Finite Group Databasebeta

← #24392

Subgroups — #24392

Order 256.

693 conjugacy classes of subgroups.

The subgroup lattice has 437 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#536C211yesmaximal, characteristic
2C2×#470C211yesmaximal, characteristic, complemented
3C2×#467C211yesmaximal, characteristic, complemented
4C2×#451C211yesmaximal, characteristic, complemented
5C2×#408C211yesmaximal, characteristic
6#2616C211yesmaximal, characteristic
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2001C211yesmaximal, characteristic, complemented
11#1894C211yesmaximal, characteristic, complemented
12#1892C211yesmaximal, characteristic, complemented
13#1873C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C23×C8C2211yescharacteristic
22#536—24no—
23#536—24no—
24C2×C4⋊Q8—21no—
25C2×#124—21no—
26C2×#123—21no—
27C2×#123—21no—
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2211yescharacteristic
30C2×C22⋊Q8C2211yescharacteristic
31C2×C22⋊Q8C2211yescharacteristic
32C2×C22⋊Q8C2211yescharacteristic
33C2×C22⋊Q8C2211yescharacteristic
34C2×C22⋊Q8C2211yescharacteristic
35C2×#117C2211yescharacteristic
36#470—24no—
37#467—24no—
38#451—24no—
39C2×#107C2211yescharacteristic
40C2×#106C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#103C2211yescharacteristic
47C2×#103C2211yescharacteristic
48C2×#103C2211yescharacteristic
49#408—24no—
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×C22⋊C8C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54#395—21no—
55#394—21no—
56#394—21no—
57#394—21no—
58#380C2211yescharacteristic
59#380C2211yescharacteristic
60#380C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63#340C2211yescharacteristic
64#340C2211yescharacteristic
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—41no—
70C2×#41—41no—
71C22×Q8—22no—
72C22×Q8—22no—
73C22×Q8—21no—
74C22×Q8—22no—
75C22×Q8—22no—
76C22×Q8—41no—
77C22×Q8—41no—
78C22×Q8D411yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×Q8C2311yescharacteristic
81C22×Q8C2311yescharacteristic
82C22×D4D411yescharacteristic
83C22×D4D411yescharacteristic
84C22×D4D411yescharacteristic
85C23×C4—21no—
86C23×C4C2311yescharacteristic
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16—41no—
94C2×Q16—41no—
95C2×M16—22no—
96C2×M16—22no—
97C2×M16—41no—
98C22×C8—22no—
99C22×C8—22no—
100C22×C8—41no—
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C4⋊Q8—28no—
106#124—28no—
107#123—28no—
108#123—28no—
109C22⋊Q8—22no—
110C22⋊Q8—22no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C22⋊Q8—22no—
116C22⋊Q8—22no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C22⋊Q8—41no—
120C22⋊Q8—41no—
121C22⋊Q8D414yes—
122C22⋊Q8D414yes—
123#117—22no—
124#117—22no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4C2311yescharacteristic
133C2×C4⋊C4C2311yescharacteristic
134C2×C4⋊C4C2311yescharacteristic
135C2×C4⋊C4C2311yescharacteristic
136C2×C4⋊C4C2311yescharacteristic
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C42—21no—
149#107—22no—
150#106—22no—
151#103—22no—
152#103—22no—
153#103—22no—
154#103—22no—
155#103—22no—
156#103—22no—
157#103—41no—
158#103—41no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161C22⋊C8—22no—
162C22⋊C8D414yes—
163#95—21no—
164#95—21no—
165#95—21no—
166C24C2×D411yescharacteristic
167C24C2×D411yescharacteristic
168C24C2×D411yescharacteristic
169#41—44no—
170#41—44no—
171#41—44no—
172#41—44no—
173C2×Q8—24no—
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—21no—
188C2×Q8—21no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—41no—
193C2×Q8—44no—
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—44no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8C2×D412yes—
200C2×Q8C2×D412yes—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—24no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4C2×D411yescharacteristic
236C22×C4C2×D411yescharacteristic
237C22×C4C2×D412yes—
238C22×C4C2×D412yes—
239C22×C4C2×D411yescharacteristic
240C22×C4C2×D412yes—
241C22×C4#4112yes—
242C22×C4C2411yescharacteristic
243Q16—44no—
244Q16—44no—
245M16—44no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—44no—
257C2×C8C2×D412yes—
258C2×C8C2×D412yes—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4C2×D412yes—
275C4⋊C4C2×D412yes—
276C22⋊C4—24no—
277C22⋊C4—24no—
278C22⋊C4—24no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—42no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C22⋊C4—81no—
287C42—24no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23#12012yes—
308C23#12112yes—
309C23C22×D411yescharacteristic
310Q8—22no—
311Q8—22no—
312Q8—22no—
313Q8—22no—
314Q8—44no—
315Q8—48no—
316Q8—41no—
317Q8—41no—
318Q8—41no—
319Q8—41no—
320Q8—41no—
321Q8—41no—
322Q8—41no—
323Q8—41no—
324D4—44no—
325D4—44no—
326D4—44no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—48no—
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—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4#12012yes—
372C2×C4#12112yes—
373C2×C4C22×D411yescharacteristic
374C2×C4C22×D411yescharacteristic
375C2×C4C22×D411yescharacteristic
376C2×C4C2×#4111yescharacteristic
377C2×C4H(5,𝔽2)11yescharacteristic
378C2×C4H(5,𝔽2)11yescharacteristic
379C8—22no—
380C8—22no—
381C8—41no—
382C8—41no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22C2×#12011yescharacteristic
405C22C2×#12111yescharacteristic
406C22#53411yescharacteristic
407C22#57711yescharacteristic
408C22#57811yescharacteristic
409C22#57811yescharacteristic
410C22#57811yescharacteristic
411C4—22no—
412C4—22no—
413C4—41no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4—81no—
422C4D4211yescharacteristic
423C4D4211yescharacteristic
424C4#54611yescharacteristic
425C4#54611yescharacteristic
426C2—42no—
427C2—42no—
428C2—81no—
429C2#241011yescharacteristic
430C2#301111yescharacteristic
431C2#301311yescharacteristic
432C2#305911yescharacteristic
433C2#306311yescharacteristic
434C2#320111yescharacteristic
435C2#320211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#470C2semidirect
3. C2×#467C2semidirect
4. C2×#451C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #2001C2semidirect
10. #2001C2semidirect
11. #1894C2semidirect
12. #1892C2semidirect
13. #1873C2semidirect
14. #1854C2semidirect
15. #1787C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5362
C2×#4702
C2×#4672
C2×#4512
C2×#4082
#26162
#20362
#20352
#20012
#18942
#18922
#18732
#18542
#17872