Finite Group Databasebeta

← #27252

Subgroups — #27252

Order 256.

478 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#477C211yesmaximal, characteristic, complemented
2C2×#452C211yesmaximal, characteristic
3#2492C211yesmaximal, characteristic, complemented
4#2341C211yesmaximal, characteristic
5#2075C211yesmaximal, characteristic, complemented
6#2055C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2023C211yesmaximal, characteristic, complemented
9#1973C211yesmaximal, characteristic, complemented
10#1925C211yesmaximal, characteristic, complemented
11#1924C211yesmaximal, characteristic, complemented
12#1878C211yesmaximal, characteristic, complemented
13#1878C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic
15#1763C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C4⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23#477—24no—
24#452—24no—
25C2×#107C2211yescharacteristic
26C2×#107C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39#400—21no—
40#393—21no—
41#391—21no—
42#391C2211yescharacteristic
43#391C2211yescharacteristic
44#390—21no—
45#390—21no—
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#388—21no—
49#380—21no—
50#380C2211yescharacteristic
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8—21no—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C4⋊Q8—22no—
85C4⋊Q8—41no—
86C22⋊Q8—22no—
87C22⋊Q8—41no—
88C4×Q8—22no—
89C4×Q8—22no—
90C4×Q8—41no—
91C4×Q8—41no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—21no—
95C2×C4⋊C4—21no—
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4D411yescharacteristic
109C2×C22⋊C4D411yescharacteristic
110C2×C42—21no—
111C2×C42—21no—
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115#107—22no—
116#107—22no—
117C4⋊C8—22no—
118#103—22no—
119#103—22no—
120#103—41no—
121#103—41no—
122#103D414yes—
123#102—22no—
124#102—22no—
125#102—41no—
126C22⋊C8—22no—
127C22⋊C8—41no—
128#97—41no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95D411yescharacteristic
137#95D411yescharacteristic
138C24—21no—
139C24C2×D411yescharacteristic
140#41—44no—
141#41—44no—
142#41—44no—
143#41—44no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—42no—
154C2×Q8—42no—
155C2×Q8—81no—
156C2×Q8C2×D412yes—
157C2×Q8#4112yes—
158C2×Q8#4112yes—
159C2×D4—24no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—42no—
167C2×D4#4112yes—
168C22×C4—21no—
169C22×C4—24no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2411yescharacteristic
190Q16—44no—
191SD16—44no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8C2×D412yes—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4C2×D412yes—
218C22⋊C4—22no—
219C22⋊C4—41no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—42no—
232C42—42no—
233C42—81no—
234C23—21no—
235C23—22no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23C22×D411yescharacteristic
249Q8—22no—
250Q8—22no—
251Q8—22no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256Q8—84no—
257Q8#12414yes—
258D4—22no—
259D4—22no—
260D4—48no—
261D4—41no—
262D4—41no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—48no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4#13712yes—
304C2×C4C22×D411yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4C2×#4111yescharacteristic
307C2×C4C2×#4111yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C2×C4H(5,𝔽2)11yescharacteristic
310C8—22no—
311C8—41no—
312C8—41no—
313C8—41no—
314C22—41no—
315C22—41no—
316C22—42no—
317C22—41no—
318C22—42no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22#52511yescharacteristic
329C22D4211yescharacteristic
330C22#54611yescharacteristic
331C22C2×#13711yescharacteristic
332C22#57511yescharacteristic
333C22#57611yescharacteristic
334C22#57811yescharacteristic
335C4—24no—
336C4—22no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—42no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4C2×#12411yescharacteristic
348C4#53811yescharacteristic
349C4#54611yescharacteristic
350C4#55111yescharacteristic
351C2—81no—
352C2—82no—
353C2#247211yescharacteristic
354C2#311611yescharacteristic
355C2#312411yescharacteristic
356C2#326811yescharacteristic
357C2#327011yescharacteristic
358C2#331311yescharacteristic
359C2#333311yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#477C2semidirect
3. #2492C2semidirect
5. #2075C2semidirect
6. #2055C2semidirect
7. #2036C2semidirect
8. #2023C2semidirect
9. #1973C2semidirect
10. #1925C2semidirect
11. #1924C2semidirect
12. #1878C2semidirect
13. #1878C2semidirect
15. #1763C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4772
C2×#4522
#24922
#23412
#20752
#20552
#20362
#20232
#19732
#19252
#19242
#18782
#17882
#17632