Finite Group Databasebeta

← #28492

Subgroups — #28492

Order 256.

571 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#497C211yesmaximal, characteristic, complemented
2C2×#470C211yesmaximal, characteristic, complemented
3C2×#468C211yesmaximal, characteristic, complemented
4C2×#467C211yesmaximal, characteristic, complemented
5C2×#465C211yesmaximal, characteristic, complemented
6C2×#435C211yesmaximal, characteristic, complemented
7#2542C211yesmaximal, characteristic, complemented
8#2540C211yesmaximal, characteristic
9#2001C211yesmaximal, characteristic, complemented
10#2001C211yesmaximal, characteristic, complemented
11#2000C211yesmaximal, characteristic, complemented
12#2000C211yesmaximal, characteristic, complemented
13#1983C211yesmaximal, characteristic, complemented
14#1981C211yesmaximal, characteristic, complemented
15#1776C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×Q16C2211yescharacteristic
18C22×SD16C2211yescharacteristic
19C22×SD16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C23×C8C2211yescharacteristic
22C2×C4⋊Q8C2211yescharacteristic
23C2×#124C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×C22⋊Q8C2211yescharacteristic
29C2×C22⋊Q8C2211yescharacteristic
30C2×#121C2211yescharacteristic
31C2×#121C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33#497—24no—
34#470—24no—
35#468—24no—
36#467—24no—
37#465—24no—
38#435—24no—
39C2×#107C2211yescharacteristic
40C2×#106C2211yescharacteristic
41C2×C4⋊C8C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×#103C2211yescharacteristic
46C2×#103C2211yescharacteristic
47C2×#103C2211yescharacteristic
48C2×#102C2211yescharacteristic
49C2×#102C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52C2×#97C2211yescharacteristic
53#399—21no—
54#399—21no—
55#399—21no—
56#392—21no—
57#390—21no—
58#390—21no—
59#390—21no—
60#384—21no—
61C4×C4⋊C4C2211yescharacteristic
62C4×C4⋊C4C2211yescharacteristic
63#340C2211yescharacteristic
64#340C2211yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×Q8C2311yescharacteristic
67C22×Q8C2311yescharacteristic
68C22×D4—21no—
69C22×D4—21no—
70C22×D4D411yescharacteristic
71C22×D4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C2×Q16—22no—
75C2×Q16—22no—
76C2×Q16—22no—
77C2×Q16—22no—
78C2×Q16—41no—
79C2×Q16—41no—
80C2×SD16—22no—
81C2×SD16—22no—
82C2×SD16—22no—
83C2×SD16—22no—
84C2×SD16—41no—
85C2×SD16—41no—
86C2×M16—22no—
87C2×M16—22no—
88C2×M16—22no—
89C22×C8—22no—
90C22×C8—22no—
91C22×C8—22no—
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C4⋊Q8—22no—
97C4⋊Q8—22no—
98#124—22no—
99#124—22no—
100C22⋊Q8—22no—
101C22⋊Q8—22no—
102C22⋊Q8—22no—
103C22⋊Q8—22no—
104C22⋊Q8—22no—
105C22⋊Q8—22no—
106C22⋊Q8—41no—
107C22⋊Q8—41no—
108C22⋊Q8—41no—
109C22⋊Q8—41no—
110C22⋊Q8—41no—
111C22⋊Q8—41no—
112#121—22no—
113#121—22no—
114#121—41no—
115#121—41no—
116C4×D4—22no—
117C4×D4—22no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4D411yescharacteristic
127C2×C4⋊C4C2311yescharacteristic
128C2×C4⋊C4C2311yescharacteristic
129C2×C4⋊C4C2311yescharacteristic
130C2×C4⋊C4C2311yescharacteristic
131C2×C22⋊C4—21no—
132C2×C22⋊C4—21no—
133C2×C22⋊C4—21no—
134C2×C22⋊C4—21no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—21no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4D411yescharacteristic
142C2×C22⋊C4D411yescharacteristic
143C2×C42—21no—
144C2×C42—21no—
145C2×C42C2311yescharacteristic
146#107D414yescomplemented
