Finite Group Databasebeta

← #25119

Subgroups — #25119

Order 256.

565 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#467C211yesmaximal, characteristic, complemented
2C2×#452C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4C2×#417C211yesmaximal, characteristic, complemented
5#2417C211yesmaximal, characteristic, complemented
6#2407C211yesmaximal, characteristic
7#2067C211yesmaximal, characteristic, complemented
8#2056C211yesmaximal, characteristic, complemented
9#2036C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1973C211yesmaximal, characteristic, complemented
12#1935C211yesmaximal, characteristic, complemented
13#1894C211yesmaximal, characteristic, complemented
14#1893C211yesmaximal, characteristic, complemented
15#1852C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×C22⋊Q8C2211yescharacteristic
24C2×#121C2211yescharacteristic
25C2×C4×D4C2211yescharacteristic
26C22×C4⋊C4C2211yescharacteristic
27#467—24no—
28#452—24no—
29#435—24no—
30C2×#107C2211yescharacteristic
31C2×#107C2211yescharacteristic
32C2×C4⋊C8C2211yescharacteristic
33#417—24no—
34C2×#103C2211yescharacteristic
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×#97C2211yescharacteristic
45#398—21no—
46#396—21no—
47#394—21no—
48#392—21no—
49#391C2211yescharacteristic
50#391C2211yescharacteristic
51#390C2211yescharacteristic
52#389C2211yescharacteristic
53#386—21no—
54#386—21no—
55#385—21no—
56#380—21no—
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—41no—
68C2×#41—41no—
69C22×Q8C2311yescharacteristic
70C22×Q8C2311yescharacteristic
71C22×D4—21no—
72C22×D4—21no—
73C22×D4D411yescharacteristic
74C22×D4C2311yescharacteristic
75C23×C4—21no—
76C23×C4—21no—
77C23×C4C2311yescharacteristic
78C23×C4C2311yescharacteristic
79C2×Q16—22no—
80C2×Q16—22no—
81C2×Q16—41no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—41no—
85C2×M16—22no—
86C2×M16—22no—
87C2×M16—41no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8—22no—
99C22⋊Q8—41no—
100C22⋊Q8—41no—
101#121—22no—
102#121—41no—
103C4×D4—22no—
104C4×D4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—41no—
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⋊C4D411yescharacteristic
127C2×C22⋊C4D411yescharacteristic
128C2×C42C2311yescharacteristic
129#107—22no—
130#107—41no—
131C4⋊C8—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103—41no—
136#103D414yescomplemented
137#102—22no—
138#102—22no—
139#102—22no—
140C22⋊C8—22no—
141C22⋊C8D414yescomplemented
142#97—22no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150C24—21no—
151C24C2×D411yescharacteristic
152C24C2×D411yescharacteristic
153#41—28no—
154#41—44no—
155#41—44no—
156#41—44no—
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×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8C2×D412yes—
169C2×D4—22no—
170C2×D4—22no—
171C2×D4—21no—
172C2×D4—21no—
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—41no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—42no—
180C2×D4—42no—
181C2×D4—42no—
182C2×D4—81no—
183C2×D4#4112yes—
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—21no—
200C22×C4—21no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—81no—
207C22×C4C2×D411yescharacteristic
208C22×C4C2×D411yescharacteristic
209C22×C4C2×D411yescharacteristic
210C22×C4C2×D411yescharacteristic
211C22×C4C2×D412yes—
212C22×C4#4112yes—
213C22×C4C2411yescharacteristic
214Q16—44no—
215SD16—44no—
216M16—44no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—44no—
227C2×C8C2×D412yes—
228C2×C8C2×D412yes—
229C2×C8C2×D412yes—
230C4⋊C4—22no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—21no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—81no—
245C4⋊C4C2×D412yes—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—81no—
254C22⋊C4—81no—
255C22⋊C4—81no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—22no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23#12112yes—
278C23#13712yes—
279C23C22×D411yescharacteristic
280Q8—22no—
281Q8—22no—
282Q8—41no—
283Q8—41no—
284Q8—41no—
285Q8—41no—
286Q8—41no—
287Q8—41no—
288D4—22no—
289D4—22no—
290D4—44no—
291D4—48no—
292D4—41no—
293D4—41no—
294D4—84no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—48no—
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—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—84no—
337C2×C4#12112yes—
338C2×C4C22×D411yescharacteristic
339C2×C4C22×D411yescharacteristic
340C2×C4C2×#4111yescharacteristic
341C2×C4C2×#4111yescharacteristic
342C2×C4H(5,𝔽2)11yescharacteristic
343C2×C4H(5,𝔽2)11yescharacteristic
344C8—22no—
345C8—22no—
346C8—22no—
347C8—41no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22C2×#12111yescharacteristic
370C22D4211yescharacteristic
371C22#54611yescharacteristic
372C22C2×#13711yescharacteristic
373C22#57511yescharacteristic
374C22#57611yescharacteristic
375C22#57811yescharacteristic
376C4—22no—
377C4—41no—
378C4—42no—
379C4—42no—
380C4—41no—
381C4—41no—
382C4—81no—
383C4—81no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4#52511yescharacteristic
388C4#53811yescharacteristic
389C4D4211yescharacteristic
390C4#54611yescharacteristic
391C2—42no—
392C2—42no—
393C2—82no—
394C2#242411yescharacteristic
395C2#305611yescharacteristic
396C2#306111yescharacteristic
397C2#326811yescharacteristic
398C2#327011yescharacteristic
399C2#331411yescharacteristic
400C2#331511yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2407C2
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×#467C2semidirect
2. C2×#452C2semidirect
3. C2×#435C2semidirect
4. C2×#417C2semidirect
5. #2417C2semidirect
7. #2067C2semidirect
8. #2056C2semidirect
9. #2036C2semidirect
10. #2000C2semidirect
11. #1973C2semidirect
12. #1935C2semidirect
13. #1894C2semidirect
14. #1893C2semidirect
15. #1852C2semidirect
136. #103D4semidirect
141. C22⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4672
C2×#4522
C2×#4352
C2×#4172
#24172
#24072
#20672
#20562
#20362
#20002
#19732
#19352
#18942
#18932
#18522