Finite Group Databasebeta

← #22472

Subgroups — #22472

Order 256.

581 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#529C211yesmaximal, characteristic
2C2×#436C211yesmaximal, characteristic, complemented
3C2×#435C211yesmaximal, characteristic, complemented
4#1985C211yesmaximal, characteristic, complemented
5#1984C211yesmaximal, characteristic, complemented
6#1870C211yesmaximal, characteristic, complemented
7#1870C211yesmaximal, characteristic, complemented
8#1870C211yesmaximal, characteristic, complemented
9#1869C211yesmaximal, characteristic, complemented
10#1818C211yesmaximal, characteristic, complemented
11#1817C211yesmaximal, characteristic, complemented
12#1798C211yesmaximal, characteristic, complemented
13#1750C211yesmaximal, characteristic, complemented
14#1729C211yesmaximal, characteristic, complemented
15#1728C211yesmaximal, characteristic, complemented
16C22×M16C2211yescharacteristic
17C22×M16C2211yescharacteristic
18C22×M16C2211yescharacteristic
19C23×C8C2211yescharacteristic
20#529—24no—
21#529—24no—
22C2×#126C411yescharacteristic
23C2×#126C411yescharacteristic
24C2×#125C411yescharacteristic
25C2×#124C411yescharacteristic
26C2×#123C411yescharacteristic
27C2×#123C411yescharacteristic
28C2×C22⋊Q8C411yescharacteristic
29C2×#121C411yescharacteristic
30C2×C4×Q8C2211yescharacteristic
31C2×C4×D4C2211yescharacteristic
32C2×C4×D4C2211yescharacteristic
33C2×C4×D4C2211yescharacteristic
34C2×#117C2211yescharacteristic
35C2×#117C2211yescharacteristic
36C22×C42C2211yescharacteristic
37#436—24no—
38#435—24no—
39C2×C4⋊C8C2211yescharacteristic
40C2×C4⋊C8C2211yescharacteristic
41C2×C22⋊C8C2211yescharacteristic
42C2×C22⋊C8C2211yescharacteristic
43C2×C22⋊C8C2211yescharacteristic
44C2×C22⋊C8C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×#97C2211yescharacteristic
50C2×C4×C8C2211yescharacteristic
51#336C2211yescharacteristic
52#336C2211yescharacteristic
53#336C2211yescharacteristic
54#336C2211yescharacteristic
55#336C2211yescharacteristic
56#336C2211yescharacteristic
57#336C2211yescharacteristic
58#336C2211yescharacteristic
59#336C2211yescharacteristic
60#336C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C22×Q8C2×C411yescharacteristic
64C22×D4C2×C411yescharacteristic
65C22×D4C2×C411yescharacteristic
66C22×D4C2×C411yescharacteristic
67C23×C4C2×C411yescharacteristic
68C23×C4C2×C411yescharacteristic
69C23×C4C2311yescharacteristic
70C23×C4C2311yescharacteristic
71C23×C4C2311yescharacteristic
72C2×M16—22no—
73C2×M16—22no—
74C2×M16—22no—
75C2×M16—22no—
76C2×M16—22no—
77C2×M16—22no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C22×C8—22no—
82C22×C8—22no—
83C22×C8—41no—
84C22×C8C2311yescharacteristic
85C22×C8C2311yescharacteristic
86C22×C8C2311yescharacteristic
87C22×C8C2311yescharacteristic
88C22×C8C2311yescharacteristic
89C22×C8C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92#126—24no—
93#126—24no—
94#125—24no—
95#124—24no—
96#123—24no—
97#123—24no—
98C22⋊Q8—24no—
99#121—24no—
100C4×Q8—22no—
101C4×Q8—22no—
102C4×D4—22no—
103C4×D4—22no—
104C4×D4—22no—
105C4×D4—22no—
106C4×D4D414yes—
107C4×D4D414yes—
108#117—22no—
109#117—22no—
110#117—22no—
111#117—22no—
112C2×C4⋊C4C2×C411yescharacteristic
113C2×C4⋊C4C2×C411yescharacteristic
114C2×C4⋊C4C2×C411yescharacteristic
115C2×C4⋊C4C2×C411yescharacteristic
116C2×C4⋊C4C2×C411yescharacteristic
117C2×C4⋊C4C2×C411yescharacteristic
118C2×C4⋊C4C2×C411yescharacteristic
119C2×C4⋊C4C2×C411yescharacteristic
120C2×C4⋊C4C2×C411yescharacteristic
121C2×C4⋊C4C2×C411yescharacteristic
122C2×C22⋊C4C2×C411yescharacteristic
123C2×C22⋊C4C2×C411yescharacteristic
124C2×C22⋊C4C2×C411yescharacteristic
125C2×C22⋊C4C2×C411yescharacteristic
126C2×C22⋊C4C2×C411yescharacteristic
127C2×C22⋊C4C2×C411yescharacteristic
128C2×C22⋊C4C2×C411yescharacteristic
129C2×C22⋊C4C2×C411yescharacteristic
130C2×C22⋊C4C2×C411yescharacteristic
131C2×C22⋊C4C2×C411yescharacteristic
132C2×C42—22no—
133C2×C42—22no—