147#106D414yescomplemented
148C4⋊C8—22no—
149#103—22no—
150#103—22no—
151#103—22no—
152#103—22no—
153#103—22no—
154#103—22no—
155#102—22no—
156#102—22no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159#97D414yescomplemented
160#95—21no—
161#95—21no—
162C24—21no—
163C24C2×D411yescharacteristic
164C24C2×D411yescharacteristic
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—21no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×D4—22no—
184C2×D4—22no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—81no—
197C2×D4—81no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4C2×D411yescharacteristic
218C22×C4C2×D411yescharacteristic
219C22×C4C2×D411yescharacteristic
220C22×C4C2×D411yescharacteristic
221C22×C4C2411yescharacteristic
222Q16—44no—
223Q16—44no—
224SD16—44no—
225SD16—44no—
226M16—44no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—44no—
236C2×C8C2×D412yes—
237C2×C8C2×D412yes—
238C2×C8C2×D412yes—
239C2×C8C2×D412yes—
240C4⋊C4—22no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4C2×D412yes—
254C4⋊C4C2×D412yes—
255C4⋊C4C2×D412yes—
256C4⋊C4C2×D412yes—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C42—42no—
270C42—81no—
271C42C2×D412yes—
272C42#4112yes—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#13712yes—
294C23C22×D411yescharacteristic
295Q8—41no—
296Q8—41no—
297Q8—41no—
298Q8—41no—
299Q8—41no—
300Q8—41no—
301Q8—41no—
302Q8—41no—
303Q8—41no—
304Q8—41no—
305Q8—41no—
306Q8—41no—
307D4—44no—
308D4—41no—
309D4—41no—
310D4—41no—
311D4—41no—
312D4—84no—
313D4—84no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
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—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4#12112yes—
354C2×C4#12112yes—
355C2×C4C22×D411yescharacteristic
356C2×C4C22×D411yescharacteristic
357C2×C4C22×D411yescharacteristic
358C2×C4C2×#4111yescharacteristic
359C2×C4H(5,𝔽2)11yescharacteristic
360C2×C4H(5,𝔽2)11yescharacteristic
361C8—22no—
362C8—41no—
363C8—41no—
364C8#12714yes—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22C2×#12111yescharacteristic
387C22D4211yescharacteristic
388C22D4211yescharacteristic
389C22C2×#13711yescharacteristic
390C22#57511yescharacteristic
391C22#57711yescharacteristic
392C22#57811yescharacteristic
393C4—22no—
394C4—22no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—41no—
400C4—41no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4C2×#12711yescharacteristic
405C4#53511yescharacteristic
406C4#55011yescharacteristic
407C4#55011yescharacteristic
408C2—42no—
409C2—81no—
410C2—161no—
411C2#254111yescharacteristic
412C2#305611yescharacteristic
413C2#306311yescharacteristic
414C2#326611yescharacteristic
415C2#326711yescharacteristic
416C2#326911yescharacteristic
417C2#327011yescharacteristic
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×#497C2semidirect
2. C2×#470C2semidirect
3. C2×#468C2semidirect
4. C2×#467C2semidirect
5. C2×#465C2semidirect
6. C2×#435C2semidirect
7. #2542C2semidirect
9. #2001C2semidirect
10. #2001C2semidirect
11. #2000C2semidirect
12. #2000C2semidirect
13. #1983C2semidirect
14. #1981C2semidirect
15. #1776C2semidirect
146. #107D4semidirect
147. #106D4semidirect
159. #97D4semidirect

Maximal subgroups

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

subgroupindex
C2×#4972
C2×#4702
C2×#4682
C2×#4672
C2×#4652
C2×#4352
#25422
#25402
#20012
#20002
#19832
#19812
#17762