Finite Group Databasebeta

← #192479

Subgroups — #192479

Order 512.

874 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60285C211yesmaximal, characteristic
2#21994C211yesmaximal, characteristic, complemented
3#21978C211yesmaximal, characteristic, complemented
4#21856C211yesmaximal, characteristic
5#17543C211yesmaximal, characteristic
6#8875C214yesmaximal, complemented
7#8874C214yesmaximal, complemented
8#8048C211yesmaximal, characteristic, complemented
9#7930C211yesmaximal, characteristic, complemented
10#3444C2211yescharacteristic
11#2995—21no—
12#2995—21no—
13#2994—21no—
14#2994—21no—
15#2985—21no—
16#2981—21no—
17#2976C2212yes—
18#2976C2212yes—
19#2971—21no—
20#2970—21no—
21#2970—22no—
22#2923—22no—
23#2920C2211yescharacteristic
24C2×#432C2211yescharacteristic
25C2×#432C2211yescharacteristic
26C2×#423C2211yescharacteristic
27C4×#117C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1994—21no—
30#1962—21no—
31#1950—21no—
32#1950—21no—
33#1940—22no—
34#1871C2214yes—
35#1855C2212yes—
36#1855C2212yes—
37#1851C2214yes—
38#1849—21no—
39#1842—21no—
40#1838—21no—
41#1831—21no—
42#1806—22no—
43#1509C414yescomplemented
44#1506C411yescharacteristic, complemented
45#1506C411yescharacteristic, complemented
46#1505C411yescharacteristic, complemented
47#1505C411yescharacteristic, complemented
48#1501C2214yes—
49#1297C2212yes—
50#1297C2212yes—
51#1282C2214yes—
52C22×M16C2312yes—
53#533C2×C411yescharacteristic
54#533D411yescharacteristic
55#533Q811yescharacteristic
56#532C2×C411yescharacteristic
57#532D411yescharacteristic
58#532Q811yescharacteristic
59#529—21no—
60#529—21no—
61#529—21no—
62#529—21no—
63#529C2×C412yes—
64#529Q812yes—
65#525—21no—
66#525—21no—
67#525C2312yes—
68C4×#41—21no—
69C4×#41—21no—
70C4×#41D412yes—
71C2×#117—21no—
72C2×#117—21no—
73C22×C42C2311yescharacteristic
74#446—21no—
75#446—21no—
76#446—22no—
77#446—22no—
78C8×Q8—21no—
79C8×Q8—21no—
80#436—21no—
81#436—21no—
82#436—22no—
83#436—22no—
84#435—22no—
85#435—22no—
86#435—22no—
87#435—22no—
88#435—22no—
89#435—22no—
90C8×D4—21no—
91C8×D4—21no—
92C8×D4—22no—
93C8×D4—22no—
94#433—22no—
95#433—22no—
96#433—22no—
97#433—22no—
98#432—22no—
99#432—22no—
100#432—21no—
101#432—21no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432—21no—
108#432C2×C411yescharacteristic
109#432C2×C411yescharacteristic
110#432C2×C412yes—
111#431—21no—
112#431—21no—
113C2×#108—22no—
114C2×#108—22no—
115#424—22no—
116#424—22no—
117#423—21no—
118#423—21no—
119#423—21no—
120#423—21no—
121#423D411yescharacteristic
122#423D411yescharacteristic
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—24no—
127C2×C4⋊C8D411yescharacteristic
128C2×C4⋊C8D411yescharacteristic
129C2×#104—24no—
130#408—22no—
131#408—22no—
132#407C2311yescharacteristic
133#407C2311yescharacteristic
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C4×M16—22no—
137C4×M16D411yescharacteristic
138C4×M16D411yescharacteristic
139C2×#97—22no—
140C2×#97—21no—
141C2×#97D412yes—
142C4×C4⋊C4—21no—
143C4×C4⋊C4—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2×C412yes—
146C4×C22⋊C4—21no—
147C4×C22⋊C4—21no—
148C4×C22⋊C4C2312yes—
149C4×C22⋊C4C2312yes—
150#376—21no—
151#376C2×C411yescharacteristic
152#376C2×C411yescharacteristic
