Finite Group Databasebeta

← #22057

Subgroups — #22057

Order 256.

484 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#452C211yesmaximal, characteristic
2#2323C211yesmaximal, characteristic, complemented
3#2311C211yesmaximal, characteristic
4#1973C211yesmaximal, characteristic, complemented
5#1969C211yesmaximal, characteristic, complemented
6#1935C211yesmaximal, characteristic, complemented
7#1934C211yesmaximal, characteristic, complemented
8#1928C211yesmaximal, characteristic, complemented
9#1925C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1878C211yesmaximal, characteristic, complemented
12#1872C211yesmaximal, characteristic, complemented
13#1852C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C4×Q8C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#452C418yescomplemented
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×#97C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#391C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#389—21no—
44#389C2211yescharacteristic
45#388—21no—
46#388—21no—
47#387—21no—
48#386—21no—
49#386—21no—
50#380—21no—
51#380—21no—
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60C2×#41—22no—
61C2×#41—41no—
62C2×#41—41no—
63C2×#41—41no—
64C2×#41—41no—
65C2×#41C2×C414yes—
66C22×Q8C2311yescharacteristic
67C22×Q8C2311yescharacteristic
68C22×D4—21no—
69C22×D4C2311yescharacteristic
70C23×C4—21no—
71C23×C4—21no—
72C23×C4C2311yescharacteristic
73C2×Q16—41no—
74C2×Q16—41no—
75C2×Q16C2×C414yescomplemented
76C2×SD16—41no—
77C2×SD16—41no—
78C2×SD16C2×C414yescomplemented
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22⋊Q8—22no—
84C22⋊Q8C2×C414yes—
85C4×Q8—41no—
86C4×Q8—41no—
87#117—41no—
88#117—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—21no—
93C2×C4⋊C4—41no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4D411yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
101C2×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#107—41no—
115#107—41no—
116#103—41no—
117#103—41no—
118#103—41no—
119#103—41no—
120#103C2×C414yescomplemented
121#102—41no—
122#102—41no—
123#102C2×C414yescomplemented
124C22⋊C8—22no—
125C22⋊C8C2×C414yes—
126#97—41no—
127C4×C8—41no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137C24—21no—
138C24C2×D411yescharacteristic
139#41—44no—
140#41—44no—
141#41—44no—
142#41—44no—
143C2×Q8—21no—
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×Q8C22×C412yes—
154C2×Q8C22×C412yes—
155C2×D4—24no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4C22×C412yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—24no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4C22×C412yes—
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4#4112yes—
188C22×C4C2411yescharacteristic
189Q16—44no—
190SD16—44no—
191C2×C8—21no—
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×C8C22×C412yes—
202C2×C8C22×C412yes—
203C4⋊C4—24no—
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—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—81no—
218C4⋊C4C22×C412yes—
219C22⋊C4—24no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—81no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C42—21no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—42no—
232C42—81no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—22no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23C4×D412yes—
247C23C22×D411yescharacteristic
248Q8—22no—
249Q8—22no—
250Q8—22no—
251Q8—22no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255Q8—41no—
256D4—22no—
257D4—22no—
258D4—48no—
259D4—41no—
260D4—41no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
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—21no—
282C2×C4—48no—
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—84no—
302C2×C4C4×D412yes—
303C2×C4C23×C411yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4H(5,𝔽2)11yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C4H(5,𝔽2)11yescharacteristic
308C2×C4H(5,𝔽2)11yescharacteristic
309C8—22no—
310C8—22no—
311C8—41no—
312C8—41no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22C2×C4×D411yescharacteristic
328C22#53411yescharacteristic
329C22#53511yescharacteristic
330C22#57611yescharacteristic
331C22#57711yescharacteristic
332C22#57811yescharacteristic
333C22#57811yescharacteristic
334C4—22no—
335C4—42no—
336C4—42no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—41no—
341C4—42no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4#51811yescharacteristic
347C4#51811yescharacteristic
348C4#53811yescharacteristic
349C4#53811yescharacteristic
350C2—42no—
351C2—82no—
352C2#232111yescharacteristic
353C2#294711yescharacteristic
354C2#295111yescharacteristic
355C2#320111yescharacteristic
356C2#320311yescharacteristic
357C2#322311yescharacteristic
358C2#322411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C2×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. #2323C2semidirect
4. #1973C2semidirect
5. #1969C2semidirect
6. #1935C2semidirect
7. #1934C2semidirect
8. #1928C2semidirect
9. #1925C2semidirect
10. #1880C2semidirect
11. #1878C2semidirect
12. #1872C2semidirect
13. #1852C2semidirect
15. #1750C2semidirect
23. #452C4semidirect
75. C2×Q16C2×C4semidirect
78. C2×SD16C2×C4semidirect
120. #103C2×C4semidirect
123. #102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
#23232
#23112
#19732
#19692
#19352
#19342
#19282
#19252
#18802
#18782
#18722
#18522
#17882
#17502