Finite Group Databasebeta

← #22877

Subgroups — #22877

Order 256.

579 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#449C211yesmaximal, characteristic
2C2×C4×SD16C211yesmaximal, characteristic, complemented
3C2×#416C211yesmaximal, characteristic, complemented
4#2363C211yesmaximal, characteristic, complemented
5#2341C211yesmaximal, characteristic, complemented
6#1970C211yesmaximal, characteristic, complemented
7#1934C211yesmaximal, characteristic, complemented
8#1926C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1880C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1863C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14C4×#103C211yesmaximal, characteristic, complemented
15C8×C22⋊C4C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×D8C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×C4×Q8C2211yescharacteristic
22C2×C4×Q8C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25#449C418yescomplemented
26C4×SD16—24no—
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29#416—24no—
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#390—21no—
42#390—21no—
43#390—21no—
44#390C2211yescharacteristic
45#390C2211yescharacteristic
46#390C2211yescharacteristic
47#389C2211yescharacteristic
48#388—21no—
49#386—21no—
50#380—21no—
51#380C2211yescharacteristic
52C4×C4⋊C4C2211yescharacteristic
53C4×C22⋊C4—21no—
54C4×C22⋊C4—21no—
55C4×C22⋊C4C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C2×#41C2×C414yescomplemented
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4—21no—
71C22×D4C2311yescharacteristic
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4C2311yescharacteristic
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16C2×C414yes—
79C2×D8—41no—
80C2×D8—41no—
81C2×D8C2×C414yes—
82C22×C8—22no—
83C22×C8—22no—
84C22×C8—41no—
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89#121—22no—
90#121C2×C414yes—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—41no—
94C4×Q8—41no—
95C4×D4—22no—
96C4×D4—41no—
97#117—22no—
98#117—41no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4D411yescharacteristic
118C2×C22⋊C4D411yescharacteristic
119C2×C42—21no—
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42C2311yescharacteristic
124C2×C42C2311yescharacteristic
125#107—41no—
126#106—22no—
127#103—22no—
128#103C2×C414yescomplemented
129#103D414yescomplemented
130#102—22no—
131#102—22no—
132#102—22no—
133#102—41no—
134#102C2×C414yescomplemented
135C22⋊C8C2×C414yescomplemented
136C4×C8—22no—
137C4×C8—41no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95D411yescharacteristic
142#95D411yescharacteristic
143C24—21no—
144C24—21no—
145C24C2×D411yescharacteristic
146#41—28no—
147#41—44no—
148#41—44no—
149#41—44no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8C22×C412yes—
154C2×Q8C2×D412yes—
155C2×Q8#4112yes—
156C2×D4—24no—
157C2×D4—24no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—42no—
169C2×D4—42no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4C22×C412yes—
173C2×D4C22×C412yes—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—24no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4—42no—
189C22×C4—81no—
190C22×C4C22×C412yes—
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2×D411yescharacteristic
196C22×C4#4112yes—
197C22×C4C2411yescharacteristic
198SD16—28no—
199D8—44no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—44no—
210C2×C8C22×C412yes—
211C2×C8C22×C412yes—
212C2×C8C2×D412yes—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—81no—
224C4⋊C4C22×C412yes—
225C4⋊C4C2×D412yes—
226C22⋊C4—24no—
227C22⋊C4—22no—
228C22⋊C4—22no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C22⋊C4—81no—
237C22⋊C4—81no—
238C42—21no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—42no—
243C42—42no—
244C42—42no—
245C42—81no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—22no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23C4×D412yes—
267C23C22×D411yescharacteristic
268Q8—22no—
269Q8—22no—
270Q8—22no—
271Q8C4×D414yes—
272D4—22no—
273D4—22no—
274D4—22no—
275D4—22no—
276D4—48no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—84no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—48no—
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—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—84no—
318C2×C4C4×D412yes—
319C2×C4#13512yes—
320C2×C4#13512yes—
321C2×C4C23×C411yescharacteristic
322C2×C4C22×D411yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—22no—
328C8—22no—
329C8—22no—
330C8—41no—
331C22—22no—
332C22—22no—
333C22—22no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22C2×C4×D411yescharacteristic
353C22D4211yescharacteristic
354C22#54611yescharacteristic
355C22C2×#13511yescharacteristic
356C22C2×#13511yescharacteristic
357C22#57611yescharacteristic
358C22#57711yescharacteristic
359C4—22no—
360C4—24no—
361C4—22no—
362C4—22no—
363C4—42no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4C2×C4×D411yescharacteristic
371C4#51811yescharacteristic
372C4#54611yescharacteristic
373C4#55111yescharacteristic
374C2—42no—
375C2—82no—
376C2—82no—
377C2#235811yescharacteristic
378C2C4×#13511yescharacteristic
379C2#294611yescharacteristic
380C2#328211yescharacteristic
381C2#328711yescharacteristic
382C2#330111yescharacteristic
383C2#330511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#449C2
LCSG ▷ C2×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
2. C2×C4×SD16C2semidirect
3. C2×#416C2semidirect
4. #2363C2semidirect
5. #2341C2semidirect
6. #1970C2semidirect
7. #1934C2semidirect
8. #1926C2semidirect
9. #1893C2semidirect
10. #1880C2semidirect
11. #1879C2semidirect
12. #1863C2semidirect
13. #1854C2semidirect
14. C4×#103C2semidirect
15. C8×C22⋊C4C2semidirect
25. #449C4semidirect
67. C2×#41C2×C4semidirect
128. #103C2×C4semidirect
129. #103D4semidirect
134. #102C2×C4semidirect
135. 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
C2×C4×SD162
C2×#4162
#23632
#23412
#19702
#19342
#19262
#18932
#18802
#18792
#18632
#18542
C4×#1032
C8×C22⋊C42