Finite Group Databasebeta

← #217454

Subgroups — #217454

Order 512.

602 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21926C211yesmaximal, characteristic
2#21922C211yesmaximal, characteristic
3C4×#424C211yesmaximal, characteristic
4#9606C211yesmaximal, characteristic, complemented
5#9606C211yesmaximal, characteristic, complemented
6#9605C211yesmaximal, characteristic, complemented
7#9604C211yesmaximal, characteristic, complemented
8#9603C211yesmaximal, characteristic, complemented
9#9603C211yesmaximal, characteristic, complemented
10#9602C211yesmaximal, characteristic, complemented
11#9601C211yesmaximal, characteristic, complemented
12#8597C211yesmaximal, characteristic
13#8597C211yesmaximal, characteristic
14#8596C211yesmaximal, characteristic
15#8595C211yesmaximal, characteristic
16C2×#424C411yescharacteristic
17C2×#424C411yescharacteristic
18C2×C4×M16C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C4×#117C2211yescharacteristic
21#1924—21no—
22#1924—21no—
23#1923—21no—
24#1923—21no—
25#1917—21no—
26#1917—21no—
27#1916—21no—
28#1916—21no—
29#1842C2211yescharacteristic
30#1842C2211yescharacteristic
31#1831C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2211yescharacteristic
34#1797C2211yescharacteristic
35#1797C2211yescharacteristic
36#1797C2211yescharacteristic
37#1768C411yescharacteristic
38#1768C411yescharacteristic
39#1768C411yescharacteristic
40#1768C411yescharacteristic
41#1767C2211yescharacteristic
42#1767C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49#1729C411yescharacteristic
50#1729C411yescharacteristic
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×M16C2×C411yescharacteristic
68C22×M16C2×C411yescharacteristic
69C23×C8C2×C411yescharacteristic
70C23×C8C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C22×C42C2311yescharacteristic
74#424—24no—
75#424—24no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2×C411yescharacteristic
81C2×C4⋊C8C2×C411yescharacteristic
82C2×C4⋊C8C2×C411yescharacteristic
83C2×C4⋊C8C2×C411yescharacteristic
84C2×C4⋊C8C2×C411yescharacteristic
85C2×C4⋊C8C2×C411yescharacteristic
86C2×C4⋊C8C2×C411yescharacteristic
87C2×C4⋊C8C2×C411yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C4×M16—41no—
101C4×M16—41no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97C2×C411yescharacteristic
105C2×#97C2×C411yescharacteristic
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—41no—
109C2×C4×C8—41no—
110C2×C4×C8C2×C411yescharacteristic
111C2×C4×C8C2×C411yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4Q811yescharacteristic
119C4×C4⋊C4Q811yescharacteristic
120C4×C22⋊C4D411yescharacteristic
121C4×C22⋊C4D411yescharacteristic
122C4×C22⋊C4Q811yescharacteristic
123C4×C22⋊C4Q811yescharacteristic
124#376C2×C411yescharacteristic
125#376C2×C411yescharacteristic
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#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
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—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—41no—
164C2×M16—41no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—41no—
178C22×C8—41no—
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C22×C411yescharacteristic
184C22×C8C22×C411yescharacteristic
185C22×C8C22×C411yescharacteristic
186C22×C8C22×C411yescharacteristic
187#117—22no—
188#117—22no—
189#117—22no—
190#117—22no—
191C2×C4⋊C4C4⋊C411yescharacteristic
192C2×C4⋊C4C4⋊C411yescharacteristic
193C2×C4⋊C4C4⋊C411yescharacteristic
194C2×C4⋊C4C4⋊C411yescharacteristic
195C2×C22⋊C4C4⋊C411yescharacteristic
196C2×C22⋊C4C4⋊C411yescharacteristic
197C2×C22⋊C4C4⋊C411yescharacteristic
198C2×C22⋊C4C4⋊C411yescharacteristic
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×Q811yescharacteristic
214C2×C42C2×Q811yescharacteristic
215C2×C42C2×Q811yescharacteristic
216C2×C42C2×Q811yescharacteristic
217C2×C42C2×Q811yescharacteristic
218C2×C42C2411yescharacteristic
