Finite Group Databasebeta

← #25400

Subgroups — #25400

Order 256.

524 conjugacy classes of subgroups.

The subgroup lattice has 397 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×#461C211yesmaximal, characteristic, complemented
4C2×#435C211yesmaximal, characteristic, complemented
5#2542C211yesmaximal, characteristic, complemented
6#2535C211yesmaximal, characteristic
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2000C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1960C211yesmaximal, characteristic, complemented
12#1949C211yesmaximal, characteristic, complemented
13#1894C211yesmaximal, characteristic, complemented
14#1892C211yesmaximal, characteristic, complemented
15#1762C211yesmaximal, characteristic, complemented
16C22×SD16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C2×C4⋊Q8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×D4C2211yescharacteristic
28C2×C4×D4C2211yescharacteristic
29#468—24no—
30#465—24no—
31#461—24no—
32#435—24no—
33C2×#107C2211yescharacteristic
34C2×#106C2211yescharacteristic
35C2×C4⋊C8C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×#97C2211yescharacteristic
47#399—21no—
48#399—21no—
49#397—21no—
50#392—21no—
51#390—21no—
52#386—21no—
53#386—21no—
54#384—21no—
55#382C2211yescharacteristic
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×C4—21no—
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×SD16—22no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C2×SD16—41no—
77C2×SD16—41no—
78C2×M16—22no—
79C2×M16—22no—
80C2×M16—41no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C4⋊Q8—22no—
89C4⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92C22⋊Q8—22no—
93C22⋊Q8—22no—
94C22⋊Q8—41no—
95C22⋊Q8—41no—
96C22⋊Q8—41no—
97C22⋊Q8—41no—
98#121—22no—
99#121—22no—
100#121—41no—
101#121—41no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
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⋊C4D411yescharacteristic
128C2×C22⋊C4D411yescharacteristic
129C2×C42—21no—
130C2×C42C2311yescharacteristic
131#107—22no—
132#106—22no—
133C4⋊C8D414yescomplemented
134#103—22no—
135#103—22no—
136#103—22no—
137#103—22no—
138#103—41no—
139#103—41no—
140#102—22no—
141#102D414yescomplemented
142C22⋊C8—22no—
143C22⋊C8—22no—
144#97—22no—
145#95—21no—
146#95—21no—
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×Q8—21no—
165C2×D4—22no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—21no—
172C2×D4—41no—
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—81no—
178C2×D4—81no—
179C2×D4C2×D412yes—
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×C4—42no—
200C22×C4—81no—
201C22×C4—81no—
202C22×C4C2×D411yescharacteristic
203C22×C4C2×D411yescharacteristic
204C22×C4C2×D411yescharacteristic
205C22×C4C2×D411yescharacteristic
206C22×C4C2411yescharacteristic
207SD16—44no—
208SD16—44no—
209M16—44no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—44no—
220C2×C8C2×D412yes—
221C2×C8C2×D412yes—
222C2×C8C2×D412yes—
223C4⋊C4—22no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4C2×D412yes—
239C4⋊C4#4112yes—
240C22⋊C4—22no—
241C22⋊C4—22no—
242C22⋊C4—42no—
243C22⋊C4—42no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22⋊C4—42no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—81no—
250C22⋊C4—81no—
251C42—42no—
252C42C2×D412yes—
253C42#4112yes—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—22no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23#13712yes—
275C23C22×D411yescharacteristic
276Q8—41no—
277Q8—41no—
278Q8—41no—
279Q8—41no—
280Q8—41no—
281Q8—41no—
282Q8—41no—
283Q8—41no—
284D4—22no—
285D4—22no—
286D4—44no—
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—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
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—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—84no—
331C2×C4#12112yes—
332C2×C4#12112yes—
333C2×C4C22×D411yescharacteristic
334C2×C4C22×D411yescharacteristic
335C2×C4C2×#4111yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4H(5,𝔽2)11yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C8—22no—
340C8—22no—
341C8—22no—
342C8—41no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—42no—
352C22—42no—
353C22—41no—
354C22—41no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22C2×#12111yescharacteristic
365C22D4211yescharacteristic
366C22#54611yescharacteristic
367C22C2×#13711yescharacteristic
368C22#57511yescharacteristic
369C22#57711yescharacteristic
370C22#57711yescharacteristic
371C4—22no—
372C4—22no—
373C4—41no—
374C4—41no—
375C4—42no—
376C4—42no—
377C4—81no—
378C4—81no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4#53211yescharacteristic
383C4#54011yescharacteristic
384C4D4211yescharacteristic
385C4#55011yescharacteristic
386C2—42no—
387C2—81no—
388C2—82no—
389C2#242811yescharacteristic
390C2#305611yescharacteristic
391C2#305711yescharacteristic
392C2#326611yescharacteristic
393C2#326711yescharacteristic
394C2#330711yescharacteristic
395C2#332011yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2535C2
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×#461C2semidirect
4. C2×#435C2semidirect
5. #2542C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #2000C2semidirect
10. #2000C2semidirect
11. #1960C2semidirect
12. #1949C2semidirect
13. #1894C2semidirect
14. #1892C2semidirect
15. #1762C2semidirect
133. C4⋊C8D4semidirect
141. #102D4semidirect

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×#4612
C2×#4352
#25422
#25352
#20362
#20352
#20002
#19602
#19492
#18942
#18922
#17622