Finite Group Databasebeta

← #25403

Subgroups — #25403

Order 256.

517 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#470C211yesmaximal, characteristic, complemented
2C2×#465C211yesmaximal, characteristic, complemented
3C2×#464C211yesmaximal, characteristic, complemented
4C2×#435C211yesmaximal, characteristic, complemented
5#2540C211yesmaximal, characteristic
6#2538C211yesmaximal, characteristic, complemented
7#2035C211yesmaximal, characteristic, complemented
8#2034C211yesmaximal, characteristic, complemented
9#2001C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1959C211yesmaximal, characteristic, complemented
12#1948C211yesmaximal, characteristic, complemented
13#1895C211yesmaximal, characteristic, complemented
14#1892C211yesmaximal, characteristic, complemented
15#1763C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×#124C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29#470—24no—
30#465—24no—
31#464—24no—
32#435—24no—
33C2×#106C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×#97C2211yescharacteristic
47#399—21no—
48#396—21no—
49#396—21no—
50#392—21no—
51#390—21no—
52#390—21no—
53#390—21no—
54#389—21no—
55#384C2211yescharacteristic
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58C4×C4⋊C4C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4—21no—
66C22×D4—21no—
67C22×D4D411yescharacteristic
68C22×D4C2311yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C2×Q16—22no—
72C2×Q16—22no—
73C2×Q16—41no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—41no—
77C2×M16—22no—
78C2×M16—22no—
79C2×M16—41no—
80C22×C8—22no—
81C22×C8—22no—
82C22×C8—41no—
83C22×C8C2311yescharacteristic
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87#124—22no—
88#124—22no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92C22⋊Q8—22no—
93C22⋊Q8—41no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96C22⋊Q8—41no—
97#121—22no—
98#121—22no—
99#121—41no—
100#121—41no—
101C4×Q8—22no—
102C4×Q8—22no—
103C4×D4—22no—
104C4×D4—22no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4D411yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4D411yescharacteristic
129C2×C22⋊C4D411yescharacteristic
130C2×C42—21no—
131C2×C42—21no—
132C2×C42C2311yescharacteristic
133#106—22no—
134#106—22no—
135C4⋊C8D414yescomplemented
136#103—22no—
137#103—22no—
138#103—22no—
139#103—41no—
140#103D414yescomplemented
141#102—22no—
142#102—22no—
143#102—41no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146#97—22no—
147#95—21no—
148#95—21no—
149#95—21no—
150C24—21no—
151C24C2×D411yescharacteristic
152C24C2×D411yescharacteristic
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—21no—
163C2×Q8—21no—
164C2×Q8C2×D412yes—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—81no—
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—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4C2×D411yescharacteristic
200C22×C4C2×D411yescharacteristic
201C22×C4C2×D411yescharacteristic
202C22×C4C2×D411yescharacteristic
203C22×C4C2411yescharacteristic
204Q16—44no—
205SD16—44no—
206M16—44no—
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×C8—44no—
217C2×C8C2×D412yes—
218C2×C8C2×D412yes—
219C2×C8C2×D412yes—
220C4⋊C4—22no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—81no—
236C4⋊C4C2×D412yes—
237C4⋊C4#4112yes—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—42no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—81no—
249C22⋊C4—81no—
250C42—42no—
251C42—81no—
252C42C2×D412yes—
253C42#4112yes—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23#13712yes—
275C23C22×D411yescharacteristic
276Q8—22no—
277Q8—22no—
278Q8—41no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282Q8—41no—
283Q8—41no—
284D4—44no—
285D4—41no—
286D4—41no—
287D4—41no—
288D4—41no—
289D4—84no—
290D4—84no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
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—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4#12112yes—
331C2×C4#12112yes—
332C2×C4C22×D411yescharacteristic
333C2×C4C22×D411yescharacteristic
334C2×C4C2×#4111yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4H(5,𝔽2)11yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C8—22no—
339C8—22no—
340C8—22no—
341C8—41no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22C2×#12111yescharacteristic
364C22D4211yescharacteristic
365C22#54611yescharacteristic
366C22C2×#13711yescharacteristic
367C22#57511yescharacteristic
368C22#57711yescharacteristic
369C22#57711yescharacteristic
370C4—22no—
371C4—22no—
372C4—41no—
373C4—41no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4#53211yescharacteristic
382C4#54011yescharacteristic
383C4D4211yescharacteristic
384C4#55011yescharacteristic
385C2—42no—
386C2—81no—
387C2—161no—
388C2#242811yescharacteristic
389C2#305611yescharacteristic
390C2#306011yescharacteristic
391C2#326611yescharacteristic
392C2#327111yescharacteristic
393C2#331111yescharacteristic
394C2#331611yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2540C2
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×#470C2semidirect
2. C2×#465C2semidirect
3. C2×#464C2semidirect
4. C2×#435C2semidirect
6. #2538C2semidirect
7. #2035C2semidirect
8. #2034C2semidirect
9. #2001C2semidirect
10. #2000C2semidirect
11. #1959C2semidirect
12. #1948C2semidirect
13. #1895C2semidirect
14. #1892C2semidirect
15. #1763C2semidirect
135. C4⋊C8D4semidirect
140. #103D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4702
C2×#4652
C2×#4642
C2×#4352
#25402
#25382
#20352
#20342
#20012
#20002
#19592
#19482
#18952
#18922
#17632