Finite Group Databasebeta

← #23115

Subgroups — #23115

Order 256.

381 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3035C211yesmaximal, characteristic
2#2935C211yesmaximal, characteristic, complemented
3#2925C211yesmaximal, characteristic, complemented
4#1975C211yesmaximal, characteristic, complemented
5#1974C211yesmaximal, characteristic, complemented
6#1963C211yesmaximal, characteristic, complemented
7#1950C211yesmaximal, characteristic, complemented
8#1929C211yesmaximal, characteristic, complemented
9#1929C211yesmaximal, characteristic, complemented
10#1876C211yesmaximal, characteristic, complemented
11#1876C211yesmaximal, characteristic, complemented
12#1768C211yesmaximal, characteristic, complemented
13#1765C211yesmaximal, characteristic, complemented
14#1764C211yesmaximal, characteristic, complemented
15#1759C211yesmaximal, characteristic, complemented
16C2×#135C2211yescharacteristic
17C2×#135C2211yescharacteristic
18C2×#131C2211yescharacteristic
19C2×#131C2211yescharacteristic
20#532C2211yescharacteristic
21C4×#41C2211yescharacteristic
22#464C411yescharacteristic, complemented
23#464C411yescharacteristic, complemented
24#463C411yescharacteristic, complemented
25#463C411yescharacteristic, complemented
26#462C411yescharacteristic, complemented
27#461C411yescharacteristic, complemented
28#460C411yescharacteristic, complemented
29#459C411yescharacteristic, complemented
30#436—21no—
31#435—21no—
32#435—21no—
33C8×D4—21no—
34#433—21no—
35#433—21no—
36#432—21no—
37#431—21no—
38C2×#108C2211yescharacteristic
39C2×#108C2211yescharacteristic
40#423C2211yescharacteristic
41C2×C4⋊C8C2211yescharacteristic
42C2×C4⋊C8C2211yescharacteristic
43C2×#104C2211yescharacteristic
44C2×#104C2211yescharacteristic
45C2×#104C2211yescharacteristic
46C2×#104C2211yescharacteristic
47C2×#104C2211yescharacteristic
48C2×#104C2211yescharacteristic
49#416C2211yescharacteristic
50#416C2211yescharacteristic
51#408C2211yescharacteristic
52#408C2211yescharacteristic
53#408C2211yescharacteristic
54#408C2211yescharacteristic
55#405C2211yescharacteristic
56#405C2211yescharacteristic
57C4×M16C2211yescharacteristic
58C2×#97C2211yescharacteristic
59C2×C4×C8C2211yescharacteristic
60#376C2211yescharacteristic
61#341C2211yescharacteristic
62#341C2211yescharacteristic
63#339C2211yescharacteristic
64#339C2211yescharacteristic
65#336C2211yescharacteristic
66#336C2211yescharacteristic
67C2×#41C2311yescharacteristic
68C2×#41C2311yescharacteristic
69C2×#41C2311yescharacteristic
70#135—41no—
71#135—41no—
72#135—41no—
73#135—41no—
74C2×Q16C2×C411yescharacteristic
75C2×Q16C2×C411yescharacteristic
76C2×SD16C2×C411yescharacteristic
77C2×SD16C2×C411yescharacteristic
78C2×SD16C2×C411yescharacteristic
79C2×SD16C2×C411yescharacteristic
80C2×D8C2×C411yescharacteristic
81C2×D8C2×C411yescharacteristic
82#131—22no—
83#131—22no—
84#131—41no—
85#131—41no—
86C2×M16—21no—
87C2×M16—21no—
88C2×M16C2311yescharacteristic
89C2×M16C2311yescharacteristic
90C2×M16C2311yescharacteristic
91C2×M16C2311yescharacteristic
92C22×C8—21no—
93C22×C8—21no—
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96C22×C8C2311yescharacteristic
97C22×C8C2311yescharacteristic
98C4⋊Q8C2×C411yescharacteristic
99#127C2×C411yescharacteristic
100#124C2×C411yescharacteristic
101#124C2×C411yescharacteristic
102#121—21no—
103#121—21no—
104C4×Q8C2×C411yescharacteristic
105C4×Q8C2×C411yescharacteristic
106C4×D4—21no—
107C4×D4—21no—
108C4×D4—21no—
109C4×D4—21no—
110C4×D4C2×C411yescharacteristic
111C4×D4C2×C411yescharacteristic
112#117—21no—
113#117C2311yescharacteristic
114C2×C42—21no—
115C2×C42C2311yescharacteristic
116C2×C42C2311yescharacteristic
117C2×C42C2311yescharacteristic
118#108—41no—
119#108—41no—
120C4⋊C8—21no—
121C4⋊C8—21no—
122C4⋊C8C2×C411yescharacteristic
123C4⋊C8C2×C411yescharacteristic
124C4⋊C8C2×C411yescharacteristic
125C4⋊C8C2×C411yescharacteristic
126C4⋊C8D411yescharacteristic
127C4⋊C8D411yescharacteristic
128C4⋊C8D411yescharacteristic
129C4⋊C8D411yescharacteristic
130#104—22no—
131#104—22no—
132#104—22no—
133#104—22no—
134#104—41no—
135#104—41no—
136#103C2×C411yescharacteristic
137#103C2×C411yescharacteristic
138#103C2×C411yescharacteristic
139#103C2×C411yescharacteristic
140#102C2×C411yescharacteristic
