Finite Group Databasebeta

← #21513

Subgroups — #21513

Order 256.

560 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#459C211yesmaximal, characteristic
2C2×#442C211yesmaximal, characteristic, complemented
3C2×C4×D8C211yesmaximal, characteristic, complemented
4#2328C211yesmaximal, characteristic, complemented
5#2275C211yesmaximal, characteristic, complemented
6#1982C211yesmaximal, characteristic, complemented
7#1949C211yesmaximal, characteristic, complemented
8#1926C211yesmaximal, characteristic, complemented
9#1925C211yesmaximal, characteristic, complemented
10#1879C211yesmaximal, characteristic, complemented
11#1879C211yesmaximal, characteristic, complemented
12#1804C211yesmaximal, characteristic, complemented
13#1768C211yesmaximal, characteristic, complemented
14#1762C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×C4×D4C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24#459C418yescomplemented
25#442—24no—
26C4×D8—24no—
27C2×#107C2211yescharacteristic
28C2×#106C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×C4⋊C8C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×#97C2211yescharacteristic
40C2×C4×C8C2211yescharacteristic
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#386—21no—
44#386—21no—
45#386—21no—
46#386—21no—
47#382C2211yescharacteristic
48C4×C4⋊C4C2211yescharacteristic
49C4×C4⋊C4C2211yescharacteristic
50C4×C4⋊C4C2211yescharacteristic
51C4×C22⋊C4—21no—
52C4×C22⋊C4—21no—
53C4×C22⋊C4—21no—
54#376C2211yescharacteristic
55C2×#95—21no—
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62C22×D4—21no—
63C22×D4—21no—
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C22×D4C2311yescharacteristic
67C23×C4—21no—
68C23×C4—21no—
69C23×C4—21no—
70C23×C4—21no—
71C23×C4—21no—
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—22no—
76C2×D8C2×C414yescomplemented
77C2×D8C2×C414yescomplemented
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82#127—22no—
83#127C2×C414yescomplemented
84C4×D4—22no—
85C4×D4—22no—
86C4×D4—22no—
87C4×D4—22no—
88C4×D4—22no—
89C4×D4—22no—
90C4×D4—22no—
91C4×D4—22no—
92C4×D4C2×C414yescomplemented
93C4×D4D414yescomplemented
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
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×C42—21no—
109C2×C42—21no—
110C2×C42—21no—
111C2×C42C2311yescharacteristic
112C2×C42C2311yescharacteristic
113C2×C42C2311yescharacteristic
114#107—22no—
115#106—22no—
116C4⋊C8—22no—
117C4⋊C8C2×C414yescomplemented
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102C2×C414yescomplemented
125#102C2×C414yescomplemented
126#97—22no—
127C4×C8—22no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95D411yescharacteristic
134#95D411yescharacteristic
135#95D411yescharacteristic
136#95D411yescharacteristic
137C24—21no—
138C24—21no—
139C24—21no—
140C2×D4—24no—
141C2×D4—24no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—21no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—81no—
156C2×D4—81no—
157C2×D4—81no—
158C2×D4—81no—
159C2×D4C22×C412yes—
160C2×D4C22×C412yes—
161C2×D4C22×C412yes—
162C2×D4C2×D412yes—
163C2×D4#4112yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—24no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—81no—
175C22×C4—81no—
176C22×C4—81no—
177C22×C4—81no—
178C22×C4—81no—
179C22×C4—81no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2411yescharacteristic
191D8—28no—
192D8—28no—
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×C8C22×C412yes—
204C2×C8C22×C412yes—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4C22×C412yes—
213C4⋊C4C2×D412yes—
214C4⋊C4#4112yes—
215C22⋊C4—24no—
216C22⋊C4—24no—
217C22⋊C4—24no—
218C22⋊C4—42no—
219C22⋊C4—42no—
220C22⋊C4—81no—
221C22⋊C4—81no—
222C22⋊C4—81no—
223C22⋊C4—81no—
224C42—21no—
225C42—21no—
226C42—81no—
227C42—81no—
228C42—81no—
229C42C22×C412yes—
230C42C2×D412yes—
231C42#4112yes—
232C42#4112yes—
233C23—22no—
234C23—22no—
235C23—22no—
236C23—22no—
237C23—22no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23C22×D411yescharacteristic
255D4—22no—
256D4—22no—
257D4—22no—
258D4—22no—
259D4—22no—
260D4—22no—
261D4—22no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—84no—
267D4—84no—
268D4C4×D414yes—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
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—84no—
299C2×C4—84no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4C4×D412yes—
304C2×C4C4×D412yes—
305C2×C4#12112yes—
306C2×C4#12112yes—
307C2×C4C23×C411yescharacteristic
308C2×C4C22×D411yescharacteristic
309C2×C4C2×#4111yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4C2×#4111yescharacteristic
313C8—22no—
314C8—22no—
315C8—41no—
316C8—41no—
317C22—24no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—42no—
328C22—42no—
329C22—42no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22C2×C4×D411yescharacteristic
339C22C2×#12111yescharacteristic
340C22#52511yescharacteristic
341C22#57611yescharacteristic
342C22#57611yescharacteristic
343C22#57611yescharacteristic
344C22#57711yescharacteristic
345C4—22no—
346C4—22no—
347C4—22no—
348C4—22no—
349C4—41no—
350C4—41no—
351C4—42no—
352C4—42no—
353C4—42no—
354C4—81no—
355C4—81no—
356C4C2×C4×D411yescharacteristic
357C4C4×#4111yescharacteristic
358C4#52911yescharacteristic
359C4#53211yescharacteristic
360C2—44no—
361C2—82no—
362C2—82no—
363C2C4×#12111yescharacteristic
364C2#294611yescharacteristic
365C2#294811yescharacteristic
366C2#304111yescharacteristic
367C2#304511yescharacteristic
368C2#311911yescharacteristic
369C2#312211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
prime radicalC2×#459C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#442C2semidirect
3. C2×C4×D8C2semidirect
4. #2328C2semidirect
5. #2275C2semidirect
6. #1982C2semidirect
7. #1949C2semidirect
8. #1926C2semidirect
9. #1925C2semidirect
10. #1879C2semidirect
11. #1879C2semidirect
12. #1804C2semidirect
13. #1768C2semidirect
14. #1762C2semidirect
15. C4×#102C2semidirect
24. #459C4semidirect
76. C2×D8C2×C4semidirect
77. C2×D8C2×C4semidirect
83. #127C2×C4semidirect
92. C4×D4C2×C4semidirect
93. C4×D4D4semidirect
117. C4⋊C8C2×C4semidirect
124. #102C2×C4semidirect
125. #102C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#4592
C2×#4422
C2×C4×D82
#23282
#22752
#19822
#19492
#19262
#19252
#18792
#18042
#17682
#17622
C4×#1022