Finite Group Databasebeta

← #24221

Subgroups — #24221

Order 256.

682 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×#536C211yesmaximal, characteristic
2C2×#464C211yesmaximal, characteristic, complemented
3C2×#462C211yesmaximal, characteristic, complemented
4C2×#452C211yesmaximal, characteristic
5C2×#451C211yesmaximal, characteristic, complemented
6#2323C211yesmaximal, characteristic, complemented
7#2043C211yesmaximal, characteristic, complemented
8#2042C211yesmaximal, characteristic, complemented
9#2016C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic, complemented
11#1983C211yesmaximal, characteristic, complemented
12#1981C211yesmaximal, characteristic, complemented
13#1798C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic
15#1787C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C23×Q8C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×Q16C2211yescharacteristic
20C22×Q16C2211yescharacteristic
21C22×SD16C2211yescharacteristic
22#536—24no—
23#536—24no—
24C2×C4⋊Q8C2211yescharacteristic
25C2×C4⋊Q8C2211yescharacteristic
26C2×#124C2211yescharacteristic
27C2×#124C2211yescharacteristic
28C2×#123—21no—
29C2×#123—21no—
30C2×C22⋊Q8—21no—
31C2×C22⋊Q8—21no—
32C2×C22⋊Q8C2211yescharacteristic
33C2×C4×Q8C2211yescharacteristic
34C2×#117C2211yescharacteristic
35#464—24no—
36#462—24no—
37#452—24no—
38#451—24no—
39C2×C4⋊C8C2211yescharacteristic
40C2×C4⋊C8C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#103C2211yescharacteristic
47C2×#103C2211yescharacteristic
48C2×#103C2211yescharacteristic
49C2×#103C2211yescharacteristic
50C2×#102C2211yescharacteristic
51C2×#102C2211yescharacteristic
52C2×#102C2211yescharacteristic
53C2×C22⋊C8C2211yescharacteristic
54C2×C22⋊C8C2211yescharacteristic
55#391C2211yescharacteristic
56#389—21no—
57#389C2211yescharacteristic
58#388—21no—
59#388—21no—
60#387—21no—
61#380C2211yescharacteristic
62C4×C4⋊C4C2211yescharacteristic
63#336C2211yescharacteristic
64#336C2211yescharacteristic
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×Q8C2311yescharacteristic
79C22×Q8C2311yescharacteristic
80C22×Q8C2311yescharacteristic
81C22×Q8C2311yescharacteristic
82C22×D4—21no—
83C22×D4C2311yescharacteristic
84C23×C4—21no—
85C23×C4—21no—
86C23×C4C2311yescharacteristic
87C2×Q16—22no—
88C2×Q16—22no—
89C2×Q16—22no—
90C2×Q16—22no—
91C2×Q16—22no—
92C2×Q16—22no—
93C2×Q16—41no—
94C2×Q16—41no—
95C2×Q16—41no—
96C2×SD16—22no—
97C2×SD16—22no—
98C2×SD16—41no—
99C22×C8C2311yescharacteristic
100C22×C8C2311yescharacteristic
101C22×C8C2311yescharacteristic
102C22×C8C2311yescharacteristic
103C4⋊Q8—22no—
104C4⋊Q8—22no—
105C4⋊Q8—22no—
106C4⋊Q8D414yes—
107#124—22no—
108#124—22no—
109#124—22no—
110#124D414yes—
111#123—28no—
112#123—28no—
113C22⋊Q8—28no—
114C22⋊Q8—28no—
115C22⋊Q8—22no—
116C22⋊Q8D414yes—
117C4×Q8—22no—
118C4×Q8—22no—
119#117—22no—
120#117—22no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4D411yescharacteristic
129C2×C4⋊C4D411yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C4⋊C4C2311yescharacteristic
132C2×C4⋊C4C2311yescharacteristic
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4D411yescharacteristic
140C2×C22⋊C4D411yescharacteristic
141C2×C42—21no—
142C2×C42—21no—
143C2×C42C2311yescharacteristic
144C2×C42C2311yescharacteristic
145C4⋊C8—22no—
146C4⋊C8—22no—
147#103—22no—
148#103—22no—