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
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—81no—
242C4⋊C8—81no—
243C4⋊C8—81no—
244C4⋊C8—81no—
245C4⋊C8M1614yes—
246C4⋊C8M1614yes—
247C22⋊C8—81no—
248C22⋊C8—81no—
249C22⋊C8—81no—
250C22⋊C8—81no—
251#97—41no—
252#97—41no—
253#97—81no—
254#97—81no—
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—41no—
264C4×C8—41no—
265C4×C8—81no—
266C4×C8—81no—
267#95C4⋊C411yescharacteristic
268#95C4⋊C411yescharacteristic
269#95C4⋊C411yescharacteristic
270#95C4⋊C411yescharacteristic
271#95C4⋊C411yescharacteristic
272#95C4⋊C411yescharacteristic
273#95C4⋊C411yescharacteristic
274#95C4⋊C411yescharacteristic
275C24C2×C4⋊C411yescharacteristic
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—41no—
289C22×C4—41no—
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×C4C2×C4⋊C411yescharacteristic
298C22×C4C2×C4⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300C22×C4C2×C4⋊C411yescharacteristic
301C22×C4#13112yes—
302C22×C4#13112yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×Q811yescharacteristic
306M16—44no—
307M16—44no—
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—44no—
329C2×C8—44no—
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—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8C2×M1612yes—
355C2×C8C2×M1612yes—
356C2×C8C2×M1612yes—
357C2×C8C2×M1612yes—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
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—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42C2×M1612yes—
391C42C2×M1612yes—
392C42H(5,𝔽2)11yescharacteristic
393C42H(5,𝔽2)11yescharacteristic
394C422−1+411yescharacteristic
395C422−1+411yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23#42412yes—
403C23C22×C4⋊C411yescharacteristic
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—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
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—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4#42312yes—
447C2×C4#42312yes—
448C2×C4#42412yes—
449C2×C4#51811yescharacteristic
450C2×C4#51911yescharacteristic
451C2×C4#53711yescharacteristic
452C2×C4#53711yescharacteristic
453C2×C4#54411yescharacteristic
454C2×C4#54411yescharacteristic
455C2×C4C22×M1611yescharacteristic
456C2×C4C22×M1611yescharacteristic
457C2×C4C2×#13111yescharacteristic
458C2×C4C2×#13111yescharacteristic
459C2×C4#57611yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57811yescharacteristic
463C8—22no—
464C8—22no—
465C8—22no—
466C8—22no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—41no—
471C8—81no—
472C8—81no—
473C8—81no—
474C8—81no—
475C8—81no—
476C8—81no—
477C8—81no—
478C8—81no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22#231611yescharacteristic
487C22C2×#42311yescharacteristic
488C22C2×#42411yescharacteristic
489C22#291211yescharacteristic
490C22#291311yescharacteristic
491C22#291611yescharacteristic
492C22#291611yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—21no—
497C4—21no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4#297411yescharacteristic
507C4#297411yescharacteristic
508C4#298411yescharacteristic
509C4#298411yescharacteristic
510C4#325711yescharacteristic
511C4#325711yescharacteristic
512C4#325811yescharacteristic
513C4#325811yescharacteristic
514C2—42no—
515C2#1779811yescharacteristic
516C2#1779911yescharacteristic
517C2#1781111yescharacteristic
518C2#1781211yescharacteristic
519C2#2195311yescharacteristic
520C2#2196611yescharacteristic
521C2#2197111yescharacteristic
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. #9606C2semidirect
5. #9606C2semidirect
6. #9605C2semidirect
7. #9604C2semidirect
8. #9603C2semidirect
9. #9603C2semidirect
10. #9602C2semidirect
11. #9601C2semidirect

Maximal subgroups

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

subgroupindex
#219262
#219222
C4×#4242
#96062
#96052
#96042
#96032
#96022
#96012
#85972
#85962
#85952