Finite Group Databasebeta

← #24786

Subgroups — #24786

Order 256.

495 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#468C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4C2×#427C211yesmaximal, characteristic, complemented
5#2415C211yesmaximal, characteristic, complemented
6#2414C211yesmaximal, characteristic
7#2066C211yesmaximal, characteristic, complemented
8#2065C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#2034C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1943C211yesmaximal, characteristic, complemented
13#1875C211yesmaximal, characteristic, complemented
14#1873C211yesmaximal, characteristic, complemented
15#1809C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C23×C8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×C22⋊Q8C2211yescharacteristic
21C2×#121C2211yescharacteristic
22C2×#121C2211yescharacteristic
23C2×C4×D4C2211yescharacteristic
24C2×#117C2211yescharacteristic
25C2×#117C2211yescharacteristic
26#468—24no—
27#465—24no—
28#435—24no—
29#427—24no—
30C2×#107C2211yescharacteristic
31C2×#107C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×#106C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45#397—21no—
46#394—21no—
47#394—21no—
48#392—21no—
49#391C2211yescharacteristic
50#390—21no—
51#390C2211yescharacteristic
52#389—21no—
53#389C2211yescharacteristic
54#389C2211yescharacteristic
55#388—21no—
56#388—21no—
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4—21no—
65C22×D4—21no—
66C22×D4D411yescharacteristic
67C22×D4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C2×SD16—22no—
71C2×SD16—22no—
72C2×SD16—22no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—22no—
76C22×C8—22no—
77C22×C8—22no—
78C22×C8D414yescomplemented
79C22×C8C2311yescharacteristic
80C22×C8C2311yescharacteristic
81C22×C8C2311yescharacteristic
82C22×C8C2311yescharacteristic
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86C22⋊Q8—41no—
87#121—22no—
88#121—22no—
89#121—22no—
90#121—41no—
91C4×D4—22no—
92C4×D4—22no—
93#117—22no—
94#117—22no—
95#117—41no—
96#117—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4D411yescharacteristic
102C2×C4⋊C4C2311yescharacteristic
103C2×C4⋊C4C2311yescharacteristic
104C2×C4⋊C4C2311yescharacteristic
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4D411yescharacteristic
119C2×C22⋊C4D411yescharacteristic
120C2×C42—21no—
121C2×C42—21no—
122C2×C42C2311yescharacteristic
123#107—22no—
124#107D414yescomplemented
125#106—22no—
126#106—22no—
127#106—22no—
128C4⋊C8—22no—
129#103—22no—
130#103—22no—
131#103—41no—
132#102—22no—
133#102—22no—
134#102—41no—
135C22⋊C8—22no—
136C22⋊C8—22no—
137#97—22no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144C24—21no—
145C24C2×D411yescharacteristic
146C24C2×D411yescharacteristic
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×D4—22no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—81no—
166C2×D4—81no—
167C2×D4—81no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
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×C4—21no—
184C22×C4—21no—
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D412yes—
190C22×C4#4112yes—
191C22×C4C2411yescharacteristic
192SD16—44no—
193M16—44no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—28no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8C2×D412yes—
204C2×C8C2×D412yes—
205C2×C8C2×D412yes—
206C2×C8#4112yes—
207C4⋊C4—22no—
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—21no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C4⋊C4—42no—
221C4⋊C4—81no—
222C4⋊C4C2×D412yes—
223C4⋊C4C2×D412yes—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22⋊C4—81no—
235C22⋊C4—81no—
236C42—21no—
237C42—21no—
238C42—42no—
239C42—42no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23#12112yes—
261C23C22×D411yescharacteristic
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266D4—44no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—84no—
272D4—84no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
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—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#12112yes—
314C2×C4C22×D411yescharacteristic
315C2×C4C22×D411yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4H(5,𝔽2)11yescharacteristic
319C2×C42−1+411yescharacteristic
320C8—22no—
321C8—41no—
322C8—41no—
323C8#12114yes—
324C22—22no—
325C22—22no—
326C22—22no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22—81no—
345C22C2×#12111yescharacteristic
346C22D4211yescharacteristic
347C22#54711yescharacteristic
348C22#57511yescharacteristic
349C22#57511yescharacteristic
350C22#57711yescharacteristic
351C22#57711yescharacteristic
352C4—22no—
353C4—41no—
354C4—42no—
355C4—42no—
356C4—41no—
357C4—41no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4C2×#12111yescharacteristic
364C4#53711yescharacteristic
365C4#54811yescharacteristic
366C4#55311yescharacteristic
367C2—42no—
368C2—81no—
369C2—161no—
370C2#241611yescharacteristic
371C2#305411yescharacteristic
372C2#305711yescharacteristic
373C2#326711yescharacteristic
374C2#327111yescharacteristic
375C2#334811yescharacteristic
376C2#335011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2414C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#468C2semidirect
2. C2×#465C2semidirect
3. C2×#435C2semidirect
4. C2×#427C2semidirect
5. #2415C2semidirect
7. #2066C2semidirect
8. #2065C2semidirect
9. #2035C2semidirect
10. #2034C2semidirect
11. #1973C2semidirect
12. #1943C2semidirect
13. #1875C2semidirect
14. #1873C2semidirect
15. #1809C2semidirect
78. C22×C8D4semidirect
124. #107D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4682
C2×#4652
C2×#4352
C2×#4272
#24152
#24142
#20662
#20652
#20352
#20342
#19732
#19432
#18752
#18732
#18092