Finite Group Databasebeta

← #24515

Subgroups — #24515

Order 256.

472 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×#452C211yesmaximal, characteristic
2C2×#433C211yesmaximal, characteristic, complemented
3#2625C211yesmaximal, characteristic, complemented
4#2311C211yesmaximal, characteristic
5#2068C211yesmaximal, characteristic, complemented
6#2067C211yesmaximal, characteristic, complemented
7#2056C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#1973C211yesmaximal, characteristic, complemented
10#1969C211yesmaximal, characteristic, complemented
11#1935C211yesmaximal, characteristic, complemented
12#1934C211yesmaximal, characteristic, complemented
13#1872C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22#452—24no—
23#433—24no—
24C2×#107C2211yescharacteristic
25C2×#107C2211yescharacteristic
26C2×C4⋊C8C2211yescharacteristic
27C2×C4⋊C8C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#102C2211yescharacteristic
34C2×#102C2211yescharacteristic
35C2×#102C2211yescharacteristic
36C2×C22⋊C8C2211yescharacteristic
37C2×C22⋊C8C2211yescharacteristic
38C2×#97C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40#400—21no—
41#398—21no—
42#397—21no—
43#397—21no—
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#390C2211yescharacteristic
49#390C2211yescharacteristic
50#389C2211yescharacteristic
51#389C2211yescharacteristic
52#386—21no—
53#386—21no—
54#380—21no—
55#380—21no—
56#380C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61C2×#41—22no—
62C2×#41—22no—
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—41no—
66C2×#41—41no—
67C22×Q8C2311yescharacteristic
68C22×Q8C2311yescharacteristic
69C22×D4—21no—
70C22×D4C2311yescharacteristic
71C23×C4—21no—
72C23×C4—21no—
73C23×C4C2311yescharacteristic
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—41no—
77C2×SD16—22no—
78C2×SD16—22no—
79C2×SD16—41no—
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22×C8C2311yescharacteristic
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86#117—22no—
87#117—22no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—21no—
91C2×C4⋊C4—21no—
92C2×C4⋊C4—41no—
93C2×C4⋊C4—41no—
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⋊C4D411yescharacteristic
108C2×C22⋊C4D411yescharacteristic
109C2×C42C2311yescharacteristic
110C2×C42C2311yescharacteristic
111#107—41no—
112#107—41no—
113C4⋊C8—22no—
114C4⋊C8—22no—
115#103—22no—
116#103—22no—
117#103—41no—
118#103—41no—
119#103—41no—
120#102—22no—
121#102—22no—
122#102—41no—
123C22⋊C8—22no—
124C22⋊C8D414yes—
125#97—22no—
126C4×C8—22no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133#95—21no—
134#95—21no—
135#95—21no—
136#95—21no—
137#95—21no—
138C24—21no—
139C24C2×D411yescharacteristic
140#41—44no—
141#41—44no—
142#41—44no—
143#41—44no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8#4112yes—
156C2×D4—24no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4#4112yes—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—24no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—81no—
181C22×C4—81no—
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D412yes—
188C22×C4#4112yes—
189C22×C4C2411yescharacteristic
190Q16—44no—
191SD16—44no—
192C2×C8—21no—
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×C8C2×D412yes—
203C2×C8C2×D412yes—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—21no—
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—42no—
217C4⋊C4—42no—
218C4⋊C4—81no—
219C4⋊C4—81no—
220C22⋊C4—22no—
221C22⋊C4—22no—
222C22⋊C4—42no—
223C22⋊C4—81no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23#12112yes—
246C23C22×D411yescharacteristic
247Q8—22no—
248Q8—22no—
249Q8—41no—
250Q8—41no—
251Q8—41no—
252Q8—41no—
253Q8—41no—
254Q8—41no—
255D4—22no—
256D4—22no—
257D4—48no—
258D4—41no—
259D4—41no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—48no—
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—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—84no—
301C2×C4#12112yes—
302C2×C4C22×D411yescharacteristic
303C2×C4C2×#4111yescharacteristic
304C2×C4C2×#4111yescharacteristic
305C2×C4C2×#4111yescharacteristic
306C2×C4H(5,𝔽2)11yescharacteristic
307C2×C4H(5,𝔽2)11yescharacteristic
308C8—22no—
309C8—22no—
310C8—41no—
311C8—41no—
312C22—22no—
313C22—22no—
314C22—22no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22C2×#12111yescharacteristic
327C22#52511yescharacteristic
328C22#53511yescharacteristic
329C22#57611yescharacteristic
330C22#57711yescharacteristic
331C22#57811yescharacteristic
332C22#57811yescharacteristic
333C4—22no—
334C4—42no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4#54611yescharacteristic
346C4#54611yescharacteristic
347C4#55111yescharacteristic
348C4#55111yescharacteristic
349C2—42no—
350C2—82no—
351C2#241111yescharacteristic
352C2#306111yescharacteristic
353C2#306311yescharacteristic
354C2#312011yescharacteristic
355C2#312411yescharacteristic
356C2#322311yescharacteristic
357C2#322411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
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. #2068C2semidirect
6. #2067C2semidirect
7. #2056C2semidirect
8. #2055C2semidirect
9. #1973C2semidirect
10. #1969C2semidirect
11. #1935C2semidirect
12. #1934C2semidirect
13. #1872C2semidirect
14. #1852C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4522
C2×#4332
#26252
#23112
#20682
#20672
#20562
#20552
#19732
#19692
#19352
#19342
#18722
#18522
#17882