Finite Group Databasebeta

← #203969

Subgroups — #203969

Order 512.

569 conjugacy classes of subgroups.

The subgroup lattice has 477 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21993C211yesmaximal, characteristic
2#20537C211yesmaximal, characteristic
3#17822C211yesmaximal, characteristic
4#17816C211yesmaximal, characteristic
5#17803C211yesmaximal, characteristic
6#9632C211yesmaximal, characteristic, complemented
7#9632C211yesmaximal, characteristic, complemented
8#9631C211yesmaximal, characteristic, complemented
9#9630C211yesmaximal, characteristic, complemented
10#9629C211yesmaximal, characteristic, complemented
11#9629C211yesmaximal, characteristic, complemented
12#9628C211yesmaximal, characteristic, complemented
13#9627C211yesmaximal, characteristic, complemented
14#8095C211yesmaximal, characteristic
15#8094C211yesmaximal, characteristic
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22#2285C2211yescharacteristic
23#1950—21no—
24#1950—21no—
25#1855—21no—
26#1855—21no—
27#1848C2211yescharacteristic
28#1848C2211yescharacteristic
29#1837C2211yescharacteristic
30#1837C2211yescharacteristic
31#1771—21no—
32#1771—21no—
33#1770C2211yescharacteristic
34#1770C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36C8×C4⋊C4C2211yescharacteristic
37#1750—21no—
38#1750—21no—
39#1573—22no—
40#1572—22no—
41#1572C412yescomplemented
42#1572C412yescomplemented
43#1571C412yescomplemented
44#1571C2211yescharacteristic
45#1571C2211yescharacteristic
46#1571C2211yescharacteristic
47#1571C2211yescharacteristic
48#1570C412yescomplemented
49#1570C2211yescharacteristic
50#1570C2211yescharacteristic
51#1570C2211yescharacteristic
52#1570C2211yescharacteristic
53#1569—22no—
54#1568—22no—
55#1564C2211yescharacteristic
56#1564C2211yescharacteristic
57#1564C2211yescharacteristic
58#1564C2211yescharacteristic
59#1303C2211yescharacteristic
60#1303C2211yescharacteristic
61#1303C2211yescharacteristic
62#1303C2211yescharacteristic
63#1303C2211yescharacteristic
64#1303C2211yescharacteristic
65#1303C2211yescharacteristic
66#1303C2211yescharacteristic
67C22×M16—21no—
68C2×#117—21no—
69C22×C42C2311yescharacteristic
70#432—21no—
71#432—21no—
72#432—21no—
73#432—21no—
74#432—21no—
75#432C2×C412yes—
76#432D412yes—
77#432D412yes—
78#431—21no—
79#431—21no—
80#431—21no—
81#431—21no—
82#431—21no—
83#431C2×C412yes—
84#431C2×C411yescharacteristic
85#431C2×C411yescharacteristic
86#431C2×C411yescharacteristic
87#431C2×C411yescharacteristic
88#431Q811yescharacteristic
89#431Q811yescharacteristic
90#431Q811yescharacteristic
91#431Q811yescharacteristic
92#431C2311yescharacteristic
93#431C2311yescharacteristic
94#431C2311yescharacteristic
95#431C2311yescharacteristic
96C2×#108—21no—
97C2×#108—21no—
98C2×#108—21no—
99C2×#108—21no—
100#423—21no—
101#423—21no—
102#423—21no—
103#423C2×C412yes—
104#423C2311yescharacteristic
105#423C2311yescharacteristic
106#423C2311yescharacteristic
107#423C2311yescharacteristic
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8C2×C412yes—
111C2×C4⋊C8D411yescharacteristic
112C2×C4⋊C8D411yescharacteristic
113C2×C4⋊C8C2311yescharacteristic
114C2×C4⋊C8C2311yescharacteristic
115#407—21no—
116#407—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×C22⋊C8D411yescharacteristic
120C2×C22⋊C8D411yescharacteristic
121C4×M16—21no—
122C4×M16—21no—
