Finite Group Databasebeta

← #483229

Subgroups — #483229

Order 512.

694 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60715C211yesmaximal, characteristic
2#33423C211yesmaximal, characteristic
3#17799C211yesmaximal, characteristic
4#11988C211yesmaximal, characteristic, complemented
5#11987C211yesmaximal, characteristic, complemented
6#11984C211yesmaximal, characteristic, complemented
7#11983C211yesmaximal, characteristic, complemented
8#11976C211yesmaximal, characteristic, complemented
9#11973C211yesmaximal, characteristic, complemented
10#11965C211yesmaximal, characteristic, complemented
11#11962C211yesmaximal, characteristic, complemented
12#11954C211yesmaximal, characteristic, complemented
13#11953C211yesmaximal, characteristic, complemented
14#11950C211yesmaximal, characteristic, complemented
15#11949C211yesmaximal, characteristic, complemented
16#3509—21no—
17#3185—21no—
18#3185—21no—
19#3177—21no—
20#3177—21no—
21#3177C412yes—
22#3177C412yes—
23#3170C2211yescharacteristic
24#3170C2211yescharacteristic
25#3170C2211yescharacteristic
26#3170C2211yescharacteristic
27#3160C2211yescharacteristic
28C2×#494C2211yescharacteristic
29#3139—21no—
30#3059—21no—
31#3059C412yes—
32#2989—21no—
33#2950—21no—
34#2950C412yes—
35C4×#131C2211yescharacteristic
36#2164—21no—
37#2164—21no—
38#2163C2211yescharacteristic
39#2163C2211yescharacteristic
40#2162C2211yescharacteristic
41#2162C2211yescharacteristic
42#2150C2211yescharacteristic
43C2×C4⋊C16C2211yescharacteristic
44#2113—21no—
45#2108—21no—
46#1569—21no—
47#1569—21no—
48#1568C2211yescharacteristic
49#1568C2211yescharacteristic
50#1567C2211yescharacteristic
51#1567C2211yescharacteristic
52#1565—21no—
53#1564C2211yescharacteristic
54C2×#335C2211yescharacteristic
55#1365C2211yescharacteristic
56#1365C2211yescharacteristic
57#1365C2211yescharacteristic
58#1365C2211yescharacteristic
59#1364C2211yescharacteristic
60#1364C2211yescharacteristic
61#1364C2211yescharacteristic
62#1364C2211yescharacteristic
63#1362C2211yescharacteristic
64#1362C2211yescharacteristic
65#1362C2211yescharacteristic
66#1362C2211yescharacteristic
67#1362C2211yescharacteristic
68#1362C2211yescharacteristic
69#1362C2211yescharacteristic
70#1362C2211yescharacteristic
71C22×Q16—21no—
72C22×Q16C2×C412yes—
73C2×#131C2×C411yescharacteristic
74C2×#131D411yescharacteristic
75#557—41no—
76#557—41no—
77#556—41no—
78#556—41no—
79#552—41no—
80#552—41no—
81#552—41no—
82#552—41no—
83#541—22no—
84#541—22no—
85#536—22no—
86#533—21no—
87#533—21no—
88#533—21no—
89#533—21no—
90#533—21no—
91C2×C4⋊Q8—21no—
92#519—22no—
93C4×#41D411yescharacteristic
94C2×M32—21no—
95C22×C16—21no—
96#500C2311yescharacteristic
97#500C2311yescharacteristic
98#499—21no—
99#499C2×C412yes—
100#498—21no—
101#494—21no—
102#494—21no—
103#494C2×C412yes—
104#494C2311yescharacteristic
105#494C2311yescharacteristic
106#491—22no—
107#491—22no—
108#487—21no—
109#487—21no—
110#487—21no—
111#487—21no—
112#470—22no—
113#470—22no—
114#470—22no—
115#470—22no—
116#467—21no—
117#467—21no—
118#467—21no—
119#467—21no—
120#467C2×C412yes—
121#467C2×C412yes—
122#467C2×C412yes—
123#467C2×C412yes—
124#446—41no—
125#441—22no—
126#441—22no—
127#441—22no—
128#441—22no—
129C4×Q16—21no—
130C4×Q16—21no—
131C4×Q16—21no—
132C4×Q16—21no—
133C4×Q16C2×C412yes—
134C4×Q16C2×C412yes—
135C4×Q16C2×C412yes—
136C4×Q16C2×C412yes—
137#436—41no—
138#435—41no—
139#435—41no—
140#432—21no—
141#432—21no—
142#431—21no—
143#427—21no—
144#427C2×C412yes—
145#423—21no—
146#423—21no—
147C2×C4⋊C8—21no—
148#419—22no—
149#419—22no—
150#405C2×C411yescharacteristic
151#405C2×C411yescharacteristic
152#405C2×C411yescharacteristic
153#405D411yescharacteristic
154#405D411yescharacteristic
155#405D411yescharacteristic
156C4×M16D411yescharacteristic
157C4×M16D411yescharacteristic
158C4×M16D411yescharacteristic
