Finite Group Databasebeta

← #23712

Subgroups — #23712

Order 256.

466 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#442C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic, complemented
3#2324C211yesmaximal, characteristic, complemented
4#2324C211yesmaximal, characteristic, complemented
5#2088C211yesmaximal, characteristic, complemented
6#2077C211yesmaximal, characteristic, complemented
7#2067C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2054C211yesmaximal, characteristic, complemented
10#2034C211yesmaximal, characteristic, complemented
11#2033C211yesmaximal, characteristic, complemented
12#1999C211yesmaximal, characteristic
13#1776C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15C4×#102C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C2×#121C2211yescharacteristic
18C2×#121C2211yescharacteristic
19C2×C4×D4C2211yescharacteristic
20C2×C4×D4C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#442—24no—
23#433—24no—
24C2×#107C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#106C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#102C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×#97C2211yescharacteristic
38C2×C4×C8C2211yescharacteristic
39#390—21no—
40#390—21no—
41#390C2211yescharacteristic
42#390C2211yescharacteristic
43#389C2211yescharacteristic
44#389C2211yescharacteristic
45#388—21no—
46#388—21no—
47#386—21no—
48#386—21no—
49#385—21no—
50#385—21no—
51#380C2211yescharacteristic
52#380C2211yescharacteristic
53C4×C4⋊C4C2211yescharacteristic
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C23×C4—21no—
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C2×D8—22no—
69C2×D8—22no—
70C2×D8—22no—
71C22×C8C2311yescharacteristic
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75#121—22no—
76#121—22no—
77#121—22no—
78#121—22no—
79C4×D4—22no—
80C4×D4—22no—
81C4×D4—22no—
82C4×D4—22no—
83#117—22no—
84#117—22no—
85C2×C4⋊C4—21no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4D411yescharacteristic
88C2×C4⋊C4D411yescharacteristic
89C2×C4⋊C4C2311yescharacteristic
90C2×C4⋊C4C2311yescharacteristic
91C2×C4⋊C4C2311yescharacteristic
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C22⋊C4—21no—
96C2×C22⋊C4—21no—
97C2×C22⋊C4—21no—
98C2×C22⋊C4—21no—
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⋊C4D411yescharacteristic
106C2×C22⋊C4D411yescharacteristic
107C2×C42—21no—
108C2×C42—21no—
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#107—22no—
112#106—22no—
113#106—22no—
114C4⋊C8—22no—
115C4⋊C8—22no—
116#102—22no—
117#102—22no—
118#102—22no—
119#102—22no—
120#102—22no—
121#102—22no—
122C22⋊C8—22no—
123C22⋊C8—22no—
124#97D414yescomplemented
125C4×C8—22no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24—21no—
133C24—21no—
134C24C2×D411yescharacteristic
135C2×D4—24no—
136C2×D4—24no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—21no—
140C2×D4—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—81no—
148C2×D4—81no—
149C2×D4—81no—
150C2×D4—81no—
151C2×D4#4112yes—
152C2×D4#4112yes—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—42no—
166C22×C4—42no—
167C22×C4—81no—
168C22×C4—81no—
169C22×C4—81no—
170C22×C4—81no—
171C22×C4C2×D411yescharacteristic
172C22×C4C2×D411yescharacteristic
173C22×C4C2×D411yescharacteristic
174C22×C4C2×D411yescharacteristic
175C22×C4C2×D411yescharacteristic
176C22×C4C2411yescharacteristic
177D8—28no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—21no—
182C2×C8—21no—
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—21no—
187C2×C8—21no—
188C2×C8C2×D412yes—
189C2×C8C2×D412yes—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—21no—
193C4⋊C4—21no—
194C4⋊C4—21no—
195C4⋊C4—21no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—81no—
203C4⋊C4—81no—
204C4⋊C4#4112yes—
205C4⋊C4#4112yes—
206C22⋊C4—24no—
207C22⋊C4—24no—
208C22⋊C4—22no—
209C22⋊C4—22no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C22⋊C4—81no—
215C22⋊C4—81no—
216C22⋊C4—81no—
217C22⋊C4—81no—
218C42—21no—
219C42—21no—
220C42—81no—
221C42—81no—
222C42C2×D412yes—
223C42#4112yes—
224C23—21no—
225C23—22no—
226C23—22no—
227C23—21no—
228C23—21no—
229C23—21no—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23—41no—
236C23—41no—
237C23—41no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23C22×D411yescharacteristic
246D4—22no—
247D4—22no—
248D4—22no—
249D4—22no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—84no—
255D4—84no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
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—84no—
292C2×C4—84no—
293C2×C4#12112yes—
294C2×C4#12112yes—
295C2×C4C22×D411yescharacteristic
296C2×C4C2×#4111yescharacteristic
297C2×C4C2×#4111yescharacteristic
298C2×C4C2×#4111yescharacteristic
299C2×C4C2×#4111yescharacteristic
300C2×C4C2×#4111yescharacteristic
301C8—22no—
302C8—22no—
303C8—41no—
304C8—41no—
305C22—41no—
306C22—42no—
307C22—41no—
308C22—42no—
309C22—41no—
310C22—42no—
311C22—41no—
312C22—42no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22C2×#12111yescharacteristic
327C22#52511yescharacteristic
328C22#52511yescharacteristic
329C22#57611yescharacteristic
330C22#57611yescharacteristic
331C22#57711yescharacteristic
332C22#57711yescharacteristic
333C4—22no—
334C4—22no—
335C4—42no—
336C4—42no—
337C4—41no—
338C4—41no—
339C4—41no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4#52511yescharacteristic
345C4#52911yescharacteristic
346C4#52911yescharacteristic
347C4#53211yescharacteristic
348C2—81no—
349C2—82no—
350C2—82no—
351C2#239211yescharacteristic
352C2#306211yescharacteristic
353C2#306211yescharacteristic
354C2#311811yescharacteristic
355C2#311911yescharacteristic
356C2#312211yescharacteristic
357C2#312511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#1999C2
LCSG ▷ C22×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
1. C2×#442C2semidirect
2. C2×#433C2semidirect
3. #2324C2semidirect
4. #2324C2semidirect
5. #2088C2semidirect
6. #2077C2semidirect
7. #2067C2semidirect
8. #2065C2semidirect
9. #2054C2semidirect
10. #2034C2semidirect
11. #2033C2semidirect
13. #1776C2semidirect
14. C4×#102C2semidirect
15. C4×#102C2semidirect
124. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4422
C2×#4332
#23242
#20882
#20772
#20672
#20652
#20542
#20342
#20332
#19992
#17762
C4×#1022