Finite Group Databasebeta

← #23148

Subgroups — #23148

Order 256.

401 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3054C211yesmaximal, characteristic
2#2933C211yesmaximal, characteristic, complemented
3#2933C211yesmaximal, characteristic, complemented
4#1938C211yesmaximal, characteristic, complemented
5#1937C211yesmaximal, characteristic, complemented
6#1922C211yesmaximal, characteristic, complemented
7#1922C211yesmaximal, characteristic, complemented
8#1876C211yesmaximal, characteristic, complemented
9#1876C211yesmaximal, characteristic, complemented
10#1871C211yesmaximal, characteristic, complemented
11#1871C211yesmaximal, characteristic, complemented
12#1765C211yesmaximal, characteristic, complemented
13#1764C211yesmaximal, characteristic, complemented
14#1753C211yesmaximal, characteristic, complemented
15#1738C211yesmaximal, characteristic, complemented
16C2×#135C2211yescharacteristic
17C2×#131C2211yescharacteristic
18C2×#131C2211yescharacteristic
19C22×M16C2211yescharacteristic
20C22×M16C2211yescharacteristic
21C22×M16C2211yescharacteristic
22#525C2211yescharacteristic
23#525C2211yescharacteristic
24#470C411yescharacteristic, complemented
25#470C411yescharacteristic, complemented
26#469C411yescharacteristic, complemented
27#469C411yescharacteristic, complemented
28#468C411yescharacteristic, complemented
29#468C411yescharacteristic, complemented
30#468C411yescharacteristic, complemented
31#468C411yescharacteristic, complemented
32#436—21no—
33#436—21no—
34#435—21no—
35#435—21no—
36#435—21no—
37#435—21no—
38C8×D4—21no—
39C8×D4—21no—
40#427C2211yescharacteristic
41#424C2211yescharacteristic
42#424C2211yescharacteristic
43C2×#104C2211yescharacteristic
44C2×#104C2211yescharacteristic
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic
47#416C2211yescharacteristic
48#416C2211yescharacteristic
49#408C2211yescharacteristic
50#408C2211yescharacteristic
51C2×C22⋊C8C2211yescharacteristic
52#405C2211yescharacteristic
53#405C2211yescharacteristic
54C2×#97C2211yescharacteristic
55C2×C4×C8C2211yescharacteristic
56C4×C22⋊C4C2211yescharacteristic
57#341C2211yescharacteristic
58#341C2211yescharacteristic
59#341C2211yescharacteristic
60#341C2211yescharacteristic
61#339C2211yescharacteristic
62#339C2211yescharacteristic
63#339C2211yescharacteristic
64#339C2211yescharacteristic
65#336C2211yescharacteristic
66#336C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C2×#41C2311yescharacteristic
69C23×C4C2311yescharacteristic
70#135—41no—
71#135—41no—
72C2×Q16C2×C411yescharacteristic
73C2×SD16C2×C411yescharacteristic
74C2×SD16C2×C411yescharacteristic
75C2×D8C2×C411yescharacteristic
76#131—22no—
77#131—22no—
78#131—41no—
79#131—41no—
80C2×M16—21no—
81C2×M16—21no—
82C2×M16—21no—
83C2×M16—21no—
84C2×M16—22no—
85C2×M16—22no—
86C2×M16—41no—
87C2×M16—41no—
88C2×M16—41no—
89C2×M16—41no—
90C2×M16C2×C411yescharacteristic
91C2×M16C2×C411yescharacteristic
92C2×M16C2×C411yescharacteristic
93C2×M16C2×C411yescharacteristic
94C2×M16C2311yescharacteristic
95C2×M16C2311yescharacteristic
96C2×M16C2311yescharacteristic
97C2×M16C2311yescharacteristic
98C22×C8—21no—
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8C2311yescharacteristic
103C22×C8C2311yescharacteristic
104C22×C8C2311yescharacteristic
105C22×C8C2311yescharacteristic
106#123—21no—
107#123—21no—
108C22⋊Q8C2×C411yescharacteristic
109C22⋊Q8C2×C411yescharacteristic
110C22⋊Q8C2×C411yescharacteristic
111C22⋊Q8C2×C411yescharacteristic
112#121C2×C411yescharacteristic
113#121C2×C411yescharacteristic
114#121C2×C411yescharacteristic
115#121C2×C411yescharacteristic
116#120—21no—
117#120—21no—
118C4×D4—21no—
119C4×D4—21no—
120C4×D4—21no—
121C4×D4—21no—
122#117C2311yescharacteristic
123#117C2311yescharacteristic
124C2×C22⋊C4—21no—
125C2×C42C2311yescharacteristic
126C2×C42C2311yescharacteristic
127#107C2×C411yescharacteristic
128#107C2×C411yescharacteristic
129#106C2×C411yescharacteristic
130#106C2×C411yescharacteristic
131C4⋊C8—21no—
132C4⋊C8—21no—
133C4⋊C8—21no—
134C4⋊C8—21no—
135#104—22no—
136#104—22no—
137#104—41no—
138#104—41no—
139#103C2×C411yescharacteristic
140#103C2×C411yescharacteristic
141#103C2×C411yescharacteristic