159C2×C4×C8C2311yescharacteristic
160C2×C4×C8C2311yescharacteristic
161C2×C4×C8C2311yescharacteristic
162C4⋊C16—21no—
163C4⋊C16—21no—
164C4⋊C16C2311yescharacteristic
165C4⋊C16C2311yescharacteristic
166C4⋊C16C2311yescharacteristic
167C4⋊C16C2311yescharacteristic
168C22⋊C16—21no—
169C22⋊C16—21no—
170C22⋊C16—21no—
171C22⋊C16—21no—
172#346—21no—
173#346—21no—
174C4×C16—21no—
175C4×C16—21no—
176#335—21no—
177#335—21no—
178#335—21no—
179#335C2311yescharacteristic
180#335C2311yescharacteristic
181#335C2311yescharacteristic
182#335C2311yescharacteristic
183#334—21no—
184#334—21no—
185#334—21no—
186C2×#41C22⋊C411yescharacteristic
187C22×Q8—22no—
188C2×Q16—22no—
189C2×Q16—22no—
190C2×Q16—21no—
191C2×Q16—21no—
192C2×Q16—42no—
193C2×Q16—42no—
194C2×Q16—44no—
195C2×Q16C22×C412yes—
196C2×Q16C22×C412yes—
197#131—41no—
198#131—41no—
199#131—42no—
200C2×M16—41no—
201C2×M16—41no—
202C2×M16C22⋊C411yescharacteristic
203C2×M16C22⋊C411yescharacteristic
204C2×M16C22⋊C411yescharacteristic
205C2×M16C22⋊C411yescharacteristic
206C2×M16C22⋊C411yescharacteristic
207C2×M16C22⋊C411yescharacteristic
208C22×C8—41no—
209C22×C8C22×C411yescharacteristic
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C2×D411yescharacteristic
213C22×C8C2×D411yescharacteristic
214C22×C8C2×D411yescharacteristic
215C4⋊Q8—22no—
216C4⋊Q8—21no—
217C4⋊Q8—21no—
218C4⋊Q8—21no—
219C4⋊Q8—21no—
220C4⋊Q8—42no—
221C4⋊Q8—41no—
222C4⋊Q8—41no—
223#126—42no—
224#126—42no—
225#126—42no—
226#126—42no—
227#125—22no—
228#125—42no—
229#125—42no—
230#125—42no—
231#125—42no—
232#125—41no—
233#125—41no—
234#125—41no—
235#125—41no—
236#124—42no—
237#123—42no—
238#123—42no—
239C22⋊Q8—22no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—41no—
244C22⋊Q8—41no—
245C22⋊Q8—41no—
246C22⋊Q8—41no—
247C22⋊Q8—41no—
248C22⋊Q8—41no—
249C4×Q8—22no—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×Q8—22no—
253C4×Q8—42no—
254C4×Q8—42no—
255C4×Q8—41no—
256C4×Q8—41no—
257C4×Q8—41no—
258C4×Q8—41no—
259C4×Q8—41no—
260C4×Q8—41no—
261C4×Q8D811yescharacteristic
262C4×Q8SD1611yescharacteristic
263C4×D4D811yescharacteristic
264C4×D4D811yescharacteristic
265C4×D4D811yescharacteristic
266C4×D4SD1611yescharacteristic
267C4×D4SD1611yescharacteristic
268C4×D4SD1611yescharacteristic
269#117—22no—
270#117—41no—
271#117—41no—
272#117—41no—
273#117—41no—
274#117—42no—
275#117—42no—
276#117C22⋊C411yescharacteristic
277#117C22⋊C411yescharacteristic
278#117C22⋊C411yescharacteristic
279C2×C4⋊C4—41no—
280C2×C4⋊C4—41no—
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284M32—42no—
285C2×C16—21no—
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—21no—
289C2×C16—42no—
290#107—21no—
291#107—21no—
292#107C22×C412yes—
293#107C22×C412yes—
294#106—22no—
295#106—22no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—21no—
299C4⋊C8—21no—
300C4⋊C8—41no—
301C4⋊C8—41no—
302C4⋊C8—41no—
303#103—22no—
304#103—22no—
305#103—22no—
306#103—22no—
307#103—22no—
308#103—22no—
309#103—22no—
310#103—22no—
311C22⋊C8—41no—
312C22⋊C8—41no—
313C22⋊C8—41no—
314#97—41no—
315#97—41no—
316#97C22⋊C411yescharacteristic
317#97C22⋊C411yescharacteristic
318#97C22⋊C411yescharacteristic
319#97C22⋊C411yescharacteristic
320#97C22⋊C411yescharacteristic
321#97C22⋊C411yescharacteristic
322C4×C8—41no—
323C4×C8C22×C411yescharacteristic
324C4×C8C22×C411yescharacteristic
325C4×C8C22×C411yescharacteristic
326C4×C8C2×D411yescharacteristic
327C4×C8C2×D411yescharacteristic
328C4×C8C2×D411yescharacteristic
329C4×C8C2×D411yescharacteristic
330C4×C8C2×D411yescharacteristic
331C4×C8C2×D411yescharacteristic
332C4×C8C2411yescharacteristic
333#41—41no—
334#41—41no—
335C2×Q8—22no—
336C2×Q8—22no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8—42no—
342C2×Q8#10211yescharacteristic
343C2×D4#10211yescharacteristic
344C2×D4#10211yescharacteristic
345C2×D4#10211yescharacteristic
