Finite Group Databasebeta

← #25114

Subgroups — #25114

Order 256.

596 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#465C211yesmaximal, characteristic, complemented
2C2×#448C211yesmaximal, characteristic, complemented
3C2×C8×D4C211yesmaximal, characteristic, complemented
4C22×#103C211yesmaximal, characteristic, complemented
5#2416C211yesmaximal, characteristic
6#2409C211yesmaximal, characteristic, complemented
7#2066C211yesmaximal, characteristic, complemented
8#2055C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#2000C211yesmaximal, characteristic, complemented
11#1968C211yesmaximal, characteristic, complemented
12#1933C211yesmaximal, characteristic, complemented
13#1893C211yesmaximal, characteristic, complemented
14#1892C211yesmaximal, characteristic, complemented
15#1853C211yesmaximal, characteristic, complemented
16C23×Q8C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20C23×C8C2211yescharacteristic
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121—21no—
24C2×#121C2211yescharacteristic
25C2×#121C2211yescharacteristic
26C2×C4×D4C2211yescharacteristic
27C22×C4⋊C4C2211yescharacteristic
28#465—24no—
29#448—24no—
30C8×D4—24no—
31C2×#106C2211yescharacteristic
32C2×#106C2211yescharacteristic
33C2×C4⋊C8C2211yescharacteristic
34C2×#103—24no—
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C4×C8C2211yescharacteristic
46#396—21no—
47#395—21no—
48#394—21no—
49#393—21no—
50#391C2211yescharacteristic
51#391C2211yescharacteristic
52#390C2211yescharacteristic
53#389C2211yescharacteristic
54#386—21no—
55#386—21no—
56#385—21no—
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8—22no—
65C22×Q8—22no—
66C22×Q8—22no—
67C22×Q8—22no—
68C22×Q8—41no—
69C22×Q8—41no—
70C22×Q8C2311yescharacteristic
71C22×Q8C2311yescharacteristic
72C22×D4—21no—
73C22×D4—21no—
74C22×D4D411yescharacteristic
75C22×D4C2311yescharacteristic
76C23×C4—21no—
77C23×C4—21no—
78C23×C4C2311yescharacteristic
79C23×C4C2311yescharacteristic
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—41no—
85C2×SD16—41no—
86C22×C8—22no—
87C22×C8—22no—
88C22×C8—22no—
89C22×C8—22no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8—41no—
99C22⋊Q8—41no—
100#121—22no—
101#121—22no—
102#121—22no—
103#121—41no—
104#121—44no—
105C4×D4—22no—
106C4×D4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4D411yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
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×C42C2311yescharacteristic
130#106—22no—
131#106—41no—
132C4⋊C8—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#103—22no—
137#103—22no—
138#103D414yescomplemented
139#102—22no—
140#102—41no—
141C22⋊C8—22no—
142C22⋊C8D414yescomplemented
143C4×C8—22no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148#95—21no—
149#95—21no—
150#95—21no—
151C24—21no—
152C24C2×D411yescharacteristic
153C24C2×D411yescharacteristic
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—28no—
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—42no—
166C2×Q8—42no—
167C2×Q8—44no—
168C2×Q8—44no—
169C2×Q8—44no—
170C2×Q8—42no—
171C2×Q8C2×D412yes—
172C2×Q8#4112yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—42no—
182C2×D4—42no—
183C2×D4—42no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4—81no—
187C2×D4SD1614yescomplemented
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—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—81no—
211C22×C4C2×D411yescharacteristic
212C22×C4C2×D411yescharacteristic
213C22×C4C2×D411yescharacteristic
214C22×C4C2×D412yes—
215C22×C4C2×D411yescharacteristic
216C22×C4#4112yes—
217C22×C4C2411yescharacteristic
218SD16—44no—
219SD16—44no—
220C2×C8—21no—
221C2×C8—21no—
222C2×C8—21no—
223C2×C8—21no—
224C2×C8—21no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—44no—
230C2×C8—44no—
231C2×C8C2×D412yes—
232C2×C8C2×D412yes—
233C2×C8C2×D412yes—
234C4⋊C4—22no—
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4C2×D412yes—
249C22⋊C4—22no—
250C22⋊C4—22no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—81no—
259C42—21no—
260C42—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23#12112yes—
279C23C2×SD1612yes—
280C23C2×SD1612yes—
281C23#13512yes—
282C23C22×D411yescharacteristic
283Q8—22no—
284Q8—22no—
285Q8—22no—
286Q8—22no—
287Q8—48no—
288Q8—41no—
289Q8—41no—
290Q8—41no—
291Q8—41no—
292D4—28no—
293D4—41no—
294D4—41no—
295D4—41no—
296D4—41no—
297D4—84no—
298D4—84no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—48no—
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—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—84no—
340C2×C4#12112yes—
341C2×C4C2×SD1612yes—
342C2×C4C22×D411yescharacteristic
343C2×C4C22×D411yescharacteristic
344C2×C4C2×#4111yescharacteristic
345C2×C4C2×#4111yescharacteristic
346C2×C4H(5,𝔽2)11yescharacteristic
347C2×C4H(5,𝔽2)11yescharacteristic
348C8—22no—
349C8—22no—
350C8—22no—
351C8—41no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#46514yescomplemented
373C22C2×#12111yescharacteristic
374C22D4211yescharacteristic
375C22#54611yescharacteristic
376C22C22×SD1611yescharacteristic
377C22C2×#13511yescharacteristic
378C22#57711yescharacteristic
379C22#57811yescharacteristic
380C4—22no—
381C4—22no—
382C4—41no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—41no—
387C4—81no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4#52511yescharacteristic
392C4#53811yescharacteristic
393C4D4211yescharacteristic
394C4#54611yescharacteristic
395C2—24no—
396C2—42no—
397C2—161no—
398C2#242411yescharacteristic
399C2C2×#46511yescharacteristic
400C2#306311yescharacteristic
401C2D4×SD1611yescharacteristic
402C2#328511yescharacteristic
403C2#329811yescharacteristic
404C2#330111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radical#2416C2
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×#465C2semidirect
2. C2×#448C2semidirect
3. C2×C8×D4C2semidirect
4. C22×#103C2semidirect
6. #2409C2semidirect
7. #2066C2semidirect
8. #2055C2semidirect
9. #2035C2semidirect
10. #2000C2semidirect
11. #1968C2semidirect
12. #1933C2semidirect
13. #1893C2semidirect
14. #1892C2semidirect
15. #1853C2semidirect
138. #103D4semidirect
142. C22⋊C8D4semidirect
187. C2×D4SD16semidirect
372. C22#465semidirect

Maximal subgroups

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

subgroupindex
C2×#4652
C2×#4482
C2×C8×D42
C22×#1032
#24162
#24092
#20662
#20552
#20352
#20002
#19682
#19332
#18932
#18922
#18532