Finite Group Databasebeta

← #161584

Subgroups — #161584

Order 512.

969 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35170C211yesmaximal, characteristic
2#22700C211yesmaximal, characteristic, complemented
3#20555C211yesmaximal, characteristic, complemented
4#7685C212yesmaximal, complemented
5#7678C212yesmaximal, complemented
6#7665C212yesmaximal, complemented
7#7658C212yesmaximal, complemented
8#7588C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12C2×#542—21no—
13C2×#436—21no—
14C2×C8×D4—21no—
15#2857C2212yes—
16#2854C2212yes—
17#2663—22no—
18#2551C411yescharacteristic
19#2547C411yescharacteristic
20#2542C411yescharacteristic
21#2537C411yescharacteristic
22#2521—22no—
23#2496C412yes—
24#2492—21no—
25#2491C412yes—
26#2427—21no—
27#2323—21no—
28C4×C4⋊Q8C2211yescharacteristic
29#2293C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1939—21no—
32#1851—22no—
33#1831C2211yescharacteristic
34#1797C2212yes—
35C8×C4⋊C4—21no—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C8×C22⋊C4—22no—
39C4×C22⋊C8C2212yes—
40#1744C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×M16—22no—
54C23×C8—22no—
55#542—44no—
56#542—44no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8C2×C412yes—
60C2×#126—22no—
61C2×#124—21no—
62C2×#124—22no—
63C2×#123—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×#121—21no—
68C2×#121C2×C412yes—
69C2×C4×Q8—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×#117—21no—
75C22×C42C2312yes—
76#436—44no—
77C8×D4—28no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88#400—22no—
89#399—22no—
90#397—22no—
91#397—22no—
92#395—22no—
93#394—22no—
94#394—22no—
95#393—22no—
96#391—21no—
97#391—21no—
98#391—22no—
99#390C2×C412yes—
100#390C2×C411yescharacteristic
101#390C2311yescharacteristic
102#389—21no—
103#389—21no—
104#389—22no—
105#389C2×C412yes—
106#389C2×C411yescharacteristic
107#389C2312yes—
108#389C2311yescharacteristic
109#388—22no—
110#388—22no—
111#388—21no—
112#388—21no—
113#388—22no—
114#387—22no—
115#386—22no—
116#384C2×C411yescharacteristic
117#384C2×C412yes—
118#380—22no—
119#380—21no—
120#380C2×C412yes—
121#380C2×C412yes—
122#380C2×C412yes—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4D411yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4C2×C412yes—
131C4×C22⋊C4D412yes—
132C4×C22⋊C4D412yes—
133C43C2311yescharacteristic
134#336—21no—
135#336—22no—
136#336—22no—
137#336—21no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C22×Q8—21no—
145C22×Q8—22no—
146C22×D4—22no—
147C22×D4C22⋊C411yescharacteristic
148C22×D4C22×C411yescharacteristic
149C23×C4—21no—
150C23×C4C22×C412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D412yes—
153C2×M16—44no—
154C2×M16—44no—
155C2×M16—82no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—28no—
161C22×C8—22no—
162C22×C8—21no—
163C22×C8—82no—
164C22×C8—82no—
165C4⋊Q8—24no—
166C4⋊Q8—44no—
167C4⋊Q8—82no—
168#126—48no—
169#124—44no—
170#124—48no—
171#123—44no—
172#123—44no—
173C22⋊Q8—44no—
174C22⋊Q8—44no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177#121—44no—
178#121—82no—
179C4×Q8—42no—
180C4×Q8—42no—
181C4×D4—24no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4D814yes—
185C4×D4SD1614yes—
186#117—42no—
187#117—42no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—82no—
223C2×C42C22×C412yes—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D411yescharacteristic
232C2×C42C2411yescharacteristic
233C4⋊C8—24no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236C4⋊C8—82no—
237C22⋊C8—28no—
238C22⋊C8—44no—
239C4×C8—24no—
240C4×C8—42no—
241C4×C8—81no—
242C4×C8—82no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—21no—
251#95—22no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255C24—21no—
256C24C2×C22⋊C412yes—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—44no—
269C2×D4—82no—
270C2×D4—82no—
271C2×D4#10214yes—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×D814yes—
297C22×C4C2×SD1614yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301M16—88no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—88no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—81no—
338C4⋊C4—82no—
339C4⋊C4#10214yes—
340C4⋊C4#10414yes—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—81no—
355C22⋊C4—82no—
356C22⋊C4#10218yes—
357C42—21no—
358C42—21no—
359C42—22no—
360C42—42no—
361C42—41no—
362C42—42no—
363C42—41no—
364C42—42no—
365C42—41no—
366C42—42no—
367C42—44no—
368C42#13114yes—
369C42C2×D812yes—
370C42C2×SD1612yes—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23C2×#10214yes—
385C23C22×C22⋊C411yescharacteristic
386Q8—84no—
387Q8—164no—
388D4—28no—
389D4—82no—
390D4—164no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—84no—
431C2×C4—84no—
432C2×C4#40812yes—
433C2×C4#40812yes—
434C2×C4C2×#10212yes—
435C2×C4C2×#10214yes—
436C2×C4C2×#10212yes—
437C2×C4C2×#10414yes—
438C2×C4C2×#10412yes—
439C2×C4C2×#13112yes—
440C2×C4C22×D811yescharacteristic
441C2×C4C22×SD1611yescharacteristic
442C2×C4#57711yescharacteristic
443C2×C4#57811yescharacteristic
444C8—42no—
445C8—44no—
446C8—42no—
447C8—42no—
448C8—81no—
449C8—82no—
450C22—24no—
451C22—24no—
452C22—24no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22C2×#40811yescharacteristic
465C22#288311yescharacteristic
466C22#288911yescharacteristic
467C22C22×#10211yescharacteristic
468C22#289811yescharacteristic
469C22C22×#10411yescharacteristic
470C22#290111yescharacteristic
471C4—22no—
472C4—22no—
473C4—21no—
474C4—21no—
475C4—24no—
476C4—21no—
477C4—21no—
478C4—24no—
479C4—22no—
480C4—42no—
481C4—41no—
482C4—81no—
483C4—81no—
484C4—82no—
485C4—82no—
486C4—81no—
487C4#149214yes—
488C2—44no—
489C2—161no—
490C2#1738811yescharacteristic
491C2#1746911yescharacteristic
492C2C2×#149211yescharacteristic
493C2#1754811yescharacteristic
494C2#1761311yescharacteristic
495C2#1765411yescharacteristic
496C2#1772111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22700C2semidirect
3. #20555C2semidirect
4. #7685C2semidirect
5. #7678C2semidirect
6. #7665C2semidirect
7. #7658C2semidirect
8. #7588C2semidirect
9. #7574C2semidirect
10. #7570C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#351702
#227002
#205552
#76852
#76782
#76652
#76582
#75882
#75742
#75702
#75602