Finite Group Databasebeta

← #29900

Subgroups — #29900

Order 256.

432 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2616C211yesmaximal, characteristic
2#2311C211yesmaximal, characteristic
3#2077C211yesmaximal, characteristic, complemented
4#2077C211yesmaximal, characteristic, complemented
5#2077C211yesmaximal, characteristic, complemented
6#2076C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2035C211yesmaximal, characteristic, complemented
9#2035C211yesmaximal, characteristic, complemented
10#2035C211yesmaximal, characteristic, complemented
11#1873C211yesmaximal, characteristic, complemented
12#1872C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1853C211yesmaximal, characteristic, complemented
15#1780C211yesmaximal, characteristic
16C22×#41C2211yescharacteristic
17C2×C22⋊Q8C2211yescharacteristic
18C2×C22⋊Q8C2211yescharacteristic
19C2×C22⋊Q8C2211yescharacteristic
20C2×#117C2211yescharacteristic
21C22×C4⋊C4C2211yescharacteristic
22C2×#107C2211yescharacteristic
23C2×#106C2211yescharacteristic
24C2×#106C2211yescharacteristic
25C2×#106C2211yescharacteristic
26C2×#103C2211yescharacteristic
27C2×#103C2211yescharacteristic
28C2×#103C2211yescharacteristic
29C2×#103C2211yescharacteristic
30C2×C22⋊C8C2211yescharacteristic
31C2×C22⋊C8C2211yescharacteristic
32C2×C22⋊C8C2211yescharacteristic
33C2×C22⋊C8C2211yescharacteristic
34C2×C22⋊C8C2211yescharacteristic
35C2×C22⋊C8C2211yescharacteristic
36#395—21no—
37#394—21no—
38#394—21no—
39#394—21no—
40#391—21no—
41#390—21no—
42#386—21no—
43#386—21no—
44#380C2211yescharacteristic
45#380C2211yescharacteristic
46#380C2211yescharacteristic
47#380C2211yescharacteristic
48#380C2211yescharacteristic
49#380C2211yescharacteristic
50#380C2211yescharacteristic
51#340C2211yescharacteristic
52#340C2211yescharacteristic
53#340C2211yescharacteristic
54#340C2211yescharacteristic
55#340C2211yescharacteristic
56#340C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59C2×#41—22no—
60C2×#41—22no—
61C2×#41—41no—
62C2×#41—41no—
63C2×#41—41no—
64C2×#41—41no—
65C22×Q8D411yescharacteristic
66C22×Q8C2311yescharacteristic
67C22×D4D411yescharacteristic
68C22×D4D411yescharacteristic
69C22×D4D411yescharacteristic
70C23×C4—21no—
71C23×C4C2311yescharacteristic
72C23×C4C2311yescharacteristic
73C23×C4C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22⋊Q8—22no—
79C22⋊Q8—22no—
80C22⋊Q8—41no—
81C22⋊Q8—41no—
82C22⋊Q8—41no—
83C22⋊Q8—41no—
84#117—41no—
85#117—41no—
86C2×C4⋊C4—21no—
87C2×C4⋊C4—21no—
88C2×C4⋊C4—21no—
89C2×C4⋊C4—21no—
90C2×C4⋊C4—41no—
91C2×C4⋊C4—41no—
92C2×C4⋊C4C2311yescharacteristic
93C2×C4⋊C4C2311yescharacteristic
94C2×C4⋊C4C2311yescharacteristic
95C2×C4⋊C4C2311yescharacteristic
96C2×C4⋊C4C2311yescharacteristic
97C2×C4⋊C4C2311yescharacteristic
98C2×C4⋊C4C2311yescharacteristic
99C2×C22⋊C4—21no—
100C2×C22⋊C4—21no—
101C2×C22⋊C4—21no—
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—21no—
109C2×C22⋊C4—21no—
110C2×C42—21no—
111#107—41no—
112#106—41no—
113#106—41no—
114#106—41no—
115#103—41no—
116#103—41no—
117#103—41no—
118#103—41no—
119C22⋊C8—22no—
120C22⋊C8—22no—
121C22⋊C8—41no—
122C22⋊C8—41no—
123C22⋊C8—41no—
124C22⋊C8—41no—
125#95—21no—
126#95—21no—
127#95—21no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132C24C2×D411yescharacteristic
133C24C2×D411yescharacteristic
134C24C2×D411yescharacteristic
135#41—44no—
136#41—44no—
137#41—44no—
138#41—44no—
139C2×Q8—22no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—21no—
145C2×Q8—21no—
146C2×Q8—41no—
147C2×Q8—41no—
148C2×D4—22no—
149C2×D4—22no—
150C2×D4—22no—
151C2×D4—41no—
152C2×D4—41no—
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—41no—
156C2×D4—41no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—42no—
182C22×C4—81no—
183C22×C4—81no—
184C22×C4C2×D411yescharacteristic
185C22×C4C2×D411yescharacteristic
186C22×C4C2×D411yescharacteristic
187C22×C4#4112yes—
188C22×C4#4112yes—
189C22×C4C2411yescharacteristic
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—21no—
206C4⋊C4—21no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—42no—
217C4⋊C4—42no—
218C4⋊C4—42no—
219C4⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—81no—
225C22⋊C4—81no—
226C22⋊C4—81no—
227C22⋊C4—81no—
228C22⋊C4—81no—
229C22⋊C4—81no—
230C22⋊C4—81no—
231C42—42no—
232C23—21no—
233C23—21no—
234C23—21no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23#12312yes—
253C23C22×D411yescharacteristic
254Q8—44no—
255Q8—41no—
256Q8—41no—
257Q8—41no—
258Q8—41no—
259D4—44no—
260D4—44no—
261D4—44no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
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—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—84no—
307C2×C4#12312yes—
308C2×C4C2×#4111yescharacteristic
309C2×C4C2×#4111yescharacteristic
310C2×C4H(5,𝔽2)11yescharacteristic
311C2×C4H(5,𝔽2)11yescharacteristic
312C2×C4H(5,𝔽2)11yescharacteristic
313C2×C4H(5,𝔽2)11yescharacteristic
314C8—41no—
315C8—41no—
316C8—41no—
317C8—41no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22C2×#12311yescharacteristic
340C22#53411yescharacteristic
341C22#53411yescharacteristic
342C22#57711yescharacteristic
343C22#57711yescharacteristic
344C22#57711yescharacteristic
345C22#57811yescharacteristic
346C4—22no—
347C4—41no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4—81no—
357C4#53811yescharacteristic
358C4#53811yescharacteristic
359C4#53811yescharacteristic
360C4#53811yescharacteristic
361C2—42no—
362C2—81no—
363C2—81no—
364C2#262211yescharacteristic
365C2#309711yescharacteristic
366C2#309911yescharacteristic
367C2#320011yescharacteristic
368C2#320111yescharacteristic
369C2#320111yescharacteristic
370C2#320411yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×#41C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #2077C2semidirect
4. #2077C2semidirect
5. #2077C2semidirect
6. #2076C2semidirect
7. #2036C2semidirect
8. #2035C2semidirect
9. #2035C2semidirect
10. #2035C2semidirect
11. #1873C2semidirect
12. #1872C2semidirect
13. #1854C2semidirect
14. #1853C2semidirect

Maximal subgroups

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

subgroupindex
#26162
#23112
#20772
#20762
#20362
#20352
#18732
#18722
#18542
#18532
#17802