123C2×#97—21no—
124C2×#97—21no—
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8—21no—
128C2×C4×C8—21no—
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#384—21no—
132#382Q812yes—
133#380—21no—
134#380Q812yes—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138#341—21no—
139#341—21no—
140#341—21no—
141#341—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#335—22no—
147#335—22no—
148#335—22no—
149#335—22no—
150#335C2×C412yes—
151#335C2×C412yes—
152#335C2×C412yes—
153#335C2×C412yes—
154#334—22no—
155#334—22no—
156#334—22no—
157#334—22no—
158#334C2×C412yes—
159#334C2×C412yes—
160#334C2×C412yes—
161#334C2×C412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×Q811yescharacteristic
164C23×C4#4111yescharacteristic
165C2×M16—22no—
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—41no—
171C2×M16—81no—
172C2×M16—81no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—41no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181#117—81no—
182#117—81no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×Q811yescharacteristic
196C2×C42C2×Q811yescharacteristic
197C2×C42C2×Q811yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C2×C42C2411yescharacteristic
202#108—81no—
203#108—81no—
204#108—81no—
205#108—81no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—41no—
219C4⋊C8—41no—
220C4⋊C8C4⋊C411yescharacteristic
221C4⋊C8C4⋊C411yescharacteristic
222C4⋊C8C4⋊C411yescharacteristic
223C4⋊C8C4⋊C411yescharacteristic
224C4⋊C8C4⋊C411yescharacteristic
225C4⋊C8C4⋊C411yescharacteristic
226C4⋊C8C4⋊C411yescharacteristic
227C4⋊C8C4⋊C411yescharacteristic
228C4⋊C8C22×C412yes—
229C4⋊C8C22×C412yes—
230C4⋊C8C22×C412yes—
231C4⋊C8C22×C412yes—
232C4⋊C8C2×D412yes—
233C4⋊C8C2×D412yes—
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—21no—
239C22⋊C8—21no—
240C22⋊C8—21no—
241C22⋊C8—21no—
242C22⋊C8—41no—
243C22⋊C8—41no—
244C22⋊C8C4⋊C411yescharacteristic
245C22⋊C8C4⋊C411yescharacteristic
246C22⋊C8C4⋊C411yescharacteristic
247C22⋊C8C4⋊C411yescharacteristic
248C22⋊C8C4⋊C411yescharacteristic
249C22⋊C8C4⋊C411yescharacteristic
250C22⋊C8C4⋊C411yescharacteristic
251C22⋊C8C4⋊C411yescharacteristic
252#97—22no—
253#97—22no—
254#97—22no—
255#97—41no—
256#97—81no—
257#97—81no—
258C4×C8—22no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—41no—
264C4×C8—81no—
265C4×C8—81no—
266C4×C8C22×C411yescharacteristic
267C4×C8C22×C411yescharacteristic
268C4×C8C22×C411yescharacteristic
269C4×C8C22×C411yescharacteristic
270C4×C8C2×Q811yescharacteristic
271C4×C8C2×Q811yescharacteristic
272C4×C8C2×Q811yescharacteristic
273C4×C8C2×Q811yescharacteristic
274#95—22no—
275#95C2×Q812yes—
276#95C2×Q812yes—
277C24C22⋊Q811yescharacteristic
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—42no—
285C22×C4—81no—
286C22×C4C2×C4⋊C411yescharacteristic
287C22×C4C2×C4⋊C411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C4×D411yescharacteristic
290C22×C4C4×Q811yescharacteristic
291C22×C4C4×Q811yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4C22⋊Q811yescharacteristic
295C22×C4C22⋊Q811yescharacteristic
296C22×C4C22⋊Q811yescharacteristic
297C22×C4#12511yescharacteristic
298C22×C4#12511yescharacteristic
299C22×C4C4⋊Q811yescharacteristic
300C22×C4C4⋊Q811yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×Q811yescharacteristic
303C22×C4C2×#4111yescharacteristic
