Finite Group Databasebeta

← #24662

Subgroups — #24662

Order 256.

485 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#459C211yesmaximal, characteristic, complemented
2C2×#433C211yesmaximal, characteristic, complemented
3#2619C211yesmaximal, characteristic, complemented
4#2324C211yesmaximal, characteristic, complemented
5#2034C211yesmaximal, characteristic, complemented
6#2033C211yesmaximal, characteristic, complemented
7#1999C211yesmaximal, characteristic, complemented
8#1999C211yesmaximal, characteristic, complemented
9#1982C211yesmaximal, characteristic, complemented
10#1972C211yesmaximal, characteristic, complemented
11#1970C211yesmaximal, characteristic, complemented
12#1949C211yesmaximal, characteristic, complemented
13#1936C211yesmaximal, characteristic, complemented
14#1934C211yesmaximal, characteristic, complemented
15#1804C211yesmaximal, characteristic, complemented
16C22×D8C2211yescharacteristic
17C22×D8C2211yescharacteristic
18C2×#127C2211yescharacteristic
19C2×#121C2211yescharacteristic
20C2×#121C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×C4×D4C2211yescharacteristic
23C2×#117C2211yescharacteristic
24#459—24no—
25#433—24no—
26C2×#107C2211yescharacteristic
27C2×#106C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×C4⋊C8C2211yescharacteristic
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×C22⋊C8C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×#97C2211yescharacteristic
41C2×C4×C8C2211yescharacteristic
42#394—21no—
43#394—21no—
44#392—21no—
45#392—21no—
46#390—21no—
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#389C2211yescharacteristic
51#388—21no—
52#386—21no—
53#385—21no—
54#382C2211yescharacteristic
55#380C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C22×D4—21no—
62C22×D4—21no—
63C22×D4—21no—
64C22×D4—21no—
65C22×D4—21no—
66C22×D4C2311yescharacteristic
67C22×D4C2311yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4—21no—
70C23×C4C2311yescharacteristic
71C2×D8—22no—
72C2×D8—22no—
73C2×D8—22no—
74C2×D8—22no—
75C2×D8—41no—
76C2×D8—41no—
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81#127—22no—
82#127—22no—
83#121—22no—
84#121—22no—
85#121—41no—
86#121—41no—
87#121—41no—
88#121—41no—
89C4×D4—22no—
90C4×D4—22no—
91#117—22no—
92#117—22no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4D411yescharacteristic
95C2×C4⋊C4D411yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C4⋊C4C2311yescharacteristic
100C2×C4⋊C4C2311yescharacteristic
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×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4D411yescharacteristic
113C2×C22⋊C4D411yescharacteristic
114C2×C42—21no—
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117#107—41no—
118#106—41no—
119C4⋊C8—22no—
120C4⋊C8D414yescomplemented
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—41no—
126#102—41no—
127#102—41no—
128#102—41no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131#97—22no—
132C4×C8—22no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24—21no—
140C24—21no—
141C24C2×D411yescharacteristic
142C2×D4—24no—
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—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—81no—
165C2×D4—81no—
166C2×D4—81no—
167C2×D4—81no—
168C2×D4—81no—
169C2×D4—81no—
170C2×D4—81no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4—81no—
174C2×D4#4112yes—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2411yescharacteristic
195D8—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×C8C2×D412yes—
208C2×C8C2×D412yes—
209C4⋊C4—22no—
210C4⋊C4—22no—
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—21no—
220C4⋊C4—81no—
221C4⋊C4#4112yes—
222C22⋊C4—24no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C22⋊C4—81no—
234C22⋊C4—81no—
235C42—21no—
236C42—21no—
237C42—81no—
238C42C2×D412yes—
239C42#4112yes—
240C23—21no—
241C23—22no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23C22×D411yescharacteristic
269D4—22no—
270D4—22no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—84no—
282D4—84no—
283D4—84no—
284D4—84no—
285D4—84no—
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—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
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—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—84no—
319C2×C4#12112yes—
320C2×C4#12112yes—
321C2×C4C22×D411yescharacteristic
322C2×C4C2×#4111yescharacteristic
323C2×C4C2×#4111yescharacteristic
324C2×C4C2×#4111yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C2×C4H(5,𝔽2)11yescharacteristic
327C8—22no—
328C8—22no—
329C8—41no—
330C8—41no—
331C22—41no—
332C22—42no—
333C22—41no—
334C22—42no—
335C22—41no—
336C22—42no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22C2×#12111yescharacteristic
360C22#52511yescharacteristic
361C22#53511yescharacteristic
362C22#57611yescharacteristic
363C22#57611yescharacteristic
364C22#57611yescharacteristic
365C22#57711yescharacteristic
366C4—22no—
367C4—22no—
368C4—42no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4#54611yescharacteristic
377C4#55011yescharacteristic
378C4#55311yescharacteristic
379C4#55511yescharacteristic
380C2—81no—
381C2—82no—
382C2—161no—
383C2—161no—
384C2#241411yescharacteristic
385C2#305811yescharacteristic
386C2#306211yescharacteristic
387C2#311911yescharacteristic
388C2#312211yescharacteristic
389C2#322011yescharacteristic
390C2#322411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
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×#459C2semidirect
2. C2×#433C2semidirect
3. #2619C2semidirect
4. #2324C2semidirect
5. #2034C2semidirect
6. #2033C2semidirect
7. #1999C2semidirect
8. #1999C2semidirect
9. #1982C2semidirect
10. #1972C2semidirect
11. #1970C2semidirect
12. #1949C2semidirect
13. #1936C2semidirect
14. #1934C2semidirect
15. #1804C2semidirect
120. C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4592
C2×#4332
#26192
#23242
#20342
#20332
#19992
#19822
#19722
#19702
#19492
#19362
#19342
#18042