Finite Group Databasebeta

← #217565

Subgroups — #217565

Order 512.

590 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21431C211yesmaximal, characteristic
2#20793C211yesmaximal, characteristic
3#20786C211yesmaximal, characteristic
4#9604C211yesmaximal, characteristic, complemented
5#9604C211yesmaximal, characteristic, complemented
6#9602C211yesmaximal, characteristic, complemented
7#9602C211yesmaximal, characteristic, complemented
8#8833C211yesmaximal, characteristic
9#8832C211yesmaximal, characteristic
10#8830C211yesmaximal, characteristic
11#8829C211yesmaximal, characteristic
12#8812C211yesmaximal, characteristic, complemented
13#8808C211yesmaximal, characteristic, complemented
14#8789C211yesmaximal, characteristic, complemented
15#8786C211yesmaximal, characteristic, complemented
16C2×#424—21no—
17C2×#424—21no—
18C2×#423C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C4×#117C2211yescharacteristic
21#1924C411yescharacteristic
22#1924C411yescharacteristic
23#1924C411yescharacteristic
24#1924C411yescharacteristic
25#1917C411yescharacteristic
26#1917C411yescharacteristic
27#1917C411yescharacteristic
28#1917C411yescharacteristic
29#1843—21no—
30#1843—21no—
31#1842C2211yescharacteristic
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1840—21no—
36#1831C2211yescharacteristic
37#1831C2211yescharacteristic
38#1818—21no—
39#1797C2211yescharacteristic
40#1797C2211yescharacteristic
41#1767C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45#1750—21no—
46C4×C22⋊C8C2211yescharacteristic
47C4×C22⋊C8C2211yescharacteristic
48#1746—21no—
49#1744C2211yescharacteristic
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59#1277C2211yescharacteristic
60#1277C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C42C2311yescharacteristic
72#424—44no—
73#424—44no—
74#423—41no—
75#423—41no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—41no—
87C2×C4⋊C8—41no—
88C2×C4⋊C8C2×C411yescharacteristic
89C2×C4⋊C8C2×C411yescharacteristic
90C2×C4⋊C8C2×C411yescharacteristic
91C2×C4⋊C8C2×C411yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8C2×C411yescharacteristic
103C2×C22⋊C8C2×C411yescharacteristic
104C2×C22⋊C8C2×C411yescharacteristic
105C2×C22⋊C8C2×C411yescharacteristic
106C2×#97—21no—
107C2×#97C2×C411yescharacteristic
108C2×#97C2×C411yescharacteristic
109C2×#97C2×C411yescharacteristic
110C2×#97C2×C411yescharacteristic
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C4×C4⋊C4C2×C411yescharacteristic
117C4×C4⋊C4C2×C411yescharacteristic
118C4×C4⋊C4Q811yescharacteristic
119C4×C4⋊C4Q811yescharacteristic
120C4×C22⋊C4C2×C411yescharacteristic
121C4×C22⋊C4C2×C411yescharacteristic
122C4×C22⋊C4Q811yescharacteristic
123C4×C22⋊C4Q811yescharacteristic
124#376D411yescharacteristic
125#376D411yescharacteristic
126C43C2311yescharacteristic
127C43C2311yescharacteristic
128#336—21no—
129#336—21no—
130#336—21no—
131#336—21no—
132#336—21no—
133#336—21no—
134#336—21no—
135#336—21no—
136#336C2×C411yescharacteristic
137#336C2×C411yescharacteristic
138#336C2×C411yescharacteristic
139#336C2×C411yescharacteristic
140#336C2×C411yescharacteristic
141#336C2×C411yescharacteristic
142#336C2×C411yescharacteristic
143#336C2×C411yescharacteristic
144#336C2×C411yescharacteristic
145#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#335—41no—
149#335—41no—
150#335—41no—
151#335—41no—
152#334—41no—
153#334—41no—
154#334—41no—
155#334—41no—
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×Q811yescharacteristic
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—42no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—42no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C22×C411yescharacteristic
176C22×C8C22×C411yescharacteristic
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185C2×C4⋊C4C4⋊C411yescharacteristic
186C2×C4⋊C4C4⋊C411yescharacteristic
187C2×C4⋊C4C4⋊C411yescharacteristic
188C2×C4⋊C4C4⋊C411yescharacteristic
189C2×C22⋊C4C4⋊C411yescharacteristic
190C2×C22⋊C4C4⋊C411yescharacteristic
191C2×C22⋊C4C4⋊C411yescharacteristic
192C2×C22⋊C4C4⋊C411yescharacteristic
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×Q811yescharacteristic
208C2×C42C2×Q811yescharacteristic
209C2×C42C2×Q811yescharacteristic
210C2×C42C2×Q811yescharacteristic