149#103—22no—
150#103—22no—
151#103—22no—
152#103—22no—
153#103—41no—
154#103—41no—
155#103—41no—
156#102—22no—
157#102—22no—
158#102—41no—
159C22⋊C8—22no—
160C22⋊C8—22no—
161#95—21no—
162#95—21no—
163#95—21no—
164#95—21no—
165#95—21no—
166C24—21no—
167C24C2×D411yescharacteristic
168#41—44no—
169#41—44no—
170#41—44no—
171#41—44no—
172C2×Q8—24no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
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—21no—
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×Q8C2×D412yes—
202C2×Q8C2×D412yes—
203C2×D4—24no—
204C2×D4—21no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—21no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4C2×D412yes—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D411yescharacteristic
232C22×C4C2×D411yescharacteristic
233C22×C4C2×D411yescharacteristic
234C22×C4C2×D412yes—
235C22×C4#4112yes—
236C22×C4C2411yescharacteristic
237Q16—44no—
238Q16—44no—
239Q16—44no—
240SD16—44no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—42no—
266C4⋊C4—81no—
267C4⋊C4C2×D412yes—
268C22⋊C4—24no—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—24no—
272C22⋊C4—24no—
273C22⋊C4—22no—
274C22⋊C4—22no—
275C22⋊C4—81no—
276C42—21no—
277C42—21no—
278C42—42no—
279C42—81no—
280C42C2×D412yes—
281C42C2×D412yes—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23#12112yes—
296C23C22×D411yescharacteristic
297Q8—22no—
298Q8—22no—
299Q8—22no—
300Q8—22no—
301Q8—22no—
302Q8—22no—
303Q8—22no—
304Q8—22no—
305Q8—48no—
306Q8—41no—
307Q8—41no—
308Q8—41no—
309Q8—41no—
310Q8—41no—
311Q8—41no—
312Q8—41no—
313Q8—41no—
314D4—22no—
315D4—22no—
316D4—48no—
317D4—41no—
318D4—41no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—48no—
341C2×C4—48no—
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—41no—
357C2×C4—41no—
358C2×C4#12012yes—
359C2×C4#12012yes—
360C2×C4#12112yes—
361C2×C4C22×D411yescharacteristic
362C2×C4C22×D411yescharacteristic
363C2×C4C22×D411yescharacteristic
364C2×C4C2×#4111yescharacteristic
365C2×C4H(5,𝔽2)11yescharacteristic
366C2×C4H(5,𝔽2)11yescharacteristic
367C8—41no—
368C8—41no—
369C8—41no—
370C8—41no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22C2×#12011yescharacteristic
386C22C2×#12111yescharacteristic
387C22#53511yescharacteristic
388C22#57711yescharacteristic
389C22#57811yescharacteristic
390C22#57811yescharacteristic
391C22#57811yescharacteristic
392C4—22no—
393C4—22no—
394C4—22no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—41no—
400C4—41no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4D4211yescharacteristic
405C4D4211yescharacteristic
406C4#54611yescharacteristic
407C4#54611yescharacteristic
408C2—42no—
409C2—82no—
410C2#240811yescharacteristic
411C2#301111yescharacteristic
412C2#301311yescharacteristic
413C2#304411yescharacteristic
414C2#304611yescharacteristic
415C2#322311yescharacteristic
416C2#322511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#464C2semidirect
3. C2×#462C2semidirect
5. C2×#451C2semidirect
6. #2323C2semidirect
7. #2043C2semidirect
8. #2042C2semidirect
9. #2016C2semidirect
10. #2015C2semidirect
11. #1983C2semidirect
12. #1981C2semidirect
13. #1798C2semidirect
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×#4642
C2×#4622
C2×#4522
C2×#4512
#23232
#20432
#20422
#20162
#20152
#19832
#19812
#17982
#17882
#17872