Finite Group Databasebeta

← #219490

Subgroups — #219490

Order 512.

646 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21704C211yesmaximal, characteristic
2#17855C211yesmaximal, characteristic, complemented
3#17854C211yesmaximal, characteristic, complemented
4#9668C211yesmaximal, characteristic, complemented
5#9667C211yesmaximal, characteristic, complemented
6#9102C211yesmaximal, characteristic, complemented
7#9101C211yesmaximal, characteristic, complemented
8#9101C211yesmaximal, characteristic, complemented
9#9099C211yesmaximal, characteristic, complemented
10#9090C211yesmaximal, characteristic, complemented
11#9089C211yesmaximal, characteristic, complemented
12#9089C211yesmaximal, characteristic, complemented
13#9087C211yesmaximal, characteristic, complemented
14#8781C211yesmaximal, characteristic, complemented
15#8138C211yesmaximal, characteristic, complemented
16C2×#496C411yescharacteristic
17#2982C2211yescharacteristic
18#2972C2211yescharacteristic
19#2972C2211yescharacteristic
20#2972C2211yescharacteristic
21C2×#442C2211yescharacteristic
22C2×#440C2211yescharacteristic
23C2×C4×SD16C2211yescharacteristic
24C2×C4×D8C2211yescharacteristic
25#2328C2211yescharacteristic
26#2326C2211yescharacteristic
27#1973C411yescharacteristic
28#1972C411yescharacteristic
29#1970C411yescharacteristic
30#1968C411yescharacteristic
31#1958C411yescharacteristic
32#1957C411yescharacteristic
33#1847C411yescharacteristic
34#1726C2211yescharacteristic
35#1580—22no—
36SD16⋊C8—22no—
37C8×SD16—22no—
38C8×D8—22no—
39#1564C2211yescharacteristic
40#1564C2211yescharacteristic
41#1479C2211yescharacteristic
42#1479C2211yescharacteristic
43#1478C2211yescharacteristic
44#1478C2211yescharacteristic
45#1478C2211yescharacteristic
46#1478C2211yescharacteristic
47#1478C2211yescharacteristic
48#1478C2211yescharacteristic
49#1461C2211yescharacteristic
50#1460C2211yescharacteristic
51#1460C2211yescharacteristic
52#1460C2211yescharacteristic
53#1450C2211yescharacteristic
54#1450C2211yescharacteristic
55#1281C2211yescharacteristic
56#1281C2211yescharacteristic
57#1275C2211yescharacteristic
58#1275C2211yescharacteristic
59#1274C2211yescharacteristic
60#1274C2211yescharacteristic
61#1274C2211yescharacteristic
62#1274C2211yescharacteristic
63C22×SD16C2×C411yescharacteristic
64C22×SD16C2×C411yescharacteristic
65C22×D8C2×C411yescharacteristic
66C22×D8C2×C411yescharacteristic
67C2×#127C2×C411yescharacteristic
68C2×#124C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73#496C818yescomplemented
74C8×Q8—22no—
75C8×Q8—41no—
76#442—41no—
77#442—41no—
78#440—41no—
79#440—41no—
80C4×SD16—21no—
81C4×SD16—21no—
82C4×SD16—21no—
83C4×SD16—21no—
84C4×D8—21no—
85C4×D8—21no—
86C4×D8—21no—
87C4×D8—21no—
88C8×D4—22no—
89C8×D4—22no—
90C8×D4—22no—
91C8×D4—41no—
92C8×D4—41no—
93C8×D4—41no—
94#431—41no—
95#431C2311yescharacteristic
96#431C2311yescharacteristic
97#431C2311yescharacteristic
98#431C2311yescharacteristic
99#431C2311yescharacteristic
100#431C2311yescharacteristic
101#431C2311yescharacteristic
102#431C2311yescharacteristic
103C2×#107C2×C411yescharacteristic
104C2×#107C2×C411yescharacteristic
105C2×#106C2×C411yescharacteristic
106C2×#106C2×C411yescharacteristic
107C2×#103C2×C411yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#102C2×C411yescharacteristic
110C2×#102C2×C411yescharacteristic
111C2×#102C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×C22⋊C8—41no—
116C2×C22⋊C8—41no—
117C2×C22⋊C8—41no—
118C2×#97C2×C411yescharacteristic
119C2×#97C2×C411yescharacteristic
120C2×#97C2311yescharacteristic
121C2×C4×C8C2×C411yescharacteristic
122C2×C4×C8C2×C411yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#388—41no—
129#386—41no—
130#386—41no—
131#385—41no—
132#384C2×C411yescharacteristic
133#382C2×C411yescharacteristic
134#376C2311yescharacteristic
135#335—21no—
136#335—21no—
137#334—21no—
138#334—21no—
139Q8⋊C8—22no—
140Q8⋊C8—41no—
141D4⋊C8—22no—
142D4⋊C8—22no—
143D4⋊C8—22no—
144D4⋊C8—41no—
145D4⋊C8—41no—
146D4⋊C8—41no—
147#324—41no—
148#323—41no—
149#323—41no—
150#323—41no—
151#322—21no—
152#322D412yescomplemented
153C82—21no—
154C82D412yescomplemented
155C22×Q8C22×C411yescharacteristic
156C22×D4—41no—
157C22×D4C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C2×SD16—21no—
164C2×SD16—21no—
165C2×SD16—22no—
166C2×SD16—41no—
167C2×SD16—41no—
168C2×SD16C2×C814yescomplemented
169C2×SD16C2×C814yescomplemented
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—22no—
173C2×D8—41no—
174C2×D8—41no—
175C2×D8C2×C814yescomplemented
176C2×D8C2×C814yescomplemented
177C22×C8—42no—
178C22×C8—42no—
179C22×C8—42no—
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C22×C411yescharacteristic
184#127—41no—
185#127C2×C814yescomplemented
186#124—41no—