304M16—41no—
305M16—41no—
306M16—41no—
307M16—41no—
308M16—41no—
309M16—41no—
310M16—41no—
311M16—41no—
312M16—82no—
313M16—82no—
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—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
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—82no—
337C2×C8—82no—
338C2×C8C2×C4⋊C411yescharacteristic
339C2×C8C2×C4⋊C411yescharacteristic
340C2×C8C2×C4⋊C411yescharacteristic
341C2×C8C2×C4⋊C411yescharacteristic
342C2×C8C2×C4⋊C411yescharacteristic
343C2×C8C2×C4⋊C411yescharacteristic
344C2×C8C2×C4⋊C411yescharacteristic
345C2×C8C2×C4⋊C411yescharacteristic
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C22⋊C4—82no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42C2×C4⋊C411yescharacteristic
356C42C2×C4⋊C411yescharacteristic
357C42C4×D411yescharacteristic
358C42C4×D411yescharacteristic
359C42C4×Q811yescharacteristic
360C42C4×Q811yescharacteristic
361C42#12111yescharacteristic
362C42#12111yescharacteristic
363C42#12511yescharacteristic
364C42#12511yescharacteristic
365C42C4⋊Q811yescharacteristic
366C42C4⋊Q811yescharacteristic
367C42C23×C411yescharacteristic
368C42C22×D411yescharacteristic
369C42C22×Q811yescharacteristic
370C42C2×#4111yescharacteristic
371C23—21no—
372C23—21no—
373C23—41no—
374C23—41no—
375C23#38911yescharacteristic
376C23#38911yescharacteristic
377C23C2×C22⋊Q811yescharacteristic
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4#38911yescharacteristic
407C2×C4#38911yescharacteristic
408C2×C4#38911yescharacteristic
409C2×C4#38911yescharacteristic
410C2×C4#38911yescharacteristic
411C2×C4#38911yescharacteristic
412C2×C4C22×C4⋊C411yescharacteristic
413C2×C4C2×C4×D411yescharacteristic
414C2×C4C2×C4×Q811yescharacteristic
415C2×C4C2×#12111yescharacteristic
416C2×C4C2×#12511yescharacteristic
417C2×C4C2×C4⋊Q811yescharacteristic
418C2×C4#57611yescharacteristic
419C2×C4#57711yescharacteristic
420C2×C4#57711yescharacteristic
421C2×C4#57811yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—21no—
426C8—21no—
427C8—21no—
428C8—21no—
429C8—21no—
430C8—42no—
431C8—42no—
432C8—42no—
433C8—42no—
434C8—42no—
435C8—42no—
436C8—81no—
437C8—81no—
438C8—81no—
439C8—81no—
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—81no—
449C22—81no—
450C22C2×#38911yescharacteristic
451C22#308111yescharacteristic
452C22#308211yescharacteristic
453C4—21no—
454C4—21no—
455C4—21no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—21no—
460C4—21no—
461C4—21no—
462C4—82no—
463C4—82no—
464C4#291211yescharacteristic
465C4#291311yescharacteristic
466C4#295712yes—
467C4#304511yescharacteristic
468C4#304611yescharacteristic
469C2—21no—
470C2—21no—
471C2—41no—
472C2—41no—
473C2#2116311yescharacteristic
474C2#2117511yescharacteristic
475C2#2118411yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#389
prime radicalC24C22⋊Q8
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9632C2semidirect
7. #9632C2semidirect
8. #9631C2semidirect
9. #9630C2semidirect
10. #9629C2semidirect
11. #9629C2semidirect
12. #9628C2semidirect
13. #9627C2semidirect
41. #1572C4semidirect
42. #1572C4semidirect
43. #1571C4semidirect
48. #1570C4semidirect

Maximal subgroups

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

subgroupindex
#219932
#205372
#178222
#178162
#178032
#96322
#96312
#96302
#96292
#96282
#96272
#80952
#80942