Finite Group Databasebeta

← #192918

Subgroups — #192918

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60266C211yesmaximal, characteristic
2#21856C211yesmaximal, characteristic
3#17814C211yesmaximal, characteristic, complemented
4#17796C211yesmaximal, characteristic, complemented
5#17462C211yesmaximal, characteristic
6#8905C214yesmaximal, complemented
7#8902C214yesmaximal, complemented
8#8092C211yesmaximal, characteristic, complemented
9#7903C211yesmaximal, characteristic, complemented
10#3444C2211yescharacteristic
11#2995—22no—
12#2993—22no—
13#2991—21no—
14#2990—21no—
15#2989—22no—
16#2962—21no—
17#2935—21no—
18#2930—22no—
19#2925C2212yes—
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2906C2212yes—
23C2×#423C2211yescharacteristic
24C2×#423C2211yescharacteristic
25C2×C4×M16C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1994—21no—
28#1962—21no—
29#1871—22no—
30#1806C2214yes—
31#1571—22no—
32#1571C2214yes—
33#1569—22no—
34#1567—21no—
35#1567—21no—
36#1567C2212yes—
37#1566C2212yes—
38#1565—22no—
39#1485C411yescharacteristic, complemented
40#1485C411yescharacteristic, complemented
41#1484C411yescharacteristic, complemented
42#1484C411yescharacteristic, complemented
43#1483C414yescomplemented
44#1476C2214yes—
45#1303C2212yes—
46#1303C2212yes—
47#1279C2214yes—
48C2×#131—22no—
49C22×M16—21no—
50C22×M16—21no—
51C22×M16—21no—
52#533C2×C411yescharacteristic
53#533D411yescharacteristic
54#533Q811yescharacteristic
55#532C2×C411yescharacteristic
56#532D411yescharacteristic
57#532Q811yescharacteristic
58#529—21no—
59#529—21no—
60#529—21no—
61#529—21no—
62#529C2×C412yes—
63#529Q812yes—
64#525—21no—
65#525—21no—
66#525D412yes—
67C4×#41—21no—
68C4×#41—21no—
69C4×#41C2312yes—
70C2×#117—21no—
71C22×C42C2311yescharacteristic
72#446—24no—
73#446—22no—
74C8×Q8—22no—
75#436—24no—
76#436—24no—
77#436—22no—
78#435—24no—
79#435—24no—
80#435—24no—
81C8×D4—22no—
82#433—24no—
83#432—21no—
84#432—22no—
85#432D412yes—
86#431—21no—
87#431—21no—
88#431—21no—
89#431—22no—
90#431C2×C411yescharacteristic
91#431C2×C411yescharacteristic
92#431D412yes—
93#431C2312yes—
94#424—22no—
95#424—24no—
96#423—21no—
97#423—21no—
98#423—22no—
99#423—21no—
100#423—21no—
101#423—21no—
102#423D411yescharacteristic
103#423D411yescharacteristic
104#423D412yes—
105#423C2312yes—
106#423C2311yescharacteristic
107#423C2311yescharacteristic
108C2×C4⋊C8—21no—
109C2×C4⋊C8—22no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8C2×C412yes—
113C2×C4⋊C8C2312yes—
114C2×#104—24no—
115#408—22no—
116#407—21no—
117C2×C22⋊C8—21no—
118#405—22no—
119C4×M16—21no—
120C4×M16—21no—
121C4×M16—21no—
122C4×M16C2×C412yes—
123C4×M16C2×C411yescharacteristic
124C4×M16C2×C411yescharacteristic
125C2×#97—21no—
126C2×C4×C8—21no—
127C2×C4×C8C2312yes—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2312yes—
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132#376—21no—
133C43—21no—
134#339—24no—
135#335—24no—
136#335—24no—
137#335—24no—
138#334—24no—
139Q8⋊C8C2×C414yes—
140Q8⋊C8C2×C414yes—
141D4⋊C8C2×C414yes—
142D4⋊C8C2×C414yes—
143C2×#41—21no—
144C2×#41—21no—
145C2×#41C2×D412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4#4111yescharacteristic
148C23×C4#4111yescharacteristic
149#131—84no—
150C2×M16—21no—
151C2×M16—22no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—24no—
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—21no—
159C2×M16—22no—
160C2×M16—44no—
161C2×M16—41no—
162C2×M16—82no—
163C2×M16—82no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—24no—
167C22×C8—22no—
168C22×C8—41no—
169C4⋊Q8C4⋊C411yescharacteristic
170C4⋊Q8C4⋊C411yescharacteristic
171C4⋊Q8C4⋊C411yescharacteristic
172#127C4⋊C411yescharacteristic
173#126—22no—
174#126—22no—
175#125—21no—
176#125—21no—
177#125C4⋊C412yes—
178#124—21no—
179#124—21no—
180#124C4⋊C412yes—
181#123—22no—
182#123—22no—
183#123—22no—
184C22⋊Q8—21no—
185C22⋊Q8—21no—
186C22⋊Q8—21no—
187C22⋊Q8—21no—
188C22⋊Q8C4⋊C412yes—
189C22⋊Q8C4⋊C412yes—
190#121—21no—
191#121—21no—
192#121—21no—
193#121—21no—
194#121C4⋊C412yes—
195#121C4⋊C412yes—
196#120—22no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8C22×C412yes—
202C4×Q8C2×Q812yes—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—22no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—21no—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4C22×C412yes—
214C4×D4C2×Q812yes—
215#117—21no—
216#117—21no—
217#117—22no—
218#117—22no—
219#117—22no—
220#117—44no—
221#117C2×D412yes—
222C2×C4⋊C4—22no—
223C2×C22⋊C4—41no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×Q811yescharacteristic
