Finite Group Databasebeta

← #24017

Subgroups — #24017

Order 256.

670 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#537C211yesmaximal, characteristic
2C2×#485C211yesmaximal, characteristic, complemented
3C2×#482C211yesmaximal, characteristic, complemented
4#2311C211yesmaximal, characteristic, complemented
5#2036C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2034C211yesmaximal, characteristic, complemented
8#2033C211yesmaximal, characteristic, complemented
9#1874C211yesmaximal, characteristic, complemented
10#1872C211yesmaximal, characteristic, complemented
11#1853C211yesmaximal, characteristic, complemented
12#1852C211yesmaximal, characteristic, complemented
13#1788C211yesmaximal, characteristic, complemented
14#1788C211yesmaximal, characteristic, complemented
15#1780C211yesmaximal, characteristic, complemented
16C22×#41C2211yescharacteristic
17C22×#41C2211yescharacteristic
18#537—24no—
19#537—24no—
20C2×C22⋊Q8—21no—
21C2×C22⋊Q8C2211yescharacteristic
22C2×C22⋊Q8C2211yescharacteristic
23C2×#121—21no—
24C2×#121—21no—
25C2×#121—21no—
26C2×#121C2211yescharacteristic
27C2×#121C2211yescharacteristic
28C2×#117C2211yescharacteristic
29C22×C4⋊C4C2211yescharacteristic
30C22×C4⋊C4C2211yescharacteristic
31#485—24no—
32#482—24no—
33C2×#107C2211yescharacteristic
34C2×#107C2211yescharacteristic
35C2×#106C2211yescharacteristic
36C2×#106C2211yescharacteristic
37C2×#103C2211yescharacteristic
38C2×#103C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#102C2211yescharacteristic
42C2×#102C2211yescharacteristic
43C2×#102C2211yescharacteristic
44C2×#102C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C22⋊C8C2211yescharacteristic
50C2×C22⋊C8C2211yescharacteristic
51#391—21no—
52#390—21no—
53#386—21no—
54#386—21no—
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#380C2211yescharacteristic
58#380C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C2×#41—22no—
64C2×#41—22no—
65C2×#41—22no—
66C2×#41—22no—
67C2×#41—22no—
68C2×#41—22no—
69C2×#41—22no—
70C2×#41—22no—
71C2×#41—41no—
72C2×#41—41no—
73C2×#41—41no—
74C2×#41—41no—
75C22×Q8D411yescharacteristic
76C22×Q8C2311yescharacteristic
77C22×D4—21no—
78C22×D4—21no—
79C22×D4—21no—
80C22×D4D411yescharacteristic
81C22×D4D411yescharacteristic
82C22×D4D411yescharacteristic
83C22×D4C2311yescharacteristic
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4C2311yescharacteristic
88C23×C4C2311yescharacteristic
89C23×C4C2311yescharacteristic
90C22×C8C2311yescharacteristic
91C22×C8C2311yescharacteristic
92C22×C8C2311yescharacteristic
93C22×C8C2311yescharacteristic
94C22⋊Q8—28no—
95C22⋊Q8—22no—
96C22⋊Q8—22no—
97C22⋊Q8—22no—
98C22⋊Q8D414yes—
99#121—28no—
100#121—28no—
101#121—28no—
102#121—22no—
103#121—22no—
104#121—22no—
105#121D414yes—
106#117—22no—
107#117—22no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—41no—
116C2×C4⋊C4C2311yescharacteristic
117C2×C4⋊C4C2311yescharacteristic
118C2×C4⋊C4C2311yescharacteristic
119C2×C4⋊C4C2311yescharacteristic
120C2×C4⋊C4C2311yescharacteristic
121C2×C4⋊C4C2311yescharacteristic
122C2×C22⋊C4—21no—
123C2×C22⋊C4—21no—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—21no—
128C2×C22⋊C4—21no—
129C2×C22⋊C4—21no—
130C2×C22⋊C4—21no—
131C2×C42—21no—
132#107—22no—
133#107—22no—
134#106—22no—
135#106—22no—
136#103—22no—
137#103—22no—
138#103—41no—
139#103—41no—
140#102—22no—
141#102—22no—
142#102—41no—
143#102—41no—
144C22⋊C8—22no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—22no—
148C22⋊C8—41no—
149C22⋊C8—41no—
150#95—21no—
151#95—21no—
152#95—21no—
153#95—21no—
154C24—21no—
155C24C2×D411yescharacteristic
156C24C2×D411yescharacteristic
157C24C2×D411yescharacteristic
158#41—44no—
159#41—44no—
160#41—44no—
161#41—44no—
