Finite Group Databasebeta

← #23947

Subgroups — #23947

Order 256.

635 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×#542C211yesmaximal, characteristic, complemented
2C2×#452C211yesmaximal, characteristic, complemented
3C2×#449C211yesmaximal, characteristic
4#2311C211yesmaximal, characteristic, complemented
5#2024C211yesmaximal, characteristic, complemented
6#2023C211yesmaximal, characteristic, complemented
7#2015C211yesmaximal, characteristic, complemented
8#2012C211yesmaximal, characteristic, complemented
9#1880C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1877C211yesmaximal, characteristic, complemented
13#1874C211yesmaximal, characteristic, complemented
14#1872C211yesmaximal, characteristic, complemented
15#1818C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×D8C2211yescharacteristic
21#542—24no—
22#542—24no—
23C2×C4⋊Q8C2211yescharacteristic
24C2×#126—21no—
25C2×#124—21no—
26C2×#124C2211yescharacteristic
27C2×#123—21no—
28C2×C22⋊Q8—21no—
29C2×C22⋊Q8C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×C4×Q8C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C2×#117C2211yescharacteristic
34C22×C4⋊C4C2211yescharacteristic
35#452—24no—
36#449—24no—
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47#391C2211yescharacteristic
48#391C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#386—21no—
52#386—21no—
53#380—21no—
54#380—21no—
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
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×Q8C2311yescharacteristic
72C22×D4—21no—
73C22×D4—21no—
74C22×D4C2311yescharacteristic
75C22×D4C2311yescharacteristic
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4C2311yescharacteristic
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—41no—
83C2×SD16—22no—
84C2×SD16—22no—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—41no—
88C2×SD16—41no—
89C2×D8—22no—
90C2×D8—22no—
91C2×D8—41no—
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98#126—28no—
99#124—28no—
100#124—22no—
101#124—22no—
102#123—28no—
103C22⋊Q8—28no—
104C22⋊Q8—22no—
105C22⋊Q8D414yes—
106#121—22no—
107#121D414yes—
108C4×Q8—22no—
109C4×Q8—22no—
110C4×D4—22no—
111C4×D4—22no—
112#117—22no—
113#117—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—41no—
120C2×C4⋊C4D411yescharacteristic
121C2×C4⋊C4D411yescharacteristic
122C2×C4⋊C4C2311yescharacteristic
123C2×C4⋊C4C2311yescharacteristic
124C2×C4⋊C4C2311yescharacteristic
125C2×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
139C2×C42C2311yescharacteristic
140#103—22no—
141#103—22no—
142#103—41no—
143#103—41no—
144#102—22no—
145#102—22no—
146#102—41no—
147#102—41no—
148C22⋊C8—22no—
149C22⋊C8—22no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154C24—21no—
155C24—21no—
156C24C2×D411yescharacteristic
157#41—44no—
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—21no—
171C2×Q8—21no—
172C2×Q8—81no—
173C2×Q8—81no—
174C2×Q8C2×D412yes—
175C2×Q8#4112yes—
176C2×D4—24no—
177C2×D4—24no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—81no—
191C2×D4—81no—
192C2×D4C2×D412yes—
193C2×D4#4112yes—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—24no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—81no—
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D411yescharacteristic
215C22×C4C2×D411yescharacteristic
216C22×C4C2×D411yescharacteristic
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D412yes—
219C22×C4C2×D412yes—
220C22×C4C2411yescharacteristic
221Q16—44no—
222SD16—44no—
223SD16—44no—
224D8—44no—
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—
237C4⋊C4—24no—
238C4⋊C4—24no—
239C4⋊C4—24no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—21no—
247C4⋊C4—21no—
248C4⋊C4—42no—
249C4⋊C4C2×D412yes—
250C4⋊C4C2×D412yes—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—24no—
255C22⋊C4—22no—
256C22⋊C4—22no—
257C22⋊C4—24no—
258C22⋊C4—81no—
259C22⋊C4—81no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C42—24no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—22no—
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#12012yes—
288C23C22×D411yescharacteristic
289Q8—22no—
290Q8—22no—
291Q8—22no—
292Q8—22no—
293Q8—41no—
294Q8—41no—
295Q8—41no—
296Q8—41no—
297Q8—84no—
298D4—22no—
299D4—22no—
300D4—22no—
301D4—22no—
302D4—48no—
303D4—41no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—84no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—48no—
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—84no—
346C2×C4—84no—
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—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—42no—
366C22—42no—
367C22—42no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C2×#12011yescharacteristic
380C22#52511yescharacteristic
381C22#53511yescharacteristic
382C22#57611yescharacteristic
383C22#57711yescharacteristic
384C22#57711yescharacteristic
385C22#57811yescharacteristic
386C4—22no—
387C4—42no—
388C4—42no—
389C4—41no—
390C4—41no—
391C4—42no—
392C4—42no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—81no—
397C4#54611yescharacteristic
398C4#54611yescharacteristic
399C4#54611yescharacteristic
400C4#54611yescharacteristic
401C2—42no—
402C2—82no—
403C2—82no—
404C2#240411yescharacteristic
405C2#301211yescharacteristic
406C2#301311yescharacteristic
407C2#312411yescharacteristic
408C2#312511yescharacteristic
409C2#322411yescharacteristic
410C2#322511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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
1. C2×#542C2semidirect
2. C2×#452C2semidirect
4. #2311C2semidirect
5. #2024C2semidirect
6. #2023C2semidirect
7. #2015C2semidirect
8. #2012C2semidirect
9. #1880C2semidirect
10. #1879C2semidirect
11. #1878C2semidirect
12. #1877C2semidirect
13. #1874C2semidirect
14. #1872C2semidirect
15. #1818C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4522
C2×#4492
#23112
#20242
#20232
#20152
#20122
#18802
#18792
#18782
#18772
#18742
#18722
#18182