—22no—
404C4—22no—
405C4—41no—
406C4—42no—
407C4—42no—
408C4—41no—
409C4—42no—
410C4—42no—
411C4—81no—
412C4—81no—
413C4#52511yescharacteristic
414C4#53511yescharacteristic
415C4#54611yescharacteristic
416C4#54611yescharacteristic
417C2—42no—
418C2—81no—
419C2—82no—
420C2—82no—
421C2#240411yescharacteristic
422C2#300211yescharacteristic
423C2#301211yescharacteristic
424C2#330311yescharacteristic
425C2#330511yescharacteristic
426C2#331911yescharacteristic
427C2#332111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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
1. C2×#555C2semidirect
2. C2×#449C2semidirect
3. C2×#416C2semidirect
4. #2407C2semidirect
5. #2024C2semidirect
6. #2023C2semidirect
7. #2000C2semidirect
8. #1999C2semidirect
9. #1893C2semidirect
10. #1879C2semidirect
11. #1877C2semidirect
12. #1875C2semidirect
13. #1873C2semidirect
14. #1869C2semidirect
15. #1825C2semidirect
107. #121D4semidirect
116. #117D4semidirect

Maximal subgroups

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

subgroupindex
C2×#5552
C2×#4492
C2×#4162
#24072
#20242
#20232
#20002
#19992
#18932
#18792
#18772
#18752
#18732
#18692
#18252