153C43—21no—
154C43—21no—
155#341—22no—
156#341—22no—
157#339—24no—
158#336—24no—
159#328C2×C414yes—
160#328C2×C414yes—
161#327C2×C414yes—
162#327C2×C414yes—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41C2×D412yes—
166C23×C4#4111yescharacteristic
167C23×C4#4111yescharacteristic
168C23×C4C2411yescharacteristic
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—82no—
174C2×M16C2×D411yescharacteristic
175C2×M16C2×D411yescharacteristic
176C2×M16C2×D411yescharacteristic
177C2×M16C2×D411yescharacteristic
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C4⋊Q8C4⋊C411yescharacteristic
185C4⋊Q8C4⋊C411yescharacteristic
186C4⋊Q8C4⋊C411yescharacteristic
187#127C4⋊C411yescharacteristic
188#126—22no—
189#126—22no—
190#125—21no—
191#125—21no—
192#125C4⋊C412yes—
193#124—21no—
194#124—21no—
195#124C4⋊C412yes—
196#123—22no—
197#123—22no—
198#123—22no—
199C22⋊Q8—21no—
200C22⋊Q8—21no—
201C22⋊Q8—21no—
202C22⋊Q8—21no—
203C22⋊Q8C22×C412yes—
204C22⋊Q8C2×Q812yes—
205#121—21no—
206#121—21no—
207#121—21no—
208#121—21no—
209#121C22×C412yes—
210#121C2×Q812yes—
211#120—22no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—21no—
215C4×Q8—21no—
216C4×Q8C4⋊C412yes—
217C4×Q8C4⋊C412yes—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4—22no—
224C4×D4—21no—
225C4×D4—21no—
226C4×D4—21no—
227C4×D4—21no—
228C4×D4C4⋊C412yes—
229C4×D4C4⋊C412yes—
230#117—21no—
231#117—21no—
232#117—22no—
233#117—22no—
234#117—22no—
235#117—44no—
236#117—44no—
237#117C2×D412yes—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—41no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C42—21no—
243C2×C42—22no—
244C2×C42—21no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—41no—
249C2×C42—41no—
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×Q811yescharacteristic
259C2×C42C2×Q811yescharacteristic
260C2×C42#4111yescharacteristic
261C2×C42#4111yescharacteristic
262#108—82no—
263#108—82no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—41no—
280C4⋊C8—41no—
281C4⋊C8—41no—
282C4⋊C8—41no—
283C4⋊C8—84no—
284#104—84no—
285C22⋊C8—22no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—22no—
291C22⋊C8—21no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—22no—
296C22⋊C8C22×C412yes—
297C22⋊C8C22×C412yes—
298#97—22no—
299#97—22no—
300#97—21no—
301#97—21no—
302#97—22no—
303#97—21no—
304#97—22no—
305#97—22no—
306#97—21no—
307C4×C8—22no—
308C4×C8—22no—
309C4×C8—21no—
310C4×C8—21no—
311#95—21no—
312#95—21no—
313#95—22no—
314#95—41no—
315#95C22×C412yes—
316#95C22×C412yes—
317C24C2×#4111yescharacteristic
318#41—42no—
319#41—42no—
320#41—82no—
321#41—82no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×Q8C2×C4⋊C412yes—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4C2×C4⋊C412yes—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4C4×D411yescharacteristic
342C22×C4C4×D411yescharacteristic
343C22×C4C4×D411yescharacteristic
344C22×C4C4×D411yescharacteristic
345C22×C4C22⋊Q811yescharacteristic
346C22×C4C22⋊Q811yescharacteristic
347C22×C4C22⋊Q811yescharacteristic
348C22×C4C22⋊Q811yescharacteristic
349C22×C4#12311yescharacteristic
350C22×C4#12311yescharacteristic
351C22×C4#12311yescharacteristic
352C22×C4#12311yescharacteristic
