Finite Group Databasebeta

← #279302

Subgroups — #279302

Order 512.

642 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33199C211yesmaximal, characteristic
2#29191C211yesmaximal, characteristic, complemented
3#22696C211yesmaximal, characteristic
4C4×#474C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10653C211yesmaximal, characteristic, complemented
7#10596C211yesmaximal, characteristic, complemented
8#10588C211yesmaximal, characteristic, complemented
9#9899C211yesmaximal, characteristic, complemented
10#9602C211yesmaximal, characteristic, complemented
11#9219C211yesmaximal, characteristic
12#9219C211yesmaximal, characteristic
13#8646C211yesmaximal, characteristic
14#8642C211yesmaximal, characteristic
15#8591C211yesmaximal, characteristic, complemented
16C2×#478—21no—
17C2×#474C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20#2859C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#2066—21no—
24#2066—21no—
25#1989—21no—
26#1988C2211yescharacteristic
27#1956C2211yescharacteristic
28#1956C2211yescharacteristic
29#1949—21no—
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1939C2211yescharacteristic
34#1939C2211yescharacteristic
35#1918—21no—
36#1865—21no—
37#1865—21no—
38#1862—21no—
39#1850—21no—
40#1850—21no—
41#1844C2211yescharacteristic
42#1842C2211yescharacteristic
43#1834C2211yescharacteristic
44#1831C2211yescharacteristic
45#1804—21no—
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1797—21no—
50C4×#107—21no—
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×#102—21no—
54C2×#336—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
67C23×C8—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×C4×Q8—21no—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C22×C42—21no—
76#478—44no—
77#474—22no—
78#474Q814yes—
79C8×D4—22no—
80C8×D4—22no—
81C8×D4—41no—
82C8×D4—41no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#102—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#382—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133C43C2311yescharacteristic
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151#334—41no—
152#334—41no—
153#332—41no—
154#332—41no—
155C22×Q8—21no—
156C22×D4C2×D411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—42no—
170C22×C8—81no—
171C22×C8—81no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C2×Q812yes—
180#125—42no—
181#125—42no—
182C4×Q8—81no—
183C4×Q8—81no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—41no—
189C4×D4SD1614yes—
190C4×D4C2×Q812yes—
191C4×D4#4112yes—
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—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—41no—
216C2×C42—81no—
217C2×C42—81no—
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42#4111yescharacteristic
223C2×C42#4111yescharacteristic
224C2×C42C2411yescharacteristic
225#107—24no—
226#107—42no—
227#107—42no—
228#106—42no—
229#106—81no—
230#106—81no—
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—41no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—81no—
247C4⋊C8—81no—
248C4⋊C8C2×Q812yes—
249#103—42no—
250#103—42no—
251#103—81no—
252#103—81no—
253#102—24no—
254#102—42no—
255#102—42no—
256#102—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—42no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271C24—21no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4C22⋊Q812yes—
282C2×D4C2×SD1612yes—
283C2×D4#13712yes—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—81no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4#12111yescharacteristic
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—22no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
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—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—84no—
340C2×C8—84no—
341C2×C8#13512yes—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—21no—
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—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4—82no—
366C4⋊C4—82no—
367C4⋊C4—82no—
368C4⋊C4C22⋊Q812yes—
369C4⋊C4C2×SD1612yes—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—24no—
373C22⋊C4—42no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—81no—
385C42—81no—
386C42—82no—
387C42C2×SD1612yes—
388C42#13512yes—
389C42C22×Q811yescharacteristic
390C42C2×#4111yescharacteristic
391C42H(5,𝔽2)11yescharacteristic
392C422−1+411yescharacteristic
393C23—22no—
394C23—22no—
395C23—22no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23C2×#12111yescharacteristic
401Q8—42no—
402Q8—42no—
403Q8—81no—
404Q8—81no—
405D4—22no—
406D4—22no—
407D4—22no—
408D4#47514yes—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—84no—
436C2×C4—84no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4#46512yes—
449C2×C4#46512yes—
450C2×C4C2×C22⋊Q811yescharacteristic
451C2×C4#53711yescharacteristic
452C2×C4#54611yescharacteristic
453C2×C4D4×Q811yescharacteristic
454C2×C4#55211yescharacteristic
455C2×C4#55411yescharacteristic
456C2×C4C22×SD1611yescharacteristic
457C2×C4C2×#13511yescharacteristic
458C2×C4C2×#13511yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57511yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57611yescharacteristic
463C2×C4#57611yescharacteristic
464C8—22no—
465C8—42no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—81no—
470C8—81no—
471C8—81no—
472C8—81no—
473C8—81no—
474C22—24no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22#241711yescharacteristic
482C22#303511yescharacteristic
483C22#304111yescharacteristic
484C22C2×#46511yescharacteristic
485C22#305611yescharacteristic
486C22#306711yescharacteristic
487C22#306911yescharacteristic
488C4—21no—
489C4—21no—
490C4—22no—
491C4—22no—
492C4—41no—
493C4—42no—
494C4—41no—
495C4—42no—
496C4—81no—
497C4—82no—
498C4—161no—
499C4—161no—
500C4—161no—
501C4C2×#47511yescharacteristic
502C4#307611yescharacteristic
503C4#324611yescharacteristic
504C4#324811yescharacteristic
505C4#329711yescharacteristic
506C4#332211yescharacteristic
507C4#338011yescharacteristic
508C4#338911yescharacteristic
509C2—44no—
510C2#1912411yescharacteristic
511C2#1918111yescharacteristic
512C2#1938511yescharacteristic
513C2#1946011yescharacteristic
514C2#2482011yescharacteristic
515C2#2483811yescharacteristic
516C2#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. #29191C2semidirect
5. #10658C2semidirect
6. #10653C2semidirect
7. #10596C2semidirect
8. #10588C2semidirect
9. #9899C2semidirect
10. #9602C2semidirect
15. #8591C2semidirect

Maximal subgroups

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

subgroupindex
#331992
#291912
#226962
C4×#4742
#106582
#106532
#105962
#105882
#98992
#96022
#92192
#86462
#86422
#85912