Finite Group Databasebeta

← #24034

Subgroups — #24034

Order 256.

690 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#550C211yesmaximal, characteristic
2C2×#482C211yesmaximal, characteristic, complemented
3C2×#416C211yesmaximal, characteristic, complemented
4#2485C211yesmaximal, characteristic, complemented
5#2041C211yesmaximal, characteristic, complemented
6#2041C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1893C211yesmaximal, characteristic, complemented
10#1869C211yesmaximal, characteristic, complemented
11#1865C211yesmaximal, characteristic, complemented
12#1863C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1788C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×#41C2211yescharacteristic
18C23×C8C2211yescharacteristic
19#550—24no—
20#550—24no—
21C2×#127—21no—
22C2×#127C2211yescharacteristic
23C2×#121—21no—
24C2×#121—21no—
25C2×#121C2211yescharacteristic
26C2×#121C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4C2211yescharacteristic
30C2×#117C2211yescharacteristic
31#482—24no—
32C2×#107C2211yescharacteristic
33C2×#106C2211yescharacteristic
34#416—24no—
35C2×#103C2211yescharacteristic
36C2×#103C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#102C2211yescharacteristic
40C2×#102C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×#102C2211yescharacteristic
46C2×#102C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50#397—21no—
51#394—21no—
52#389C2211yescharacteristic
53#389C2211yescharacteristic
54#388—21no—
55#387—21no—
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#336C2211yescharacteristic
62#336C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—41no—
71C2×#41—41no—
72C2×#41—41no—
73C2×#41—41no—
74C2×#41D414yes—
75C22×Q8C2311yescharacteristic
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4—21no—
81C22×D4—21no—
82C22×D4D411yescharacteristic
83C22×D4C2311yescharacteristic
84C22×D4C2311yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C22×C8—22no—
90C22×C8—41no—
91C22×C8—41no—
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22×C8C2311yescharacteristic
95C22×C8C2311yescharacteristic
96#127—28no—
97#127—22no—
98#127—22no—
99#121—28no—
100#121—28no—
101#121—22no—
102#121—22no—
103#121—22no—
104#121D414yes—
105C4×Q8—22no—
106C4×Q8—22no—
107C4×D4—28no—
108C4×D4—22no—
109C4×D4—22no—
110#117—22no—
111#117—22no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4D411yescharacteristic
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4D411yescharacteristic
128C2×C22⋊C4D411yescharacteristic
129C2×C42—21no—
130C2×C42—21no—
131C2×C42C2311yescharacteristic
132#107—22no—
133#106—22no—
134#103—22no—
135#103—22no—
136#103—22no—
137#103—22no—
138#102—22no—
139#102—22no—
140#102—22no—
141#102—22no—
142#102—41no—
143#102—41no—
144#102—41no—
145#102—41no—
146C22⋊C8—22no—
147C22⋊C8—41no—
148C22⋊C8—41no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154C24—21no—
155C24—21no—
156C24C2×D411yescharacteristic
157C24C2×D411yescharacteristic
158#41—28no—
159#41—44no—
160#41—44no—
161#41—44no—
162#41—44no—
163#41—44no—
164#41—44no—
165#41—44no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8C2×D412yes—
170C2×Q8#4112yes—
171C2×Q8#4112yes—
172C2×D4—24no—
173C2×D4—24no—
174C2×D4—22no—
175C2×D4—24no—
176C2×D4—24no—
177C2×D4—24no—
178C2×D4—24no—
179C2×D4—22no—
180C2×D4—21no—
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—81no—
198C2×D4—81no—
199C2×D4—81no—
200C2×D4—81no—
201C2×D4—81no—
202C2×D4—81no—
203C2×D4C2×D412yes—
204C2×D4C2×D412yes—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—24no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C2×D411yescharacteristic
228C22×C4C2×D411yescharacteristic
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D412yes—
231C22×C4C2×D412yes—
232C22×C4C2×D411yescharacteristic
233C22×C4C2411yescharacteristic
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—44no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—21no—
250C4⋊C4—21no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—42no—
259C4⋊C4—81no—
260C4⋊C4C2×D412yes—
261C22⋊C4—24no—
262C22⋊C4—22no—
263C22⋊C4—24no—
264C22⋊C4—22no—
265C22⋊C4—42no—
266C22⋊C4—42no—
267C22⋊C4—81no—
268C22⋊C4—81no—
269C42—24no—
270C42—21no—
271C42—21no—
272C42—42no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—22no—
281C23—22no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23#12012yes—
301C23C22×D411yescharacteristic
302Q8—22no—
303Q8—22no—
304Q8—22no—
305Q8#12314yes—
306D4—22no—
307D4—22no—
308D4—22no—
309D4—22no—
310D4—44no—
311D4—48no—
312D4—48no—
313D4—48no—
314D4—41no—
315D4—41no—
316D4—41no—
317D4—41no—
318D4—84no—
319D4—84no—
320D4—84no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—48no—
340C2×C4—48no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4#12012yes—
360C2×C4#13512yes—
361C2×C4C22×D411yescharacteristic
362C2×C4C22×D411yescharacteristic
363C2×C4C2×#4111yescharacteristic
364C2×C4C2×#4111yescharacteristic
365C2×C4H(5,𝔽2)11yescharacteristic
366C2×C4H(5,𝔽2)11yescharacteristic
367C8—41no—
368C8—41no—
369C8—41no—
370C8—41no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—42no—
379C22—42no—
380C22—41no—
381C22—41no—
382C22—42no—
383C22—42no—
384C22—41no—
385C22—42no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22C2×#12011yescharacteristic
400C22#54611yescharacteristic
401C22#54611yescharacteristic
402C22C2×#13511yescharacteristic
403C22#57511yescharacteristic
404C22#57611yescharacteristic
405C22#57711yescharacteristic
406C4—22no—
407C4—24no—
408C4—22no—
409C4—41no—
410C4—41no—
411C4—42no—
412C4—81no—
413C4—81no—
414C4—81no—
415C4—81no—
416C4C2×#12311yescharacteristic
417C4#53411yescharacteristic
418C4#54611yescharacteristic
419C4#54611yescharacteristic
420C2—42no—
421C2—81no—
422C2—82no—
423C2—82no—
424C2#240511yescharacteristic
425C2#300211yescharacteristic
426C2#301211yescharacteristic
427C2#330411yescharacteristic
428C2#330511yescharacteristic
429C2#330911yescharacteristic
430C2#331711yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#550C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#482C2semidirect
3. C2×#416C2semidirect
4. #2485C2semidirect
5. #2041C2semidirect
6. #2041C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1893C2semidirect
10. #1869C2semidirect
11. #1865C2semidirect
12. #1863C2semidirect
13. #1788C2semidirect
14. #1788C2semidirect
15. #1788C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5502
C2×#4822
C2×#4162
#24852
#20412
#20342
#20332
#18932
#18692
#18652
#18632
#17882