134C2×C42C2×C411yescharacteristic
135C2×C42C2×C411yescharacteristic
136C2×C42C2311yescharacteristic
137C2×C42C2311yescharacteristic
138C2×C42C2311yescharacteristic
139C2×C42C2311yescharacteristic
140C4⋊C8—22no—
141C4⋊C8—22no—
142C22⋊C8—22no—
143C22⋊C8—22no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150#97—22no—
151C4×C8—22no—
152C24C22×C411yescharacteristic
153C24C22×C411yescharacteristic
154C24C22×C411yescharacteristic
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—41no—
158C2×D4—22no—
159C2×D4—22no—
160C2×D4—22no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C2×D4—41no—
165C2×D4—41no—
166C2×D4C22⋊C414yes—
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—22no—
176C22×C4—22no—
177C22×C4—22no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4C22×C411yescharacteristic
188C22×C4C22×C411yescharacteristic
189C22×C4C22×C411yescharacteristic
190C22×C4C22×C411yescharacteristic
191C22×C4C22×C411yescharacteristic
192C22×C4C22×C411yescharacteristic
193C22×C4C22×C411yescharacteristic
194C22×C4C22×C411yescharacteristic
195C22×C4C22×C411yescharacteristic
196C22×C4C22×C411yescharacteristic
197C22×C4C22×C411yescharacteristic
198C22×C4C2×D412yes—
199C22×C4C2×D412yes—
200C22×C4C2×D412yes—
201C22×C4C2×D412yes—
202C22×C4C2411yescharacteristic
203M16—44no—
204M16—44no—
205M16—44no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
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—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—44no—
229C2×C8#4112yes—
230C2×C8#4112yes—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4C22⋊C414yes—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—22no—
244C22⋊C4—22no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—22no—
248C22⋊C4—22no—
249C22⋊C4C22⋊C414yes—
250C22⋊C4C22⋊C414yes—
251C42—22no—
252C42—22no—
253C42—21no—
254C42—21no—
255C42—21no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42C2×D412yes—
260C42C2×D412yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23C2×C22⋊C412yes—
281C23C2×C22⋊C412yes—
282C23C23×C411yescharacteristic
283Q8—44no—
284D4—44no—
285D4—44no—
286D4—44no—
287C2×C4—21no—
288C2×C4—21no—
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—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4C2×C22⋊C412yes—
330C2×C4C2×C22⋊C412yes—
331C2×C4C2×C22⋊C412yes—
332C2×C4C2×C22⋊C412yes—
333C2×C4#13112yes—
334C2×C4C22×D411yescharacteristic
335C2×C4C22×D411yescharacteristic
336C2×C4C2×#4111yescharacteristic
337C2×C4C2×#4111yescharacteristic
338C2×C4H(5,𝔽2)11yescharacteristic
339C2×C42−1+411yescharacteristic
340C8—22no—
341C8—22no—
342C8—41no—
343C8—41no—
344C8—41no—
345C8—41no—
346C8—41no—
347C8—41no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22C22×C22⋊C411yescharacteristic
370C22C4×#4111yescharacteristic
371C22#52011yescharacteristic
372C22C2×#13111yescharacteristic
373C22#56811yescharacteristic
374C22#56811yescharacteristic
375C22#56811yescharacteristic
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4#54611yescharacteristic
388C4#54611yescharacteristic
389C4#54711yescharacteristic
390C4#54711yescharacteristic
391C2—42no—
392C2—42no—
393C2—81no—
394C2#233911yescharacteristic
395C2#287911yescharacteristic
396C2#288011yescharacteristic
397C2#296711yescharacteristic
398C2#296911yescharacteristic
399C2#299311yescharacteristic
400C2#299511yescharacteristic
subgroupG/H
center, Frattini, cosocleC22×C4C24
commutator, socleC23C23×C4
prime radicalC2×#121C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#436C2semidirect
3. C2×#435C2semidirect
4. #1985C2semidirect
5. #1984C2semidirect
6. #1870C2semidirect
7. #1870C2semidirect
8. #1870C2semidirect
9. #1869C2semidirect
10. #1818C2semidirect
11. #1817C2semidirect
12. #1798C2semidirect
13. #1750C2semidirect
14. #1729C2semidirect
15. #1728C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5292
C2×#4362
C2×#4352
#19852
#19842
#18702
#18692
#18182
#18172
#17982
#17502
#17292
#17282