162#41—44no—
163#41—44no—
164#41—44no—
165#41—44no—
166C2×Q8—22no—
167C2×Q8—22no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—21no—
171C2×Q8—21no—
172C2×Q8—21no—
173C2×Q8—41no—
174C2×Q8C2×D412yes—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—22no—
179C2×D4—22no—
180C2×D4—24no—
181C2×D4—24no—
182C2×D4—22no—
183C2×D4—22no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—21no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—81no—
195C2×D4—81no—
196C2×D4—81no—
197C2×D4—81no—
198C2×D4C2×D412yes—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—24no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—24no—
212C22×C4—21no—
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—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—81no—
226C22×C4—81no—
227C22×C4—81no—
228C22×C4—81no—
229C22×C4C2×D411yescharacteristic
230C22×C4C2×D411yescharacteristic
231C22×C4C2×D412yes—
232C22×C4C2×D412yes—
233C22×C4C2×D411yescharacteristic
234C22×C4#4112yes—
235C22×C4#4112yes—
236C22×C4C2411yescharacteristic
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—21no—
247C2×C8—21no—
248C2×C8—21no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—42no—
263C4⋊C4C2×D412yes—
264C4⋊C4C2×D412yes—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—42no—
270C22⋊C4—81no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C22⋊C4—81no—
274C42—42no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—22no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23#12012yes—
302C23#12312yes—
303C23C22×D411yescharacteristic
304Q8—22no—
305Q8—22no—
306Q8—44no—
307Q8—41no—
308Q8—41no—
309D4—22no—
310D4—22no—
311D4—44no—
312D4—44no—
313D4—44no—
314D4—48no—
315D4—41no—
316D4—41no—
317D4—84no—
318D4—84no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—48no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
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—41no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4#12012yes—
363C2×C4#12312yes—
364C2×C4C22×D411yescharacteristic
365C2×C4C22×D411yescharacteristic
366C2×C4C2×#4111yescharacteristic
367C2×C4C2×#4111yescharacteristic
368C2×C4H(5,𝔽2)11yescharacteristic
369C2×C4H(5,𝔽2)11yescharacteristic
370C8—41no—
371C8—41no—
372C8—41no—
373C8—41no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—42no—
396C22—41no—
397C22—42no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22C2×#12011yescharacteristic
403C22C2×#12311yescharacteristic
404C22#53411yescharacteristic
405C22#57611yescharacteristic
406C22#57711yescharacteristic
407C22#57711yescharacteristic
408C22#57811yescharacteristic
409C4—22no—
410C4—22no—
411C4—41no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—81no—
416C4—81no—
417C4—81no—
418C4—81no—
419C4#54611yescharacteristic
420C4#54611yescharacteristic
421C4#54611yescharacteristic
422C4#54611yescharacteristic
423C2—42no—
424C2—42no—
425C2—81no—
426C2—82no—
427C2#240511yescharacteristic
428C2#301211yescharacteristic
429C2#301311yescharacteristic
430C2#309811yescharacteristic
431C2#309911yescharacteristic
432C2#319811yescharacteristic
433C2#320111yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC2×#537C2
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
2. C2×#485C2semidirect
3. C2×#482C2semidirect
4. #2311C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #2034C2semidirect
8. #2033C2semidirect
9. #1874C2semidirect
10. #1872C2semidirect
11. #1853C2semidirect
12. #1852C2semidirect
13. #1788C2semidirect
14. #1788C2semidirect
15. #1780C2semidirect

Maximal subgroups

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

subgroupindex
C2×#5372
C2×#4852
C2×#4822
#23112
#20362
#20352
#20342
#20332
#18742
#18722
#18532
#18522
#17882
#17802