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D411yescharacteristic
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×Q811yescharacteristic
358C22×C4C2×#4111yescharacteristic
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—44no—
364M16—42no—
365M16—42no—
366M16—82no—
367M16—82no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C2×C8—44no—
381C2×C8—42no—
382C2×C8—82no—
383C2×C8—82no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—21no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4C2×C4⋊C412yes—
397C4⋊C4C2×C4⋊C412yes—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—22no—
403C22⋊C4—22no—
404C22⋊C4—22no—
405C22⋊C4—22no—
406C22⋊C4—22no—
407C22⋊C4—24no—
408C22⋊C4—44no—
409C22⋊C4—42no—
410C42—24no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—44no—
420C42—82no—
421C42—82no—
422C42C2×C4⋊C411yescharacteristic
423C42C2×C4⋊C411yescharacteristic
424C42C2×C4⋊C411yescharacteristic
425C42C2×C4⋊C411yescharacteristic
426C42C4×D411yescharacteristic
427C42C4×D411yescharacteristic
428C42C4×D411yescharacteristic
429C42C4×D411yescharacteristic
430C42#12011yescharacteristic
431C42#12011yescharacteristic
432C42#12011yescharacteristic
433C42#12011yescharacteristic
434C42C22⋊Q811yescharacteristic
435C42C22⋊Q811yescharacteristic
436C42C22⋊Q811yescharacteristic
437C42C22⋊Q811yescharacteristic
438C23—22no—
439C23—22no—
440C23—41no—
441C23—41no—
442C23C2×C4×D411yescharacteristic
443C23C2×C22⋊Q811yescharacteristic
444C23C2×#12311yescharacteristic
445Q8—42no—
446Q8—42no—
447Q8—81no—
448D4—42no—
449D4—44no—
450D4—42no—
451D4—81no—
452D4—82no—
453D4—82no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—24no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4#38511yescharacteristic
492C2×C4#38511yescharacteristic
493C2×C4#38511yescharacteristic
494C2×C4#38511yescharacteristic
495C2×C4#38511yescharacteristic
496C2×C4#38511yescharacteristic
497C2×C4#38511yescharacteristic
498C2×C4#38511yescharacteristic
499C2×C4C22×C4⋊C411yescharacteristic
500C2×C4C2×C4×D411yescharacteristic
501C2×C4C2×#12011yescharacteristic
502C2×C4C2×C22⋊Q811yescharacteristic
503C8—42no—
504C8—42no—
505C8—42no—
506C8—41no—
507C8—41no—
508C8—42no—
509C8—41no—
510C8—41no—
511C8—82no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—82no—
524C22C2×#38511yescharacteristic
525C22#295612yes—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—21no—
534C4—21no—
535C4—21no—
536C4—44no—
537C4—42no—
538C4—42no—
539C4—42no—
540C4—82no—
541C4—84no—
542C4#186112yes—
543C4#302711yescharacteristic
544C4#302811yescharacteristic
545C2—22no—
546C2—41no—
547C2—41no—
548C2—82no—
549C2C2×#186111yescharacteristic
550C2#2088511yescharacteristic
551C2#2089011yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#385
prime radical#3444C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21994C2semidirect
3. #21978C2semidirect
6. #8875C2semidirect
7. #8874C2semidirect
8. #8048C2semidirect
9. #7930C2semidirect
43. #1509C4semidirect
44. #1506C4semidirect
45. #1506C4semidirect
46. #1505C4semidirect
47. #1505C4semidirect

Maximal subgroups

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

subgroupindex
#602852
#219942
#219782
#218562
#175432
#88752
#88742
#80482
#79302