211C2×C42C2×Q811yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
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—
238C4⋊C8—42no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C22⋊C8—22no—
244C22⋊C8—22no—
245C22⋊C8—41no—
246C22⋊C8—41no—
247C22⋊C8—42no—
248C22⋊C8—42no—
249#97—22no—
250#97—22no—
251#97—41no—
252#97—41no—
253#97—42no—
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—42no—
263#95C4⋊C411yescharacteristic
264#95C4⋊C411yescharacteristic
265#95C4⋊C411yescharacteristic
266#95C4⋊C411yescharacteristic
267#95C4⋊C411yescharacteristic
268#95C4⋊C411yescharacteristic
269#95C4⋊C411yescharacteristic
270#95C4⋊C411yescharacteristic
271C24C2×C4⋊C411yescharacteristic
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—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—41no—
285C22×C4—41no—
286C22×C4C2×C4⋊C411yescharacteristic
287C22×C4C2×C4⋊C411yescharacteristic
288C22×C4C2×C4⋊C411yescharacteristic
289C22×C4C2×C4⋊C411yescharacteristic
290C22×C4C2×C4⋊C411yescharacteristic
291C22×C4C2×C4⋊C411yescharacteristic
292C22×C4C2×C4⋊C411yescharacteristic
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C2×C4⋊C411yescharacteristic
296C22×C4C2×C4⋊C411yescharacteristic
297C22×C4AGL(1,ℤ/8)12yes—
298C22×C4#13712yes—
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×Q811yescharacteristic
302M16—84no—
303C2×C8—21no—
304C2×C8—21no—
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—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—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—84no—
350C2×C8#13112yes—
351C2×C8#13112yes—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C22⋊C4—22no—
357C22⋊C4—22no—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42AGL(1,ℤ/8)12yes—
385C42#13712yes—
386C42H(5,𝔽2)11yescharacteristic
387C42H(5,𝔽2)11yescharacteristic
388C422−1+411yescharacteristic
389C422−1+411yescharacteristic
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23#42812yes—
397C23C22×C4⋊C411yescharacteristic
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
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—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4#42812yes—
441C2×C4#42812yes—
442C2×C4#42812yes—
443C2×C4#52011yescharacteristic
444C2×C4#52011yescharacteristic
445C2×C4#53511yescharacteristic
446C2×C4#53611yescharacteristic
447C2×C4#54411yescharacteristic
448C2×C4#54411yescharacteristic
449C2×C4C2×#13111yescharacteristic
450C2×C4C2×#13111yescharacteristic
451C2×C4#56811yescharacteristic
452C2×C4#56811yescharacteristic
453C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
454C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
455C2×C4C2×#13711yescharacteristic
456C2×C4C2×#13711yescharacteristic
457C8—22no—
458C8—22no—
459C8—41no—
460C8—41no—
461C8—41no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—81no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8—81no—
470C8—81no—
471C8—81no—
472C8—81no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22#231711yescharacteristic
481C22#290611yescharacteristic
482C22#290611yescharacteristic
483C22C2×#42811yescharacteristic
484C22C2×#42811yescharacteristic
485C22#291611yescharacteristic
486C22#291611yescharacteristic
487C4—22no—
488C4—22no—
489C4—22no—
490C4—21no—
491C4—21no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4#299211yescharacteristic
501C4#299311yescharacteristic
502C4#299411yescharacteristic
503C4#299511yescharacteristic
504C4#321711yescharacteristic
505C4#321811yescharacteristic
506C4#323811yescharacteristic
507C4#323911yescharacteristic
508C2—42no—
509C2#1783011yescharacteristic
510C2#1783011yescharacteristic
511C2#1783011yescharacteristic
512C2#1783211yescharacteristic
513C2#2198011yescharacteristic
514C2#2198711yescharacteristic
515C2#2199511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C4⋊C4
prime radicalC24C2×C4⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9604C2semidirect
5. #9604C2semidirect
6. #9602C2semidirect
7. #9602C2semidirect
12. #8812C2semidirect
13. #8808C2semidirect
14. #8789C2semidirect
15. #8786C2semidirect

Maximal subgroups

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

subgroupindex
#214312
#207932
#207862
#96042
#96022
#88332
#88322
#88302
#88292
#88122
#88082
#87892
#87862