Finite Group Databasebeta

← #22501

Subgroups — #22501

Order 256.

597 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#553C211yesmaximal, characteristic
2C2×#436C211yesmaximal, characteristic, complemented
3#2352C211yesmaximal, characteristic, complemented
4#1991C211yesmaximal, characteristic, complemented
5#1989C211yesmaximal, characteristic, complemented
6#1895C211yesmaximal, characteristic, complemented
7#1895C211yesmaximal, characteristic, complemented
8#1875C211yesmaximal, characteristic, complemented
9#1874C211yesmaximal, characteristic, complemented
10#1825C211yesmaximal, characteristic, complemented
11#1823C211yesmaximal, characteristic, complemented
12#1804C211yesmaximal, characteristic, complemented
13C4×#102C211yesmaximal, characteristic, complemented
14#1736C211yesmaximal, characteristic, complemented
15#1736C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18#553—24no—
19#553C418yescomplemented
20C2×#126—21no—
21C2×#125C2211yescharacteristic
22C2×#123—21no—
23C2×#121—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×C4×D4—21no—
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29C2×#117C2211yescharacteristic
30C2×#117C2211yescharacteristic
31C22×C4⋊C4C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33#436—24no—
34C2×C4⋊C8C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#388—21no—
43#387—21no—
44#386—21no—
45#382C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C22⋊C4—21no—
50#340C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4D411yescharacteristic
65C22×D4C2311yescharacteristic
66C23×C4—21no—
67C23×C4—21no—
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C2×M16—22no—
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—41no—
76C2×M16—41no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81#126—28no—
82#125—22no—
83#125C2×C414yescomplemented
84#123—28no—
85#121—28no—
86#121—22no—
87#121—22no—
88#121C2×C414yes—
89#121C2×C414yescomplemented
90C4×D4—28no—
91C4×D4—22no—
92C4×D4C2×C414yes—
93C4×D4C2×C414yescomplemented
94C4×D4D414yes—
95#117—22no—
96#117—22no—
97#117—41no—
98#117C2×C414yes—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—22no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—41no—
106C2×C4⋊C4C2×C414yescomplemented
107C2×C4⋊C4D411yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4D411yescharacteristic
123C2×C22⋊C4D411yescharacteristic
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42C2311yescharacteristic
128C4⋊C8—22no—
129#102—22no—
130#102—22no—
131#102—41no—
132#102—41no—
133C22⋊C8—22no—
134C22⋊C8—22no—
135C4×C8—22no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141C24—21no—
142C24C2×D411yescharacteristic
143C24C2×D411yescharacteristic
144C2×D4—24no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—81no—
156C2×D4—81no—
157C2×D4C22⋊C414yes—
158C2×D4C22×C412yes—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
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—24no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4C22×C412yes—
184C22×C4C22×C412yes—
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D412yes—
190C22×C4C2×D412yes—
191C22×C4C2411yescharacteristic
192M16—44no—
193M16—44no—
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×C8—21no—
204C2×C8—21no—
205C2×C8#4112yes—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
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—21no—
219C4⋊C4—42no—
220C4⋊C4C22⋊C414yes—
221C4⋊C4C22×C412yes—
222C4⋊C4C22×C412yes—
223C4⋊C4C22×C412yes—
224C4⋊C4#4112yes—
225C22⋊C4—24no—
226C22⋊C4—24no—
227C22⋊C4—24no—
228C22⋊C4—24no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—81no—
232C22⋊C4C22⋊C414yes—
233C22⋊C4C22⋊C414yes—
234C42—24no—
235C42—42no—
236C42—81no—
237C42C22×C412yes—
238C42C2×D412yes—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—22no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23C2×C22⋊C412yes—
259C23C2×C22⋊C412yes—
260C23C22×D411yescharacteristic
261D4—22no—
262D4—22no—
263D4—44no—
264D4—48no—
265D4—41no—
266D4—41no—
267D4—84no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
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—41no—
304C2×C4—41no—
305C2×C4—84no—
306C2×C4—84no—
307C2×C4—84no—
308C2×C4C2×C22⋊C412yes—
309C2×C4C2×C22⋊C412yes—
310C2×C4C2×C22⋊C412yes—
311C2×C4C2×C22⋊C412yes—
312C2×C4C23×C411yescharacteristic
313C2×C4C22×D411yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4C2×#4111yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C2×C42−1+411yescharacteristic
318C8—22no—
319C8—41no—
320C8—41no—
321C8—41no—
322C22—22no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22C22×C22⋊C411yescharacteristic
344C22#54611yescharacteristic
345C22#54711yescharacteristic
346C22#57511yescharacteristic
347C22#57511yescharacteristic
348C22#57611yescharacteristic
349C22#57711yescharacteristic
350C4—22no—
351C4—22no—
352C4—22no—
353C4—22no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—42no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4C4×#4111yescharacteristic
362C4#52011yescharacteristic
363C4#54611yescharacteristic
364C4#54711yescharacteristic
365C2—42no—
366C2—42no—
367C2—82no—
368C2#233911yescharacteristic
369C2#289611yescharacteristic
370C2#289711yescharacteristic
371C2#332211yescharacteristic
372C2#332611yescharacteristic
373C2#335111yescharacteristic
374C2#335511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#553C2
LCSG ▷ C2×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
2. C2×#436C2semidirect
3. #2352C2semidirect
4. #1991C2semidirect
5. #1989C2semidirect
6. #1895C2semidirect
7. #1895C2semidirect
8. #1875C2semidirect
9. #1874C2semidirect
10. #1825C2semidirect
11. #1823C2semidirect
12. #1804C2semidirect
13. C4×#102C2semidirect
14. #1736C2semidirect
15. #1736C2semidirect
19. #553C4semidirect
83. #125C2×C4semidirect
89. #121C2×C4semidirect
93. C4×D4C2×C4semidirect
106. C2×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#5532
C2×#4362
#23522
#19912
#19892
#18952
#18752
#18742
#18252
#18232
#18042
C4×#1022
#17362