141#102C2×C411yescharacteristic
142#102C2×C411yescharacteristic
143#102C2×C411yescharacteristic
144C22⋊C8—21no—
145C22⋊C8—21no—
146C22⋊C8—21no—
147C22⋊C8—21no—
148C22⋊C8—21no—
149C22⋊C8—21no—
150C22⋊C8—21no—
151C22⋊C8—21no—
152#97—21no—
153#97—21no—
154#97—21no—
155#97—41no—
156C4×C8—21no—
157C4×C8—21no—
158C4×C8—21no—
159C4×C8—41no—
160#95—21no—
161#95—21no—
162#41—21no—
163#41—21no—
164#41—21no—
165#41—21no—
166#41—21no—
167#41—21no—
168#41—21no—
169#41—21no—
170#41—21no—
171#41—21no—
172#41—21no—
173#41—21no—
174C2×Q8C22×C411yescharacteristic
175C2×Q8C22×C411yescharacteristic
176C2×Q8C22×C411yescharacteristic
177C2×D4—21no—
178C2×D4—21no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4C22×C411yescharacteristic
183C2×D4C22×C411yescharacteristic
184C2×D4C22×C411yescharacteristic
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4#4111yescharacteristic
193C22×C4#4111yescharacteristic
194C22×C4C2411yescharacteristic
195Q16—41no—
196Q16—41no—
197SD16—41no—
198SD16—41no—
199SD16—41no—
200SD16—41no—
201D8—41no—
202D8—41no—
203M16—21no—
204M16—21no—
205M16—21no—
206M16—21no—
207M16—21no—
208M16—21no—
209M16—21no—
210M16—42no—
211M16—42no—
212M16#4112yes—
213C2×C8—21no—
214C2×C8—21no—
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—42no—
226C2×C8—42no—
227C2×C8C22×C411yescharacteristic
228C2×C8C22×C411yescharacteristic
229C2×C8C22×C411yescharacteristic
230C2×C8C22×C411yescharacteristic
231C2×C8C2×D411yescharacteristic
232C2×C8C2×D411yescharacteristic
233C2×C8C2×D411yescharacteristic
234C2×C8C2×D411yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—21no—
237C4⋊C4—21no—
238C4⋊C4—21no—
239C4⋊C4C22×C411yescharacteristic
240C4⋊C4C22×C411yescharacteristic
241C22⋊C4—21no—
242C22⋊C4—21no—
243C22⋊C4—21no—
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4—21no—
247C22⋊C4—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42C22×C411yescharacteristic
255C42C22×C411yescharacteristic
256C42C2×D411yescharacteristic
257C42C2×D411yescharacteristic
258C42#4112yes—
259C23—21no—
260C23—21no—
261C23—21no—
262C23C2×#4111yescharacteristic
263Q8—21no—
264Q8—21no—
265Q8—21no—
266Q8—21no—
267Q8—21no—
268Q8—21no—
269D4—21no—
270D4—21no—
271D4—21no—
272D4—21no—
273D4—21no—
274D4—21no—
275D4—42no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—41no—
280D4—41no—
281D4—41no—
282D4—81no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4C4×D411yescharacteristic
305C2×C4C4×D411yescharacteristic
306C2×C4C4×D411yescharacteristic
307C2×C4C4×D411yescharacteristic
308C2×C4C23×C411yescharacteristic
309C2×C4C22×D411yescharacteristic
310C2×C4C2×#4111yescharacteristic
311C2×C4C2×#4111yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C2×C4H(5,𝔽2)11yescharacteristic
314C8—21no—
315C8—21no—
316C8—21no—
317C8—21no—
318C8—21no—
319C8—21no—
320C8—21no—
321C8—21no—
322C8—22no—
323C8—41no—
324C8—41no—
325C8—41no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22C2×C4×D411yescharacteristic
338C22#55311yescharacteristic
339C22#55511yescharacteristic
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—42no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4C4×#4111yescharacteristic
352C4#51811yescharacteristic
353C4#54611yescharacteristic
354C4#55011yescharacteristic
355C2—21no—
356C2—21no—
357C2—81no—
358C2—81no—
359C2—81no—
360C2#236311yescharacteristic
361C2#295511yescharacteristic
362C2#295711yescharacteristic
subgroupG/H
centerC2×C4C22×D4
commutatorC2×C4C23×C4
Frattini, cosocleC22×C4C24
socleC22C2×C4×D4
prime radical#3035C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4⋊C8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #2935C2semidirect
3. #2925C2semidirect
4. #1975C2semidirect
5. #1974C2semidirect
6. #1963C2semidirect
7. #1950C2semidirect
8. #1929C2semidirect
9. #1929C2semidirect
10. #1876C2semidirect
11. #1876C2semidirect
12. #1768C2semidirect
13. #1765C2semidirect
14. #1764C2semidirect
15. #1759C2semidirect
22. #464C4semidirect
23. #464C4semidirect
24. #463C4semidirect
25. #463C4semidirect
26. #462C4semidirect
27. #461C4semidirect
28. #460C4semidirect
29. #459C4semidirect

Maximal subgroups

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

subgroupindex
#30352
#29352
#29252
#19752
#19742
#19632
#19502
#19292
#18762
#17682
#17652
#17642
#17592