Finite Group Databasebeta

← #24516

Subgroups — #24516

Order 256.

528 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#450C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic, complemented
3#2625C211yesmaximal, characteristic, complemented
4#2309C211yesmaximal, characteristic
5#2067C211yesmaximal, characteristic, complemented
6#2065C211yesmaximal, characteristic, complemented
7#2055C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#1973C211yesmaximal, characteristic, complemented
10#1968C211yesmaximal, characteristic, complemented
11#1936C211yesmaximal, characteristic, complemented
12#1934C211yesmaximal, characteristic, complemented
13#1873C211yesmaximal, characteristic, complemented
14#1854C211yesmaximal, characteristic, complemented
15#1786C211yesmaximal, characteristic
16C23×D4C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C2×#117C2211yescharacteristic
22#450—24no—
23#433—24no—
24C2×#107C2211yescharacteristic, complemented
25C2×#106C2211yescharacteristic, complemented
26C2×C4⋊C8C2211yescharacteristic, complemented
27C2×C4⋊C8C2211yescharacteristic, complemented
28C2×#103C2211yescharacteristic, complemented
29C2×#103C2211yescharacteristic, complemented
30C2×#102C2211yescharacteristic
31C2×#102C2211yescharacteristic
32C2×#102C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic, complemented
39C2×C4×C8C2211yescharacteristic, complemented
40#400—21no—
41#398—21no—
42#397—21no—
43#397—21no—
44#391C2211yescharacteristic, complemented
45#391C2211yescharacteristic, complemented
46#390—21no—
47#390—21no—
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#390C2211yescharacteristic
52#389C2211yescharacteristic, complemented
53#389C2211yescharacteristic, complemented
54#380C2211yescharacteristic
55#379—21no—
56#379—21no—
57#340C2211yescharacteristic, complemented
58#340C2211yescharacteristic, complemented
59#340C2211yescharacteristic, complemented
60#340C2211yescharacteristic, complemented
61C25—21no—
62C22×Q8C2311yescharacteristic
63C22×D4—22no—
64C22×D4—22no—
65C22×D4—21no—
66C22×D4—22no—
67C22×D4—22no—
68C22×D4—41no—
69C22×D4—41no—
70C22×D4C2311yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—41no—
78C2×SD16—41no—
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85#117—22no—
86#117—22no—
87#117—41no—
88#117—41no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4D411yescharacteristic
93C2×C4⋊C4D411yescharacteristic
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×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4D411yescharacteristic
111C2×C22⋊C4D411yescharacteristic
112C2×C42—21no—
113C2×C42C2311yescharacteristic
114C2×C42C2311yescharacteristic
115#107—41no—
116#106—41no—
117C4⋊C8—22no—
118C4⋊C8—22no—
119#103—41no—
120#103—41no—
121#102—22no—
122#102—22no—
123#102—22no—
124#102—22no—
125#102—41no—
126#102—41no—
127C22⋊C8—22no—
128C22⋊C8D414yes—
129#97—22no—
130C4×C8—22no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—24no—
139C24—21no—
140C24—21no—
141C24—42no—
142C24—42no—
143C24—81no—
144C24—81no—
145C24—81no—
146C24—81no—
147C24C2×D411yescharacteristic
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×D4—24no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—44no—
166C2×D4—44no—
167C2×D4—44no—
168C2×D4—44no—
169C2×D4—42no—
170C2×D4—42no—
171C2×D4#4112yes—
172C2×D4#4112yes—
173C22×C4—21no—
174C22×C4—21no—
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×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D412yes—
189C22×C4#4112yes—
190C22×C4C2411yescharacteristic
191SD16—44no—
192SD16—44no—
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×C8C2×D412yes—
204C2×C8C2×D412yes—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—21no—
208C4⋊C4—21no—
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—42no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C22⋊C4—81no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—42no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—48no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—84no—
260C23—84no—
261C23—84no—
262C23—84no—
263C23—84no—
264C23—84no—
265C23#12112yes—
266C23C22×D411yescharacteristic
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271D4—22no—
272D4—22no—
273D4—22no—
274D4—22no—
275D4—48no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
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—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
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#12112yes—
317C2×C4C22×D411yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C8—22no—
324C8—22no—
325C8—41no—
326C8—41no—
327C22—22no—
328C22—22no—
329C22—22no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—42no—
340C22—816no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22C2×#12111yescharacteristic
350C22#52511yescharacteristic
351C22#53511yescharacteristic
352C22#57611yescharacteristic
353C22#57711yescharacteristic
354C22#57711yescharacteristic
355C22#57711yescharacteristic
356C4—22no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—81no—
367C4#54611yescharacteristic
368C4#54611yescharacteristic
369C4#55111yescharacteristic
370C4#55111yescharacteristic
371C2—42no—
372C2—82no—
373C2—82no—
374C2#241111yescharacteristic
375C2#305711yescharacteristic
376C2#306211yescharacteristic
377C2#311811yescharacteristic
378C2#312311yescharacteristic
379C2#322211yescharacteristic
380C2#322411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC23×D4C22
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
2. C2×#433C2semidirect
3. #2625C2semidirect
5. #2067C2semidirect
6. #2065C2semidirect
7. #2055C2semidirect
8. #2055C2semidirect
9. #1973C2semidirect
10. #1968C2semidirect
11. #1936C2semidirect
12. #1934C2semidirect
13. #1873C2semidirect
14. #1854C2semidirect
24. C2×#107C22semidirect
25. C2×#106C22semidirect
26. C2×C4⋊C8C22semidirect
27. C2×C4⋊C8C22semidirect
28. C2×#103C22semidirect
29. C2×#103C22semidirect
38. C2×#97C22semidirect
39. C2×C4×C8C22semidirect
44. #391C22semidirect
45. #391C22semidirect
52. #389C22semidirect
53. #389C22semidirect
57. #340C22semidirect
58. #340C22semidirect
59. #340C22semidirect
60. #340C22semidirect

Maximal subgroups

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

subgroupindex
C2×#4502
C2×#4332
#26252
#23092
#20672
#20652
#20552
#19732
#19682
#19362
#19342
#18732
#18542
#17862