Finite Group Databasebeta

← #30748

Subgroups — #30748

Order 256.

443 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#495C211yesmaximal, characteristic
2C2×#462C211yesmaximal, characteristic, complemented
3#2545C211yesmaximal, characteristic, complemented
4#2448C211yesmaximal, characteristic, complemented
5#2054C211yesmaximal, characteristic
6#2042C211yesmaximal, characteristic, complemented
7#2015C211yesmaximal, characteristic, complemented
8#2015C211yesmaximal, characteristic, complemented
9#2013C211yesmaximal, characteristic, complemented
10#2012C211yesmaximal, characteristic
11#1933C211yesmaximal, characteristic, complemented
12#1916C211yesmaximal, characteristic, complemented
13#1880C211yesmaximal, characteristic, complemented
14#1880C211yesmaximal, characteristic, complemented
15#1878C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×SD16C2211yescharacteristic
21C22×D8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×#124C2211yescharacteristic
25C2×#124C2211yescharacteristic
26C2×C4×Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28#495—24no—
29#462—24no—
30C2×C4⋊C8C2211yescharacteristic
31C2×#103C2211yescharacteristic
32C2×#103C2211yescharacteristic
33C2×#103C2211yescharacteristic
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#102C2211yescharacteristic
37C2×#102C2211yescharacteristic
38C2×#102C2211yescharacteristic
39C2×C4×C8C2211yescharacteristic
40#399—21no—
41#397—21no—
42#396—21no—
43#396—21no—
44#391C2211yescharacteristic
45#391C2211yescharacteristic
46#390C2211yescharacteristic
47#390C2211yescharacteristic
48#390C2211yescharacteristic
49#389C2211yescharacteristic
50#388—21no—
51#388—21no—
52#388—21no—
53#388—21no—
54C4×C4⋊C4C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61C22×Q8C2311yescharacteristic
62C22×Q8C2311yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×D4C2311yescharacteristic
65C22×D4C2311yescharacteristic
66C2×Q16—22no—
67C2×Q16—41no—
68C2×Q16—41no—
69C2×Q16—41no—
70C2×Q16—41no—
71C2×Q16D414yes—
72C2×SD16—22no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—41no—
77C2×SD16—41no—
78C2×SD16—41no—
79C2×SD16—41no—
80C2×SD16—41no—
81C2×D8—22no—
82C2×D8—41no—
83C2×D8—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C4⋊Q8—22no—
89C4⋊Q8—41no—
90#124—22no—
91#124—22no—
92#124—41no—
93#124—41no—
94#124—41no—
95#124—41no—
96C4×Q8—22no—
97C4×Q8—41no—
98C4×Q8—41no—
99C4×Q8—41no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4D411yescharacteristic
106C2×C4⋊C4D411yescharacteristic
107C2×C4⋊C4C2311yescharacteristic
108C2×C4⋊C4C2311yescharacteristic
109C2×C4⋊C4C2311yescharacteristic
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⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42C2311yescharacteristic
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127C4⋊C8—22no—
128#103—22no—
129#103—22no—
130#103—22no—
131#103—41no—
132#103—41no—
133#102—41no—
134#102—41no—
135#102—41no—
136C4×C8—22no—
137#95—21no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95D411yescharacteristic
143#95D411yescharacteristic
144C24—21no—
145C24—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—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8C2×D412yes—
163C2×Q8C2×D412yes—
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4C2×D411yescharacteristic
183C22×C4C2×D411yescharacteristic
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4C2×D411yescharacteristic
188C22×C4C2411yescharacteristic
189Q16—44no—
190Q16—44no—
191SD16—44no—
192SD16—44no—
193SD16—44no—
194D8—44no—
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×C8—21no—
204C2×C8—21no—
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—41no—
215C4⋊C4—42no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—81no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C22⋊C4—42no—
230C22⋊C4—81no—
231C22⋊C4—81no—
232C42—21no—
233C42—21no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—42no—
239C42—42no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—41no—
254C23C22×D411yescharacteristic
255Q8—22no—
256Q8—22no—
257Q8—22no—
258Q8—22no—
259Q8—41no—
260Q8—41no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267D4—41no—
268D4—41no—
269D4—41no—
270D4—41no—
271D4—41no—
272D4—41no—
273D4—41no—
274D4—41no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
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—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
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#13512yes—
311C2×C4C22×D411yescharacteristic
312C2×C4C2×#4111yescharacteristic
313C2×C4H(5,𝔽2)11yescharacteristic
314C2×C4H(5,𝔽2)11yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C2×C4H(5,𝔽2)11yescharacteristic
317C8—22no—
318C8—22no—
319C8—41no—
320C8—41no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22#53511yescharacteristic
336C22D4211yescharacteristic
337C22#55011yescharacteristic
338C22C2×#13511yescharacteristic
339C22#57511yescharacteristic
340C22#57711yescharacteristic
341C22#57811yescharacteristic
342C4—22no—
343C4—42no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4#53411yescharacteristic
355C4#54011yescharacteristic
356C4#54611yescharacteristic
357C4#56011yescharacteristic
358C2—161no—
359C2—161no—
360C2#267311yescharacteristic
361C2#321211yescharacteristic
362C2#322511yescharacteristic
363C2#327111yescharacteristic
364C2#328511yescharacteristic
365C2#339411yescharacteristic
366C2#340211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D8C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#462C2semidirect
3. #2545C2semidirect
4. #2448C2semidirect
6. #2042C2semidirect
7. #2015C2semidirect
8. #2015C2semidirect
9. #2013C2semidirect
11. #1933C2semidirect
12. #1916C2semidirect
13. #1880C2semidirect
14. #1880C2semidirect
15. #1878C2semidirect

Maximal subgroups

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

subgroupindex
C2×#4952
C2×#4622
#25452
#24482
#20542
#20422
#20152
#20132
#20122
#19332
#19162
#18802
#18782