Finite Group Databasebeta

← #228395

Subgroups — #228395

Order 512.

576 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#23756C211yesmaximal, characteristic
2#23054C211yesmaximal, characteristic
3#22626C211yesmaximal, characteristic
4#9600C211yesmaximal, characteristic, complemented
5#9553C211yesmaximal, characteristic, complemented
6#9548C211yesmaximal, characteristic, complemented
7#9539C211yesmaximal, characteristic, complemented
8#9535C211yesmaximal, characteristic, complemented
9#9077C211yesmaximal, characteristic
10#8850C211yesmaximal, characteristic, complemented
11#8831C211yesmaximal, characteristic
12#8283C211yesmaximal, characteristic, complemented
13#8213C211yesmaximal, characteristic, complemented
14#8146C211yesmaximal, characteristic
15#7896C211yesmaximal, characteristic
16C2×#446C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C4×C22⋊Q8C2211yescharacteristic
20#2285C2211yescharacteristic
21#2099C411yescharacteristic
22#2096C411yescharacteristic
23#2084C411yescharacteristic
24#2077C411yescharacteristic
25#2077C411yescharacteristic
26#2072C411yescharacteristic
27#2035C411yescharacteristic
28#2035C411yescharacteristic
29#1984C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1924—21no—
34#1924—21no—
35#1917—21no—
36#1916—21no—
37#1916—21no—
38#1798—21no—
39#1797C2211yescharacteristic
40C4×#106C2211yescharacteristic
41C4×#106C2211yescharacteristic
42#1771—21no—
43#1770C2211yescharacteristic
44#1770C2211yescharacteristic
45C8×C4⋊C4C2211yescharacteristic
46C4×#103C2211yescharacteristic
47C4×#103C2211yescharacteristic
48#1750—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×M16—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×#117—21no—
71C22×C42C2311yescharacteristic
72#446—22no—
73#446—41no—
74#431—22no—
75#431—41no—
76C2×#106C2×C411yescharacteristic
77C2×#106C2×C411yescharacteristic
78C2×#106C2×C411yescharacteristic
79C2×#106C2×C411yescharacteristic
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103C2×C411yescharacteristic
89C2×#103C2×C411yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×#103C2×C411yescharacteristic
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8C2×C411yescharacteristic
95C2×C22⋊C8C2×C411yescharacteristic
96C2×C22⋊C8D411yescharacteristic
97C2×C22⋊C8D411yescharacteristic
98C4×M16—22no—
99C4×M16—41no—
100C2×#97—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391C2×C411yescharacteristic
113#389C2×C411yescharacteristic
114#387—21no—
115#385—21no—
116#384C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118#382—21no—
119#380—21no—
120#380C2×C411yescharacteristic
121#380C2×C411yescharacteristic
122#380C2×C411yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—21no—
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—41no—
151#334—22no—
152Q8⋊C8—22no—
153Q8⋊C8—41no—
154#322—41no—
155C82—22no—
156C22×Q8C22×C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4111yescharacteristic
160C2×M16—24no—
161C2×M16—42no—
162C2×M16—42no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22⋊Q8—41no—
178C22⋊Q8—41no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183#117—81no—
184#117—81no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4111yescharacteristic
214C2×C42C2411yescharacteristic
215#106—22no—
216#106—41no—
217#106—41no—
218#106—41no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—81no—
229C4⋊C8—81no—
230C4⋊C8#4112yes—
231#103—22no—
232#103—41no—
233#103—41no—
234#103—41no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—41no—
238C22⋊C8—41no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241#97—42no—
242#97—42no—
243#97—42no—
244#97—81no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—42no—
257C4×C8—81no—
258C4×C8#4112yes—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268C24C4×D411yescharacteristic
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4C4×D411yescharacteristic
300C22×C4C4×D411yescharacteristic
301C22×C4C4×D411yescharacteristic
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4111yescharacteristic
305M16—48no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—22no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—22no—
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—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
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—
346C2×C8#13112yes—
347C2×C8#13712yes—
348C4⋊C4—21no—
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—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C22⋊C4—42no—
371C22⋊C4—81no—
372C22⋊C4—81no—
373C22⋊C4—81no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42C2×#4111yescharacteristic
389C42C2×#4111yescharacteristic
390C42H(5,𝔽2)11yescharacteristic
391C422−1+411yescharacteristic
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23C2×C4×D411yescharacteristic
400Q8—41no—
401Q8—41no—
402Q8—41no—
403Q8—41no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
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—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4#44312yes—
449C2×C4C4×#4111yescharacteristic
450C2×C4#52011yescharacteristic
451C2×C4#54611yescharacteristic
452C2×C4#54711yescharacteristic
453C2×C4#55111yescharacteristic
454C2×C4#55211yescharacteristic
455C2×C4C2×#13111yescharacteristic
456C2×C4#56811yescharacteristic
457C2×C4#56811yescharacteristic
458C2×C4#56811yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57511yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57711yescharacteristic
463C8—24no—
464C8—22no—
465C8—22no—
466C8—42no—
467C8—41no—
468C8—42no—
469C8—41no—
470C8—41no—
471C8—41no—
472C8—41no—
473C8—41no—
474C8—81no—
475C8—81no—
476C8—81no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22#235911yescharacteristic
485C22#293011yescharacteristic
486C22#293511yescharacteristic
487C22C4×#13711yescharacteristic
488C22#294911yescharacteristic
489C22C2×#44311yescharacteristic
490C22#295611yescharacteristic
491C4—21no—
492C4—21no—
493C4—22no—
494C4—41no—
495C4—41no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4#296711yescharacteristic
505C4#296811yescharacteristic
506C4#299311yescharacteristic
507C4#299511yescharacteristic
508C4#332611yescharacteristic
509C4#333111yescharacteristic
510C4#334411yescharacteristic
511C4#335011yescharacteristic
512C2—81no—
513C2#1794411yescharacteristic
514C2#1795111yescharacteristic
515C2#1807911yescharacteristic
516C2#1808511yescharacteristic
517C2#2290311yescharacteristic
518C2#2294511yescharacteristic
519C2#2294811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9600C2semidirect
5. #9553C2semidirect
6. #9548C2semidirect
7. #9539C2semidirect
8. #9535C2semidirect
10. #8850C2semidirect
12. #8283C2semidirect
13. #8213C2semidirect

Maximal subgroups

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

subgroupindex
#237562
#230542
#226262
#96002
#95532
#95482
#95392
#95352
#90772
#88502
#88312
#82832
#82132
#81462
#78962