346C22×C4—42no—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×D811yescharacteristic
351C22×C4C2×D811yescharacteristic
352C22×C4C2×D811yescharacteristic
353C22×C4C2×SD1611yescharacteristic
354C22×C4C2×SD1611yescharacteristic
355C22×C4C2×SD1611yescharacteristic
356Q16—42no—
357Q16—42no—
358Q16—44no—
359Q16—41no—
360Q16—41no—
361Q16—41no—
362Q16—41no—
363Q16—44no—
364M16—22no—
365M16—22no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—161no—
371M16—161no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—22no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—161no—
386C2×C8C2×C22⋊C411yescharacteristic
387C2×C8C2×C22⋊C411yescharacteristic
388C2×C8C2×C22⋊C411yescharacteristic
389C2×C8C2×C22⋊C411yescharacteristic
390C2×C8C2×C22⋊C411yescharacteristic
391C2×C8C2×C22⋊C411yescharacteristic
392C2×C8C23×C411yescharacteristic
393C2×C8C22×D411yescharacteristic
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4#10211yescharacteristic
415C4⋊C4#10211yescharacteristic
416C4⋊C4#10211yescharacteristic
417C4⋊C4#10211yescharacteristic
418C4⋊C4#10211yescharacteristic
419C4⋊C4#10211yescharacteristic
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4#10211yescharacteristic
426C22⋊C4#10211yescharacteristic
427C22⋊C4#10211yescharacteristic
428C22⋊C4#10211yescharacteristic
429C22⋊C4#10211yescharacteristic
430C22⋊C4#10211yescharacteristic
431C42—42no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—42no—
436C42C2×C22⋊C411yescharacteristic
437C42C2×C22⋊C411yescharacteristic
438C42C2×C22⋊C411yescharacteristic
439C42C2×D811yescharacteristic
440C42C2×D811yescharacteristic
441C42C2×D811yescharacteristic
442C42C2×SD1611yescharacteristic
443C42C2×SD1611yescharacteristic
444C42C2×SD1611yescharacteristic
445C42C22×D411yescharacteristic
446C16—41no—
447C16—41no—
448C16—41no—
449C16—41no—
450C23C2×#10211yescharacteristic
451C23C2×#10211yescharacteristic
452C23C2×#10211yescharacteristic
453Q8—41no—
454Q8—42no—
455Q8—42no—
456Q8—42no—
457Q8—42no—
458Q8—82no—
459Q8—84no—
460Q8—81no—
461Q8—81no—
462D4—41no—
463D4—41no—
464D4—41no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—84no—
485C2×C4C2×#10211yescharacteristic
486C2×C4C2×#10211yescharacteristic
487C2×C4C2×#10211yescharacteristic
488C2×C4C2×#10211yescharacteristic
489C2×C4C2×#10211yescharacteristic
490C2×C4C2×#10211yescharacteristic
491C2×C4C2×#10211yescharacteristic
492C2×C4C2×#10211yescharacteristic
493C2×C4C2×#10211yescharacteristic
494C2×C4C22×C22⋊C411yescharacteristic
495C2×C4C22×D811yescharacteristic
496C2×C4C22×SD1611yescharacteristic
497C8—21no—
498C8—21no—
499C8—21no—
500C8—21no—
501C8—21no—
502C8—21no—
503C8—81no—
504C8—81no—
505C8—81no—
506C8—81no—
507C8#41312yes—
508C8#57511yescharacteristic
509C8#57511yescharacteristic
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22C22×#10211yescharacteristic
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—82no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4C2×#41311yescharacteristic
535C4#289611yescharacteristic
536C4#341611yescharacteristic
537C4#341711yescharacteristic
538C2—41no—
539C2—41no—
540C2—41no—
541C2#1770411yescharacteristic
542C2#3338211yescharacteristic
543C2#3339311yescharacteristic
subgroupG/H
center, socleC22C22×#102
commutatorC2×C8C23×C4
Frattini, cosocleC4×C8C24
prime radicalC2×#41C22⋊C4
LCSG ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11988C2semidirect
5. #11987C2semidirect
6. #11984C2semidirect
7. #11983C2semidirect
8. #11976C2semidirect
9. #11973C2semidirect
10. #11965C2semidirect
11. #11962C2semidirect
12. #11954C2semidirect
13. #11953C2semidirect
14. #11950C2semidirect
15. #11949C2semidirect

Maximal subgroups

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

subgroupindex
#607152
#334232
#177992
#119882
#119872
#119842
#119832
#119762
#119732
#119652
#119622
#119542
#119532
#119502
#119492