142#103C2×C411yescharacteristic
143#102C2×C411yescharacteristic
144#102C2×C411yescharacteristic
145#102C2×C411yescharacteristic
146#102C2×C411yescharacteristic
147C22⋊C8—21no—
148C22⋊C8—21no—
149C22⋊C8—21no—
150C22⋊C8—21no—
151C22⋊C8D411yescharacteristic
152C22⋊C8D411yescharacteristic
153C22⋊C8D411yescharacteristic
154C22⋊C8D411yescharacteristic
155#97—21no—
156#97—21no—
157#97—41no—
158C4×C8—21no—
159C4×C8—21no—
160C4×C8—41no—
161#95—21no—
162C24#4111yescharacteristic
163#41—21no—
164#41—21no—
165#41—21no—
166#41—21no—
167#41—21no—
168#41—21no—
169#41—21no—
170#41—21no—
171C2×Q8C22×C411yescharacteristic
172C2×Q8C22×C411yescharacteristic
173C2×D4—21no—
174C2×D4—21no—
175C2×D4—21no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—21no—
179C2×D4C22×C411yescharacteristic
180C2×D4C22×C411yescharacteristic
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×C4C22×C411yescharacteristic
190C22×C4C22×C411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4#4111yescharacteristic
194C22×C4C2411yescharacteristic
195Q16—41no—
196SD16—41no—
197SD16—41no—
198D8—41no—
199M16—21no—
200M16—21no—
201M16—21no—
202M16—21no—
203M16—21no—
204M16—21no—
205M16—41no—
206M16—41no—
207M16—41no—
208M16—41no—
209M16—42no—
210M16—42no—
211M16—42no—
212M16—42no—
213M16#4112yes—
214M16#4112yes—
215C2×C8—21no—
216C2×C8—21no—
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—21no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8C22×C411yescharacteristic
232C2×C8C22×C411yescharacteristic
233C2×C8C22×C411yescharacteristic
234C2×C8C22×C411yescharacteristic
235C2×C8C2×D411yescharacteristic
236C2×C8C2×D411yescharacteristic
237C2×C8C2×D411yescharacteristic
238C2×C8C2×D411yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—21no—
243C4⋊C4C22×C411yescharacteristic
244C4⋊C4C22×C411yescharacteristic
245C4⋊C4C22×C411yescharacteristic
246C4⋊C4C22×C411yescharacteristic
247C22⋊C4—21no—
248C22⋊C4—21no—
249C22⋊C4—21no—
250C22⋊C4—21no—
251C22⋊C4—21no—
252C22⋊C4—21no—
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—42no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23C4×D411yescharacteristic
271C23C4×D411yescharacteristic
272C23C2×#4111yescharacteristic
273Q8—21no—
274Q8—21no—
275Q8—21no—
276Q8—21no—
277D4—21no—
278D4—21no—
279D4—21no—
280D4—21no—
281D4—42no—
282D4—42no—
283D4—41no—
284D4—41no—
285D4—41no—
286D4—41no—
287D4—81no—
288D4—81no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
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×C4C4×D411yescharacteristic
316C2×C4C4×D411yescharacteristic
317C2×C4C23×C411yescharacteristic
318C2×C4C22×D411yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C2×C4C2×#4111yescharacteristic
321C2×C4H(5,𝔽2)11yescharacteristic
322C2×C4H(5,𝔽2)11yescharacteristic
323C8—21no—
324C8—21no—
325C8—21no—
326C8—21no—
327C8—21no—
328C8—21no—
329C8—21no—
330C8—21no—
331C8—22no—
332C8—22no—
333C8—41no—
334C8—41no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—41no—
349C22—81no—
350C22—81no—
351C22C2×C4×D411yescharacteristic
352C22#55311yescharacteristic
353C22#55311yescharacteristic
354C4—21no—
355C4—21no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—81no—
364C4C4×#4111yescharacteristic
365C4#51811yescharacteristic
366C4#54611yescharacteristic
367C4#54611yescharacteristic
368C2—21no—
369C2—21no—
370C2—41no—
371C2—41no—
372C2—81no—
373C2—81no—
374C2#236411yescharacteristic
375C2#295511yescharacteristic
376C2#295611yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C2×C4×D4
prime radical#3054C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #2933C2semidirect
3. #2933C2semidirect
4. #1938C2semidirect
5. #1937C2semidirect
6. #1922C2semidirect
7. #1922C2semidirect
8. #1876C2semidirect
9. #1876C2semidirect
10. #1871C2semidirect
11. #1871C2semidirect
12. #1765C2semidirect
13. #1764C2semidirect
14. #1753C2semidirect
15. #1738C2semidirect
24. #470C4semidirect
25. #470C4semidirect
26. #469C4semidirect
27. #469C4semidirect
28. #468C4semidirect
29. #468C4semidirect
30. #468C4semidirect
31. #468C4semidirect

Maximal subgroups

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

subgroupindex
#30542
#29332
#19382
#19372
#19222
#18762
#18712
#17652
#17642
#17532
#17382