236C2×C42C2×Q811yescharacteristic
237C2×C42#4111yescharacteristic
238C2×C42#4111yescharacteristic
239C2×C42C2411yescharacteristic
240C4⋊C8—24no—
241C4⋊C8—24no—
242C4⋊C8—22no—
243C4⋊C8—21no—
244C4⋊C8—22no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—22no—
248C4⋊C8—21no—
249C4⋊C8—22no—
250C4⋊C8—22no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8C22×C412yes—
254C4⋊C8C22×C412yes—
255C4⋊C8C2×D412yes—
256C4⋊C8C2×D412yes—
257C4⋊C8C2×D412yes—
258C4⋊C8C2×D412yes—
259#104—84no—
260C22⋊C8—24no—
261C22⋊C8—24no—
262C22⋊C8—21no—
263C22⋊C8—21no—
264C22⋊C8—22no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267#97—24no—
268#97—21no—
269#97—21no—
270#97—22no—
271#97—22no—
272C4×C8—24no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—22no—
276C4×C8—22no—
277C4×C8—22no—
278C4×C8C22×C412yes—
279C4×C8C22×C412yes—
280#95—22no—
281#95—41no—
282C24#12311yescharacteristic
283#41—24no—
284#41—24no—
285#41—82no—
286#41—82no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8C2×C4⋊C412yes—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4C2×C4⋊C412yes—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4C2×C4⋊C411yescharacteristic
305C22×C4C2×C4⋊C411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12011yescharacteristic
312C22×C4C22⋊Q811yescharacteristic
313C22×C4C22⋊Q811yescharacteristic
314C22×C4C22⋊Q811yescharacteristic
315C22×C4C22⋊Q811yescharacteristic
316C22×C4#12311yescharacteristic
317C22×C4#12311yescharacteristic
318C22×C4#12311yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322M16—24no—
323M16—44no—
324M16—42no—
325M16—42no—
326M16—42no—
327M16—44no—
328M16—81no—
329M16—82no—
330M16—81no—
331M16—84no—
332M16—82no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—22no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—44no—
342C2×C8—44no—
343C2×C8—42no—
344C2×C8—82no—
345C2×C8—84no—
346C2×C8—82no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—42no—
359C4⋊C4C2×C4⋊C412yes—
360C4⋊C4C2×C4⋊C412yes—
361C22⋊C4—22no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—22no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—22no—
370C22⋊C4—44no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—44no—
375C42—42no—
376C42—42no—
377C42—82no—
378C42C2×C4⋊C411yescharacteristic
379C42C2×C4⋊C411yescharacteristic
380C42C4×D411yescharacteristic
381C42C4×D411yescharacteristic
382C42C4×D411yescharacteristic
383C42C4×D411yescharacteristic
384C42#12011yescharacteristic
385C42#12011yescharacteristic
386C42C22⋊Q811yescharacteristic
387C42C22⋊Q811yescharacteristic
388C42C22⋊Q811yescharacteristic
389C42C22⋊Q811yescharacteristic
390C42C23×C411yescharacteristic
391C42C22×D411yescharacteristic
392C42C22×D411yescharacteristic
393C42C22×Q811yescharacteristic
394C23—22no—
395C23—22no—
396C23—41no—
397C23—41no—
398C23#38511yescharacteristic
399C23#38511yescharacteristic
400C23C2×#12311yescharacteristic
401Q8—24no—
402Q8—42no—
403Q8—81no—
404D4—24no—
405D4—44no—
406D4—42no—
407D4—81no—
408D4—82no—
409D4—82no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4#38511yescharacteristic
440C2×C4#38511yescharacteristic
441C2×C4#38511yescharacteristic
442C2×C4#38511yescharacteristic
443C2×C4#38511yescharacteristic
444C2×C4#38511yescharacteristic
445C2×C4C22×C4⋊C411yescharacteristic
446C2×C4C2×C4×D411yescharacteristic
447C2×C4C2×C4×D411yescharacteristic
448C2×C4C2×#12011yescharacteristic
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×C22⋊Q811yescharacteristic
451C2×C4#57612yes—
452C2×C4#57812yes—
453C8—24no—
454C8—42no—
455C8—42no—
456C8—42no—
457C8—42no—
458C8—42no—
459C8—42no—
460C8—42no—
461C8—82no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—82no—
474C22C2×#38511yescharacteristic
475C22#309812yes—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—84no—
488C4#186112yes—
489C4#291312yes—
490C4#295612yes—
491C4#300811yescharacteristic
492C4#301111yescharacteristic
493C2—22no—
494C2—41no—
495C2—41no—
496C2—82no—
497C2#2084711yescharacteristic
498C2C2×#186111yescharacteristic
499C2#2088511yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#385
prime radical#3444C22
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
3. #17814C2semidirect
4. #17796C2semidirect
6. #8905C2semidirect
7. #8902C2semidirect
8. #8092C2semidirect
9. #7903C2semidirect
39. #1485C4semidirect
40. #1485C4semidirect
41. #1484C4semidirect
42. #1484C4semidirect
43. #1483C4semidirect

Maximal subgroups

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

subgroupindex
#602662
#218562
#178142
#177962
#174622
#89052
#89022
#80922
#79032