Finite Group Databasebeta

← #23585

Subgroups — #23585

Order 256.

561 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#542C211yesmaximal, characteristic
2C2×#440C211yesmaximal, characteristic, complemented
3C2×C4×Q16C211yesmaximal, characteristic, complemented
4#2323C211yesmaximal, characteristic, complemented
5#2057C211yesmaximal, characteristic, complemented
6#2042C211yesmaximal, characteristic, complemented
7#2036C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic
9#2024C211yesmaximal, characteristic, complemented
10#2015C211yesmaximal, characteristic, complemented
11#2001C211yesmaximal, characteristic, complemented
12#2000C211yesmaximal, characteristic
13#1763C211yesmaximal, characteristic, complemented
14C4×#102C211yesmaximal, characteristic, complemented
15#1750C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17C22×SD16C2211yescharacteristic
18#542—24no—
19#542—24no—
20C2×C4⋊Q8C2211yescharacteristic
21C2×#126—21no—
22C2×#124—21no—
23C2×#124C2211yescharacteristic
24C2×#123—21no—
25C2×C22⋊Q8—21no—
26C2×C22⋊Q8C2211yescharacteristic
27C2×C22⋊Q8C2211yescharacteristic
28C2×#121C2211yescharacteristic
29C2×C4×Q8C2211yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×#117C2211yescharacteristic
33#440—24no—
34C4×Q16—24no—
35C2×#107C2211yescharacteristic
36C2×#107C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×#97C2211yescharacteristic
48C2×C4×C8C2211yescharacteristic
49#391C2211yescharacteristic
50#389—21no—
51#389C2211yescharacteristic
52#388—21no—
53#388—21no—
54#387—21no—
55#380C2211yescharacteristic
56C4×C4⋊C4C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8—21no—
64C22×Q8C2311yescharacteristic
65C22×Q8C2311yescharacteristic
66C22×D4—21no—
67C22×D4C2311yescharacteristic
68C23×C4—21no—
69C23×C4C2311yescharacteristic
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—22no—
73C2×SD16—22no—
74C2×SD16—22no—
75C2×SD16—22no—
76C22×C8C2311yescharacteristic
77C22×C8C2311yescharacteristic
78C22×C8C2311yescharacteristic
79C22×C8C2311yescharacteristic
80C4⋊Q8—22no—
81C4⋊Q8—22no—
82#126—28no—
83#124—28no—
84#124—22no—
85#124—22no—
86#123—28no—
87C22⋊Q8—28no—
88C22⋊Q8—22no—
89C22⋊Q8—22no—
90C22⋊Q8—22no—
91C22⋊Q8—22no—
92#121—22no—
93#121—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—22no—
97C4×Q8D414yes—
98C4×D4—22no—
99C4×D4D414yes—
100#117—22no—
101#117—22no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4D411yescharacteristic
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4D411yescharacteristic
123C2×C22⋊C4D411yescharacteristic
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—21no—
127C2×C42C2311yescharacteristic
128C2×C42C2311yescharacteristic
129#107—22no—
130#107—22no—
131#103—22no—
132#103—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#102—22no—
137#102—22no—
138#102—22no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141#97—22no—
142C4×C8—22no—
143#95—21no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24—21no—
149C24C2×D411yescharacteristic
150C2×Q8—24no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—81no—
162C2×Q8—81no—
163C2×Q8C2×D412yes—
164C2×Q8#4112yes—
165C2×D4—24no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—21no—
170C2×D4—21no—
171C2×D4—81no—
172C2×D4—81no—
173C2×D4C2×D412yes—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—24no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2×D411yescharacteristic
193C22×C4C2×D411yescharacteristic
194C22×C4C2×D411yescharacteristic
195C22×C4C2411yescharacteristic
196Q16—28no—
197SD16—28no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8#4112yes—
209C2×C8#4112yes—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—24no—
213C4⋊C4—24no—
214C4⋊C4—22no—
215C4⋊C4—22no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—42no—
224C4⋊C4—81no—
225C4⋊C4C2×D412yes—
226C4⋊C4C2×D412yes—
227C4⋊C4#4112yes—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—22no—
235C22⋊C4—22no—
236C22⋊C4—81no—
237C42—24no—
238C42—21no—
239C42—21no—
240C42—42no—
241C42—81no—
242C42C2×D412yes—
243C42C2×D412yes—
244C23—21no—
245C23—22no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23C22×D411yescharacteristic
259Q8—22no—
260Q8—22no—
261Q8—22no—
262Q8—22no—
263Q8—41no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—84no—
268D4—22no—
269D4—22no—
270D4—41no—
271D4—41no—
272D4—84no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
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—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—84no—
312C2×C4#12012yes—
313C2×C4#12012yes—
314C2×C4C22×D411yescharacteristic
315C2×C4C22×D411yescharacteristic
316C2×C4C2×#4111yescharacteristic
317C2×C4C2×#4111yescharacteristic
318C2×C4C2×#4111yescharacteristic
319C2×C4C2×#4111yescharacteristic
320C8—22no—
321C8—22no—
322C8—41no—
323C8—41no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—42no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—42no—
332C22—42no—
333C22—41no—
334C22—81no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22C2×#12011yescharacteristic
339C22#52511yescharacteristic
340C22#53211yescharacteristic
341C22#57611yescharacteristic
342C22#57711yescharacteristic
343C22#57811yescharacteristic
344C22#57811yescharacteristic
345C4—22no—
346C4—22no—
347C4—42no—
348C4—41no—
349C4—42no—
350C4—42no—
351C4—41no—
352C4—41no—
353C4—42no—
354C4—42no—
355C4—81no—
356C4—81no—
357C4#52511yescharacteristic
358C4#52511yescharacteristic
359C4#53211yescharacteristic
360C4#53211yescharacteristic
361C2—81no—
362C2—82no—
363C2#238811yescharacteristic
364C2#301111yescharacteristic
365C2#301211yescharacteristic
366C2#312111yescharacteristic
367C2#312411yescharacteristic
368C2#317511yescharacteristic
369C2#317811yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#440C2semidirect
3. C2×C4×Q16C2semidirect
4. #2323C2semidirect
5. #2057C2semidirect
6. #2042C2semidirect
7. #2036C2semidirect
9. #2024C2semidirect
10. #2015C2semidirect
11. #2001C2semidirect
13. #1763C2semidirect
14. C4×#102C2semidirect
15. #1750C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5422
C2×#4402
C2×C4×Q162
#23232
#20572
#20422
#20362
#20332
#20242
#20152
#20012
#20002
#17632
C4×#1022
#17502