Finite Group Databasebeta

← #22053

Subgroups — #22053

Order 256.

512 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#449C211yesmaximal, characteristic
2#2324C211yesmaximal, characteristic, complemented
3#2311C211yesmaximal, characteristic, complemented
4#1972C211yesmaximal, characteristic, complemented
5#1968C211yesmaximal, characteristic, complemented
6#1935C211yesmaximal, characteristic, complemented
7#1934C211yesmaximal, characteristic, complemented
8#1927C211yesmaximal, characteristic, complemented
9#1926C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1877C211yesmaximal, characteristic, complemented
12#1874C211yesmaximal, characteristic, complemented
13#1853C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117C2211yescharacteristic
22C22×C4⋊C4C2211yescharacteristic
23#449C418yescomplemented
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×#97C2211yescharacteristic
37C2×C4×C8C2211yescharacteristic
38#391C2211yescharacteristic
39#391C2211yescharacteristic
40#390—21no—
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#390C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#386—21no—
47#386—21no—
48#386—21no—
49#385—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×C414yescomplemented
66C22×Q8C2311yescharacteristic
67C22×D4—21no—
68C22×D4—21no—
69C22×D4C2311yescharacteristic
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C2×SD16—41no—
76C2×SD16—41no—
77C2×SD16C2×C414yescomplemented
78C2×D8—41no—
79C2×D8—41no—
80C2×D8C2×C414yescomplemented
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85#121—22no—
86#121C2×C414yes—
87C4×D4—41no—
88C4×D4—41no—
89#117—41no—
90#117—41no—
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⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4D411yescharacteristic
112C2×C22⋊C4D411yescharacteristic
113C2×C42—21no—
114C2×C42C2311yescharacteristic
115C2×C42C2311yescharacteristic
116#106—41no—
117#106—41no—
118#103—41no—
119#103—41no—
120#103C2×C414yescomplemented
121#102—41no—
122#102—41no—
123#102—41no—
124#102—41no—
125#102C2×C414yescomplemented
126C22⋊C8—22no—
127C22⋊C8C2×C414yescomplemented
128#97—41no—
129C4×C8—41no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137C24—21no—
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×Q8C22×C412yes—
150C2×D4—24no—
151C2×D4—24no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—81no—
165C2×D4—81no—
166C2×D4C22×C412yes—
167C2×D4C22×C412yes—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—24no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—42no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—81no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4C22×C412yes—
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4#4112yes—
194C22×C4C2411yescharacteristic
195SD16—44no—
196D8—44no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8C22×C412yes—
208C2×C8C22×C412yes—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—42no—
221C4⋊C4—81no—
222C4⋊C4C22×C412yes—
223C22⋊C4—24no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4—42no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—81no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—22no—
246C23—22no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23C4×D412yes—
260C23C22×D411yescharacteristic
261Q8—22no—
262Q8—22no—
263Q8—41no—
264Q8—41no—
265D4—22no—
266D4—22no—
267D4—22no—
268D4—22no—
269D4—48no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—84no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—48no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4C4×D412yes—
315C2×C4C23×C411yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4H(5,𝔽2)11yescharacteristic
318C2×C4H(5,𝔽2)11yescharacteristic
319C2×C4H(5,𝔽2)11yescharacteristic
320C2×C4H(5,𝔽2)11yescharacteristic
321C8—22no—
322C8—22no—
323C8—41no—
324C8—41no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22C2×C4×D411yescharacteristic
347C22#53411yescharacteristic
348C22#53511yescharacteristic
349C22#57611yescharacteristic
350C22#57711yescharacteristic
351C22#57711yescharacteristic
352C22#57711yescharacteristic
353C4—22no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—42no—
358C4—41no—
359C4—41no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4#51811yescharacteristic
365C4#51811yescharacteristic
366C4#53811yescharacteristic
367C4#53811yescharacteristic
368C2—42no—
369C2—82no—
370C2—82no—
371C2#232111yescharacteristic
372C2#294611yescharacteristic
373C2#295211yescharacteristic
374C2#319811yescharacteristic
375C2#320411yescharacteristic
376C2#322111yescharacteristic
377C2#322411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
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. #2324C2semidirect
3. #2311C2semidirect
4. #1972C2semidirect
5. #1968C2semidirect
6. #1935C2semidirect
7. #1934C2semidirect
8. #1927C2semidirect
9. #1926C2semidirect
10. #1879C2semidirect
11. #1877C2semidirect
12. #1874C2semidirect
13. #1853C2semidirect
14. #1788C2semidirect
15. #1750C2semidirect
23. #449C4semidirect
65. C2×#41C2×C4semidirect
77. C2×SD16C2×C4semidirect
80. C2×D8C2×C4semidirect
120. #103C2×C4semidirect
125. #102C2×C4semidirect
127. C22⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4492
#23242
#23112
#19722
#19682
#19352
#19342
#19272
#19262
#18792
#18772
#18742
#18532
#17882
#17502