187#124C2×C814yescomplemented
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—21no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C42—21no—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C2411yescharacteristic
221#107—21no—
222#107—21no—
223#107—41no—
224#106—21no—
225#106—21no—
226#106—41no—
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—42no—
236#103—21no—
237#103—21no—
238#103—41no—
239#102—21no—
240#102—21no—
241#102—21no—
242#102—21no—
243#102—21no—
244#102—21no—
245#102—41no—
246#102—41no—
247#102—41no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—21no—
251C22⋊C8—21no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—21no—
255C22⋊C8—21no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259#97—22no—
260#97—21no—
261#97—21no—
262#97C2×C814yescomplemented
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—41no—
270C4×C8—41no—
271C4×C8—42no—
272C4×C8C2×D412yes—
273C4×C8C2×D412yes—
274C4×C8C2×D411yescharacteristic
275C4×C8C2×D411yescharacteristic
276C4×C8#4111yescharacteristic
277C4×C8#4111yescharacteristic
278#95—21no—
279#95—21no—
280#95—41no—
281#95—41no—
282C24—21no—
283C24—21no—
284C24—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8C22×C812yes—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—44no—
307C2×D4—161no—
308C2×D4—161no—
309C2×D4C22×C812yes—
310C2×D4C22×C812yes—
311C2×D4C22×C812yes—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—161no—
326C22×C4—161no—
327C22×C4—161no—
328C22×C4C23×C411yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331SD16—24no—
332SD16—24no—
333SD16—44no—
334D8—24no—
335D8—24no—
336D8—44no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—162no—
353C2×C8—162no—
354C2×C8—162no—
355C2×C8C4×D411yescharacteristic
356C2×C8C4×D411yescharacteristic
357C2×C8C4×D411yescharacteristic
358C2×C8C4×D411yescharacteristic
359C2×C8C22×C812yes—
360C2×C8C22×C812yes—
361C2×C8#13112yes—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—161no—
373C22⋊C4—44no—
374C22⋊C4—41no—
375C22⋊C4—41no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C22⋊C4—161no—
381C22⋊C4—161no—
382C22⋊C4—161no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—41no—
387C42—41no—
388C42C22×C812yes—
389C42C22×D411yescharacteristic
390C42C2×#4111yescharacteristic
391C42H(5,𝔽2)11yescharacteristic
392C42H(5,𝔽2)11yescharacteristic
393C23—22no—
394C23—22no—
395C23—22no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23#51811yescharacteristic
409Q8—22no—
410Q8—22no—
411Q8—41no—
412Q8—41no—
413D4—22no—
414D4—22no—
415D4—22no—
416D4—22no—
417D4—22no—
418D4—22no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—41no—
423D4—41no—
424D4—41no—
425D4—164no—
426C2×C4—22no—
427C2×C4—22no—
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—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—161no—
449C2×C4—161no—
450C2×C4—161no—
451C2×C4C2×C4×D411yescharacteristic
452C2×C4#51811yescharacteristic
453C2×C4#53511yescharacteristic
454C2×C4#53511yescharacteristic
455C2×C4#54011yescharacteristic
456C2×C4#54011yescharacteristic
457C2×C4C23×C811yescharacteristic
458C2×C4C2×#13111yescharacteristic
459C2×C4#56811yescharacteristic
460C2×C4#56811yescharacteristic
461C8—21no—
462C8—21no—
463C8—22no—
464C8—42no—
465C8—42no—
466C8—41no—
467C8—81no—
468C8—81no—
469C8—81no—
470C8—81no—
471C8—81no—
472C8—81no—
473C8C8×D412yes—
474C8C8×D412yes—
475C22—21no—
476C22—21no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—161no—
490C22—161no—
491C22—161no—
492C22#232811yescharacteristic
493C22#297211yescharacteristic
494C22#298111yescharacteristic
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—42no—
500C4—42no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4C2×C8×D411yescharacteristic
507C4#293211yescharacteristic
508C4#297211yescharacteristic
509C4#298111yescharacteristic
510C2—21no—
511C2—21no—
512C2—82no—
513C2—82no—
514C2—82no—
515C2#2226011yescharacteristic
516C2#2228411yescharacteristic
517C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
prime radicalC2×#496C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #17855C2semidirect
3. #17854C2semidirect
4. #9668C2semidirect
5. #9667C2semidirect
6. #9102C2semidirect
7. #9101C2semidirect
8. #9101C2semidirect
9. #9099C2semidirect
10. #9090C2semidirect
11. #9089C2semidirect
12. #9089C2semidirect
13. #9087C2semidirect
14. #8781C2semidirect
15. #8138C2semidirect
73. #496C8semidirect
152. #322D4semidirect
154. C82D4semidirect
168. C2×SD16C2×C8semidirect
169. C2×SD16C2×C8semidirect
175. C2×D8C2×C8semidirect
176. C2×D8C2×C8semidirect
185. #127C2×C8semidirect
187. #124C2×C8semidirect
262. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
#217042
#178552
#178542
#96682
#96672
#91022
#91012
#90992
#90902
#90892
#90872
#87812
#81382