Finite Group Databasebeta

← #226685

Subgroups — #226685

Order 512.

571 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22836C211yesmaximal, characteristic
2#22736C211yesmaximal, characteristic, complemented
3#21115C211yesmaximal, characteristic
4#9560C211yesmaximal, characteristic
5#9556C211yesmaximal, characteristic
6#9539C211yesmaximal, characteristic, complemented
7#9535C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic
9#8952C211yesmaximal, characteristic
10#8788C211yesmaximal, characteristic, complemented
11#8569C211yesmaximal, characteristic
12#8566C211yesmaximal, characteristic, complemented
13#8265C211yesmaximal, characteristic, complemented
14#8213C211yesmaximal, characteristic, complemented
15#8209C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19#2293C2211yescharacteristic
20C42×Q8C2211yescharacteristic
21#1954C2211yescharacteristic
22#1954C2211yescharacteristic
23#1940—21no—
24#1940—21no—
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1926C411yescharacteristic
28#1926C411yescharacteristic
29#1924—21no—
30#1923—21no—
31#1919C411yescharacteristic
32#1919C411yescharacteristic
33#1880C411yescharacteristic
34#1880C411yescharacteristic
35#1865C411yescharacteristic
36#1865C411yescharacteristic
37#1831—21no—
38#1831C2211yescharacteristic
39#1831C2211yescharacteristic
40#1818—21no—
41C4×#106C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44#1767—21no—
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1750—21no—
50#1744C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×#322C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73#446—22no—
74#446—22no—
75#446—41no—
76#446—41no—
77C4×SD16—22no—
78C4×SD16—22no—
79C2×#106C2×C411yescharacteristic
80C2×#106C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8D411yescharacteristic
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#103C2×C411yescharacteristic
94C2×#103C2×C411yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#390C2×C411yescharacteristic
114#390C2×C411yescharacteristic
115#389C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#388—21no—
118#388—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C43—21no—
128C43C2311yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336C2×C411yescharacteristic
145#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150Q8⋊C8—22no—
151Q8⋊C8—22no—
152D4⋊C8—41no—
153D4⋊C8—41no—
154#322—41no—
155#322—41no—
156C22×Q8C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C2×SD16—22no—
160C2×SD16—41no—
161C2×SD16—41no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8C22×C411yescharacteristic
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—22no—
179C4×Q8—41no—
180C4×Q8—41no—
181C4×Q8—41no—
182C4×Q8#4112yes—
183C4×Q8#4112yes—
184C4×D4—21no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188#117—81no—
189#117—81no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#106—22no—
221#106—22no—
222#106—41no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—41no—
232C4⋊C8—41no—
233C4⋊C8—42no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238#103—22no—
239#103—41no—
240#103—41no—
241#102—22no—
242#102—22no—
243#102—41no—
244C22⋊C8—42no—
245C22⋊C8—42no—
246C22⋊C8—81no—
247C22⋊C8—81no—
248#97—42no—
249#97—42no—
250#97—81no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—81no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8#11712yes—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4C4×D411yescharacteristic
297C22×C4C4×D411yescharacteristic
298C22×C4C4×D411yescharacteristic
299C22×C4C4×D411yescharacteristic
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303SD16—44no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
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—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
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—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—81no—
361C4⋊C4—81no—
362C4⋊C4#11712yes—
363C4⋊C4#13112yes—
364C22⋊C4—42no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—81no—
368C22⋊C4—81no—
369C22⋊C4—81no—
370C42—22no—
371C42—22no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—41no—
381C42—41no—
382C42—42no—
383C42—42no—
384C42—42no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—81no—
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C42H(5,𝔽2)11yescharacteristic
393C422−1+411yescharacteristic
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23C2×C4×D411yescharacteristic
402Q8—22no—
403Q8—22no—
404Q8—41no—
405Q8—41no—
406D4—41no—
407D4—41no—
408D4—41no—
409D4—41no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4C2×#11711yescharacteristic
450C2×C4#52011yescharacteristic
451C2×C4#54611yescharacteristic
452C2×C4#54811yescharacteristic
453C2×C4#55211yescharacteristic
454C2×C4#55311yescharacteristic
455C2×C4C2×#13111yescharacteristic
456C2×C4#56811yescharacteristic
457C2×C4#56811yescharacteristic
458C2×C4#56811yescharacteristic
459C2×C4#57511yescharacteristic
460C2×C4#57511yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57711yescharacteristic
463C8—42no—
464C8—41no—
465C8—41no—
466C8—41no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—81no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22#235211yescharacteristic
485C22#293011yescharacteristic
486C22#293311yescharacteristic
487C22#294311yescharacteristic
488C22#294911yescharacteristic
489C22#295511yescharacteristic
490C22#295611yescharacteristic
491C4—21no—
492C4—21no—
493C4—42no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—42no—
498C4—41no—
499C4—42no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4#292111yescharacteristic
505C4#292511yescharacteristic
506C4#299311yescharacteristic
507C4#299511yescharacteristic
508C4#332611yescharacteristic
509C4#332611yescharacteristic
510C4#337111yescharacteristic
511C4#337111yescharacteristic
512C2—161no—
513C2#1793711yescharacteristic
514C2#1795111yescharacteristic
515C2#1800211yescharacteristic
516C2#1803011yescharacteristic
517C2#2275511yescharacteristic
518C2#2276511yescharacteristic
519C2#2278411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22736C2semidirect
6. #9539C2semidirect
7. #9535C2semidirect
10. #8788C2semidirect
12. #8566C2semidirect
13. #8265C2semidirect
14. #8213C2semidirect
15. #8209C2semidirect

Maximal subgroups

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

subgroupindex
#228362
#227362
#211152
#95602
#95562
#95392
#95352
#89522
#87882
#85692
#85662
#82652
#82132
#82092