Finite Group Databasebeta

← #279746

Subgroups — #279746

Order 512.

596 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29998C211yesmaximal, characteristic
2#28515C211yesmaximal, characteristic, complemented
3#22701C211yesmaximal, characteristic
4#22116C211yesmaximal, characteristic
5#10654C211yesmaximal, characteristic, complemented
6#10652C211yesmaximal, characteristic, complemented
7#10596C211yesmaximal, characteristic, complemented
8#10596C211yesmaximal, characteristic, complemented
9#9907C211yesmaximal, characteristic, complemented
10#9605C211yesmaximal, characteristic, complemented
11#9227C211yesmaximal, characteristic
12#9219C211yesmaximal, characteristic
13#8652C211yesmaximal, characteristic
14#8642C211yesmaximal, characteristic
15#8591C211yesmaximal, characteristic, complemented
16C2×#479—21no—
17C2×#479—21no—
18C2×#474C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21#2859C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23#2293C2211yescharacteristic
24#2097—21no—
25#2097—21no—
26#2088—21no—
27#2088—21no—
28#2065—21no—
29#2065—21no—
30#1988C2211yescharacteristic
31#1956C2211yescharacteristic
32#1956C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1943—21no—
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1936—21no—
40#1926—21no—
41#1921—21no—
42#1851—21no—
43#1850—21no—
44#1845C2211yescharacteristic
45#1844C2211yescharacteristic
46#1842C2211yescharacteristic
47#1831C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1798—21no—
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1729—21no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×C4×Q8—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74#479—44no—
75#479—44no—
76#474—22no—
77#474Q814yes—
78#436—22no—
79#436—41no—
80C8×D4—22no—
81C8×D4—41no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8D411yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#103—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×#102—21no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391—21no—
116#390D411yescharacteristic
117#390D411yescharacteristic
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389D411yescharacteristic
125#389D411yescharacteristic
126#388—21no—
127#388—21no—
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#382—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2311yescharacteristic
135C4×C22⋊C4—21no—
136C43C2311yescharacteristic
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336D411yescharacteristic
151#336D411yescharacteristic
152#334—41no—
153#334—41no—
154#332—41no—
155#332—41no—
156C22×Q8—21no—
157C22×D4C2×D411yescharacteristic
158C23×C4—21no—
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—81no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—81no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C2×Q812yes—
179#125—42no—
180#125—42no—
181C4×Q8—42no—
182C4×Q8—42no—
183C4×D4—21no—
184C4×D4—21no—
185C4×D4C2×Q812yes—
186C4×D4#4112yes—
187#117—81no—
188#117—81no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C42—21no—
214C2×C42—41no—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#107—24no—
223#107—42no—
224#107—42no—
225#106—42no—
226#106—42no—
227#106—81no—
228#106—81no—
229#106—81no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—81no—
244C4⋊C8—81no—
245C4⋊C8—81no—
246C4⋊C8—81no—
247C4⋊C8C2×Q812yes—
248#103—42no—
249#103—42no—
250#103—42no—
251#103—42no—
252#102—24no—
253#102—81no—
254#102—81no—
255#102—81no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258#97—81no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270C24—21no—
271C2×Q8—22no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4C22⋊Q812yes—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12111yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303M16—84no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—84no—
333C2×C8AGL(1,ℤ/8)12yes—
334C2×C8AGL(1,ℤ/8)12yes—
335C4⋊C4—21no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4C22⋊Q812yes—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—42no—
368C22⋊C4—81no—
369C22⋊C4—81no—
370C22⋊C4—81no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—82no—
384C42C22×Q811yescharacteristic
385C42C2×#4111yescharacteristic
386C42H(5,𝔽2)11yescharacteristic
387C422−1+411yescharacteristic
388C23—22no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23C2×#12111yescharacteristic
396Q8—42no—
397Q8—42no—
398Q8—81no—
399Q8—81no—
400D4—22no—
401D4—22no—
402D4—41no—
403D4—41no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4C2×C22⋊Q811yescharacteristic
444C2×C4#53711yescharacteristic
445C2×C4#54611yescharacteristic
446C2×C4D4×Q811yescharacteristic
447C2×C4#55211yescharacteristic
448C2×C4#55411yescharacteristic
449C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
450C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57711yescharacteristic
453C2×C4#57711yescharacteristic
454C2×C4#57711yescharacteristic
455C2×C4#57711yescharacteristic
456C2×C4#57811yescharacteristic
457C8—22no—
458C8—22no—
459C8—42no—
460C8—41no—
461C8—41no—
462C8—81no—
463C8—81no—
464C8—81no—
465C8—81no—
466C8—81no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22#241711yescharacteristic
475C22#303811yescharacteristic
476C22#304211yescharacteristic
477C22#306011yescharacteristic
478C22#306311yescharacteristic
479C22#306711yescharacteristic
480C22#306911yescharacteristic
481C4—21no—
482C4—21no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—41no—
487C4—42no—
488C4—81no—
489C4—81no—
490C4—82no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4#308211yescharacteristic
495C4#308411yescharacteristic
496C4#325811yescharacteristic
497C4#326011yescharacteristic
498C4#330611yescharacteristic
499C4#330611yescharacteristic
500C4#338411yescharacteristic
501C4#338411yescharacteristic
502C2—82no—
503C2#1935511yescharacteristic
504C2#1937711yescharacteristic
505C2#1951711yescharacteristic
506C2#1956211yescharacteristic
507C2#2482911yescharacteristic
508C2#2485411yescharacteristic
509C2#2486011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #28515C2semidirect
5. #10654C2semidirect
6. #10652C2semidirect
7. #10596C2semidirect
8. #10596C2semidirect
9. #9907C2semidirect
10. #9605C2semidirect
15. #8591C2semidirect

Maximal subgroups

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

subgroupindex
#299982
#285152
#227012
#221162
#106542
#106522
#105962
#99072
#96052
#92272
#92192
#86522
#86422
#85912