Finite Group Databasebeta

← #30542

Subgroups — #30542

Order 256.

481 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#491C211yesmaximal, characteristic, complemented
2C2×#448C211yesmaximal, characteristic
3#2490C211yesmaximal, characteristic
4#2434C211yesmaximal, characteristic, complemented
5#2055C211yesmaximal, characteristic, complemented
6#2036C211yesmaximal, characteristic, complemented
7#2026C211yesmaximal, characteristic, complemented
8#2013C211yesmaximal, characteristic, complemented
9#2013C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic
11#1981C211yesmaximal, characteristic, complemented
12#1973C211yesmaximal, characteristic, complemented
13#1928C211yesmaximal, characteristic, complemented
14#1924C211yesmaximal, characteristic, complemented
15#1787C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×C4⋊Q8C2211yescharacteristic
22C2×#125C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25#491—24no—
26#448—24no—
27C2×#107C2211yescharacteristic
28C2×C4⋊C8C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×#103C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C22⋊C8C2211yescharacteristic
40C2×C22⋊C8C2211yescharacteristic
41C2×#97C2211yescharacteristic
42#397—21no—
43#395—21no—
44#395—21no—
45#394—21no—
46#391C2211yescharacteristic
47#391C2211yescharacteristic
48#390—21no—
49#390C2211yescharacteristic
50#390C2211yescharacteristic
51#388—21no—
52#385—21no—
53#385—21no—
54#380C2211yescharacteristic
55C4×C4⋊C4C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8—21no—
62C22×Q8—22no—
63C22×Q8—22no—
64C22×Q8—41no—
65C22×Q8—41no—
66C22×Q8—41no—
67C22×Q8—41no—
68C22×Q8C2311yescharacteristic
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4C2311yescharacteristic
73C23×C4—21no—
74C23×C4C2311yescharacteristic
75C2×SD16—22no—
76C2×SD16—22no—
77C2×SD16—22no—
78C2×SD16—41no—
79C2×SD16—41no—
80C2×SD16—41no—
81C2×SD16—41no—
82C2×SD16—41no—
83C2×SD16—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C4⋊Q8—22no—
89C4⋊Q8—22no—
90C4⋊Q8—41no—
91C4⋊Q8—41no—
92#125—22no—
93#125—41no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96#121—22no—
97#121—41no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4C2311yescharacteristic
106C2×C4⋊C4C2311yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
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⋊C4D411yescharacteristic
117C2×C22⋊C4D411yescharacteristic
118C2×C42—21no—
119C2×C42C2311yescharacteristic
120C2×C42C2311yescharacteristic
121#107—41no—
122C4⋊C8—41no—
123#103—22no—
124#103—22no—
125#103—22no—
126#103—22no—
127#103—41no—
128#103—41no—
129#103—41no—
130#103—41no—
131#103D414yes—
132#102—41no—
133C22⋊C8—22no—
134C22⋊C8—41no—
135#97—22no—
136#95—21no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24C2×D411yescharacteristic
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—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—42no—
163C2×Q8—42no—
164C2×Q8—42no—
165C2×Q8—44no—
166C2×Q8—44no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8C2×D412yes—
170C2×Q8#4112yes—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—81no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—42no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4C2×D411yescharacteristic
197C22×C4C2×D411yescharacteristic
198C22×C4C2×D411yescharacteristic
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2411yescharacteristic
202SD16—44no—
203SD16—44no—
204SD16—44no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8C2×D412yes—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4C2×D412yes—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C22⋊C4—81no—
240C22⋊C4—81no—
241C42—21no—
242C42—21no—
243C42—21no—
244C42—21no—
245C42—81no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#13712yes—
260C23C22×D411yescharacteristic
261Q8—22no—
262Q8—22no—
263Q8—22no—
264Q8—22no—
265Q8—48no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269Q8—41no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—41no—
274D4—41no—
275D4—41no—
276D4—41no—
277D4—41no—
278D4—84no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—48no—
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—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4#13712yes—
320C2×C4C22×D411yescharacteristic
321C2×C4C2×#4111yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C2×C4H(5,𝔽2)11yescharacteristic
324C2×C4H(5,𝔽2)11yescharacteristic
325C2×C4H(5,𝔽2)11yescharacteristic
326C8—22no—
327C8—41no—
328C8—41no—
329C8—41no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22—81no—
343C22—81no—
344C22#53511yescharacteristic
345C22D4211yescharacteristic
346C22#54611yescharacteristic
347C22C2×#13711yescharacteristic
348C22C2×#13711yescharacteristic
349C22#57711yescharacteristic
350C22#57711yescharacteristic
351C4—22no—
352C4—41no—
353C4—42no—
354C4—42no—
355C4—41no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—81no—
362C4—81no—
363C4#53511yescharacteristic
364C4#53811yescharacteristic
365C4#54611yescharacteristic
366C4H(3,𝔽4)11yescharacteristic
367C2—42no—
368C2—161no—
369C2#267011yescharacteristic
370C2#321811yescharacteristic
371C2#322111yescharacteristic
372C2#326611yescharacteristic
373C2#327811yescharacteristic
374C2#332011yescharacteristic
375C2#332011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#491C2semidirect
4. #2434C2semidirect
5. #2055C2semidirect
6. #2036C2semidirect
7. #2026C2semidirect
8. #2013C2semidirect
9. #2013C2semidirect
11. #1981C2semidirect
12. #1973C2semidirect
13. #1928C2semidirect
14. #1924C2semidirect
15. #1787C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4912
C2×#4482
#24902
#24342
#20552
#20362
#20262
#20132
#20002
#19812
#19732
#19282
#19242
#17872