Finite Group Databasebeta

← #30504

Subgroups — #30504

Order 256.

561 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#467C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#452C211yesmaximal, characteristic, complemented
4C2×#452C211yesmaximal, characteristic, complemented
5#2542C211yesmaximal, characteristic, complemented
6#2407C211yesmaximal, characteristic
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2016C211yesmaximal, characteristic, complemented
10#2013C211yesmaximal, characteristic, complemented
11#2001C211yesmaximal, characteristic, complemented
12#2000C211yesmaximal, characteristic, complemented
13#1927C211yesmaximal, characteristic, complemented
14#1925C211yesmaximal, characteristic, complemented
15#1869C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×Q16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C23×C8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30#467—24no—
31#465—24no—
32#452—24no—
33#452—24no—
34C2×#107C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47#399—21no—
48#399—21no—
49#394—21no—
50#392—21no—
51#391C2211yescharacteristic
52#390—21no—
53#390C2211yescharacteristic
54#386—21no—
55#385—21no—
56#384—21no—
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41—41no—
68C2×#41—41no—
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×D4—21no—
73C22×D4—21no—
74C22×D4D411yescharacteristic
75C22×D4C2311yescharacteristic
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—22no—
82C2×Q16—41no—
83C2×Q16—41no—
84C2×Q16—41no—
85C2×SD16—22no—
86C2×SD16—22no—
87C2×SD16—22no—
88C2×SD16—41no—
89C2×SD16—41no—
90C2×SD16—41no—
91C22×C8—22no—
92C22×C8—22no—
93C22×C8—41no—
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C4⋊Q8—41no—
99C4⋊Q8—41no—
100C22⋊Q8—22no—
101C22⋊Q8—22no—
102C22⋊Q8—22no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—41no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110#121—22no—
111#121—41no—
112C4×D4—41no—
113C4×D4—41no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
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
139#107—22no—
140#106—22no—
141#103—22no—
142#103—22no—
143#103—22no—
144#103—41no—
145#103—41no—
146#103D414yescomplemented
147#102—22no—
148#102D414yescomplemented
149C22⋊C8—22no—
150C22⋊C8—22no—
151C22⋊C8—41no—
152#95—21no—
153#95—21no—
154#95—21no—
155#95—21no—
156C24—21no—
157C24C2×D411yescharacteristic
158C24C2×D411yescharacteristic
159#41—44no—
160#41—44no—
161#41—44no—
162#41—44no—
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—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8C2×D412yes—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—41no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—81no—
195C2×D4C2×D412yes—
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—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×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2×D411yescharacteristic
222C22×C4C2411yescharacteristic
223Q16—44no—
224Q16—44no—
225SD16—44no—
226SD16—44no—
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—
237C2×C8—44no—
238C2×C8C2×D412yes—
239C2×C8C2×D412yes—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4C2×D412yes—
255C4⋊C4C2×D412yes—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—81no—
266C22⋊C4—81no—
267C42—21no—
268C42—21no—
269C42—42no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—22no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#13712yes—
291C23C22×D411yescharacteristic
292Q8—22no—
293Q8—22no—
294Q8—41no—
295Q8—41no—
296Q8—41no—
297Q8—41no—
298Q8—41no—
299Q8—41no—
300Q8—41no—
301Q8—41no—
302Q8—41no—
303Q8—41no—
304D4—22no—
305D4—22no—
306D4—44no—
307D4—48no—
308D4—41no—
309D4—41no—
310D4—84no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—48no—
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—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4#13512yes—
353C2×C4C22×D411yescharacteristic
354C2×C4C22×D411yescharacteristic
355C2×C4H(5,𝔽2)11yescharacteristic
356C2×C4H(5,𝔽2)11yescharacteristic
357C2×C4H(5,𝔽2)11yescharacteristic
358C2×C4H(5,𝔽2)11yescharacteristic
359C8—22no—
360C8—22no—
361C8—41no—
362C8—41no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#53411yescharacteristic
385C22D4211yescharacteristic
386C22D4211yescharacteristic
387C22C2×#13511yescharacteristic
388C22C2×#13711yescharacteristic
389C22#57711yescharacteristic
390C22#57811yescharacteristic
391C4—22no—
392C4—41no—
393C4—41no—
394C4—42no—
395C4—41no—
396C4—42no—
397C4—42no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4#53511yescharacteristic
403C4#53511yescharacteristic
404C4D4211yescharacteristic
405C4#56011yescharacteristic
406C2—42no—
407C2—81no—
408C2—82no—
409C2#266911yescharacteristic
410C2#319011yescharacteristic
411C2#320111yescharacteristic
412C2#326611yescharacteristic
413C2#327011yescharacteristic
414C2#328311yescharacteristic
415C2#328411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2407C2
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×#467C2semidirect
2. C2×#465C2semidirect
3. C2×#452C2semidirect
4. C2×#452C2semidirect
5. #2542C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #2016C2semidirect
10. #2013C2semidirect
11. #2001C2semidirect
12. #2000C2semidirect
13. #1927C2semidirect
14. #1925C2semidirect
15. #1869C2semidirect
146. #103D4semidirect
148. #102D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×#4652
C2×#4522
#25422
#24072
#20362
#20352
#20162
#20132
#20012
#20002
#19272
#19252
#18692