Finite Group Databasebeta

← #25115

Subgroups — #25115

Order 256.

592 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#465C211yesmaximal, characteristic, complemented
2C2×#449C211yesmaximal, characteristic, complemented
3C2×C8×D4C211yesmaximal, characteristic, complemented
4C2×#416C211yesmaximal, characteristic, complemented
5#2414C211yesmaximal, characteristic, complemented
6#2407C211yesmaximal, characteristic, complemented
7#2065C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2036C211yesmaximal, characteristic, complemented
10#1999C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1933C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic, complemented
14#1893C211yesmaximal, characteristic, complemented
15#1854C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×#121C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C2×#117C2211yescharacteristic
27#465—24no—
28#449—24no—
29C8×D4—24no—
30C2×#107C2211yescharacteristic
31C2×#106C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33#416—24no—
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C4×C8C2211yescharacteristic
45#394—21no—
46#394—21no—
47#392—21no—
48#392—21no—
49#391C2211yescharacteristic
50#390—21no—
51#390C2211yescharacteristic
52#390C2211yescharacteristic
53#389C2211yescharacteristic
54#388—21no—
55#386—21no—
56#385—21no—
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—41no—
68C2×#41—41no—
69C22×Q8C2311yescharacteristic
70C22×D4—21no—
71C22×D4—21no—
72C22×D4—21no—
73C22×D4—21no—
74C22×D4D411yescharacteristic
75C22×D4C2311yescharacteristic
76C22×D4C2311yescharacteristic
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—41no—
83C2×D8—22no—
84C2×D8—22no—
85C2×D8—41no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22⋊Q8—22no—
97C22⋊Q8—41no—
98#121—22no—
99#121—22no—
100#121—22no—
101#121—22no—
102#121—41no—
103#121—41no—
104C4×D4—22no—
105C4×D4—22no—
106#117—22no—
107#117—41no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4D411yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
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×C42C2311yescharacteristic
131#107—41no—
132#106—22no—
133C4⋊C8—22no—
134#103—22no—
135#103—22no—
136#103D414yescomplemented
137#102—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—41no—
142C22⋊C8—22no—
143C22⋊C8D414yescomplemented
144C4×C8—22no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150C24—21no—
151C24—21no—
152C24C2×D411yescharacteristic
153C24C2×D411yescharacteristic
154#41—28no—
155#41—44no—
156#41—44no—
157#41—44no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8C2×D412yes—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—81no—
187C2×D4—81no—
188C2×D4—81no—
189C2×D4—81no—
190C2×D4#4112yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D412yes—
214C22×C4C2×D411yescharacteristic
215C22×C4#4112yes—
216C22×C4C2411yescharacteristic
217SD16—44no—
218D8—44no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—44no—
229C2×C8—44no—
230C2×C8C2×D412yes—
231C2×C8C2×D412yes—
232C2×C8C2×D412yes—
233C4⋊C4—22no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—42no—
244C4⋊C4—81no—
245C4⋊C4C2×D412yes—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—81no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C22⋊C4—81no—
259C42—21no—
260C42—21no—
261C42—42no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—22no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#12112yes—
289C23#13512yes—
290C23C22×D411yescharacteristic
291Q8—22no—
292Q8—22no—
293Q8—41no—
294Q8—41no—
295D4—28no—
296D4—22no—
297D4—22no—
298D4—48no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—41no—
304D4—41no—
305D4—84no—
306D4—84no—
307D4—84no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—48no—
325C2×C4—41no—
326C2×C4—41no—
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#12112yes—
346C2×C4#13512yes—
347C2×C4C22×D411yescharacteristic
348C2×C4C22×D411yescharacteristic
349C2×C4C2×#4111yescharacteristic
350C2×C4C2×#4111yescharacteristic
351C2×C4H(5,𝔽2)11yescharacteristic
352C2×C4H(5,𝔽2)11yescharacteristic
353C8—22no—
354C8—22no—
355C8—22no—
356C8—41no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—42no—
372C22—42no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C2×#12111yescharacteristic
386C22D4211yescharacteristic
387C22#54611yescharacteristic
388C22C2×#13511yescharacteristic
389C22C2×#13511yescharacteristic
390C22#57611yescharacteristic
391C22#57711yescharacteristic
392C4—22no—
393C4—22no—
394C4—41no—
395C4—42no—
396C4—42no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4#52511yescharacteristic
403C4#53811yescharacteristic
404C4D4211yescharacteristic
405C4#54611yescharacteristic
406C2—42no—
407C2—42no—
408C2—82no—
409C2—161no—
410C2#242411yescharacteristic
411C2#305011yescharacteristic
412C2#306211yescharacteristic
413C2#328211yescharacteristic
414C2#328711yescharacteristic
415C2#329711yescharacteristic
416C2#330311yescharacteristic
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×#465C2semidirect
2. C2×#449C2semidirect
3. C2×C8×D4C2semidirect
4. C2×#416C2semidirect
5. #2414C2semidirect
6. #2407C2semidirect
7. #2065C2semidirect
8. #2055C2semidirect
9. #2036C2semidirect
10. #1999C2semidirect
11. #1970C2semidirect
12. #1933C2semidirect
13. #1893C2semidirect
14. #1893C2semidirect
15. #1854C2semidirect
136. #103D4semidirect
143. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
C2×#4492
C2×C8×D42
C2×#4162
#24142
#24072
#20652
#20552
#20362
#19992
#19702
#19332
#18932
#18542