Finite Group Databasebeta

← #164457

Subgroups — #164457

Order 512.

1000 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36303C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic, complemented
3#20567C211yesmaximal, characteristic, complemented
4#7710C212yesmaximal, complemented
5#7682C211yesmaximal, characteristic, complemented
6#7682C211yesmaximal, characteristic, complemented
7#7670C212yesmaximal, complemented
8#7635C211yesmaximal, characteristic, complemented
9#7609C212yesmaximal, complemented
10#7575C212yesmaximal, complemented
11#7567C211yesmaximal, characteristic, complemented
12C2×D4×Q8C411yescharacteristic
13C2×#547C411yescharacteristic
14C2×#436—21no—
15#2862C2211yescharacteristic
16#2850C2212yes—
17#2568C411yescharacteristic
18#2547C411yescharacteristic
19#2503C412yes—
20#2492—21no—
21#2492—21no—
22#2488—22no—
23#2488—22no—
24#2436C412yes—
25#2339—21no—
26#2339—21no—
27C4×#124C2211yescharacteristic
28C2×#391C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1817C2212yes—
34#1797C2212yes—
35C8×C4⋊C4—21no—
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1753—22no—
39C4×C22⋊C8C2212yes—
40#1744C2211yescharacteristic
41#1744C2211yescharacteristic
42#1728C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2211yescharacteristic
48#1272C2211yescharacteristic
49#1272C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×#41—21no—
54C23×Q8C2×C412yes—
55C22×M16—22no—
56D4×Q8—44no—
57#547—44no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C412yes—
63C2×#123—22no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8C2×C412yes—
67C2×#121C2×C412yes—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C22×C4⋊C4C2×C412yes—
75C22×C42C2312yes—
76#436—28no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×#97—21no—
84C2×#97—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—22no—
88#400—22no—
89#400—22no—
90#400—22no—
91#397—22no—
92#393—22no—
93#393—22no—
94#391—21no—
95#391—21no—
96#391—21no—
97#391—21no—
98#391—22no—
99#391—22no—
100#391C2×C412yes—
101#391C2×C412yes—
102#391C2×C412yes—
103#391C2312yes—
104#391C2311yescharacteristic
105#390C2311yescharacteristic
106#388—22no—
107#388—22no—
108#388—21no—
109#388—21no—
110#386—22no—
111#386—22no—
112#386—22no—
113#386—22no—
114#382—21no—
115#382—21no—
116#382C2×C412yes—
117#382C2×C411yescharacteristic
118#381—22no—
119#381—22no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C2×#95—22no—
131C2×#95—22no—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143C2×#41—44no—
144C2×#41—44no—
145C2×#41—44no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—24no—
149C22×Q8—42no—
150C22×Q8—44no—
151C22×Q8—82no—
152C22×Q8C22×C412yes—
153C22×Q8C22×C411yescharacteristic
154C22×D4—22no—
155C22×D4C22⋊C411yescharacteristic
156C22×D4C22×C411yescharacteristic
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—21no—
160C23×C4C22×C412yes—
161C23×C4C2×D412yes—
162C23×C4C2×D412yes—
163C2×M16—28no—
164C2×M16—82no—
165C2×M16—82no—
166C22×C8—21no—
167C22×C8—22no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—44no—
175#124—22no—
176#124—41no—
177#124—82no—
178#123—48no—
179C22⋊Q8—48no—
180C22⋊Q8—44no—
181C22⋊Q8—44no—
182#121—44no—
183C4×Q8—42no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—44no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C42—21no—
215C2×C42—82no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2411yescharacteristic
226C4⋊C8—24no—
227C4⋊C8—42no—
228C4⋊C8—82no—
229C22⋊C8—28no—
230C22⋊C8—82no—
231C22⋊C8—82no—
232#97—81no—
233#97—81no—
234C4×C8—24no—
235C4×C8—42no—
236C4×C8—82no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C24—21no—
255C24C2×C22⋊C412yes—
256#41—816no—
257C2×Q8—21no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—44no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×Q8—82no—
270C2×Q8—88no—
271C2×Q8—88no—
272C2×Q8#13112yes—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—44no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4#10414yes—
283C22×C4—21no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—82no—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C412yes—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319M16—88no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—24no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—41no—
337C2×C8—41no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—44no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4#10014yes—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—42no—
356C22⋊C4—44no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—44no—
362C22⋊C4—82no—
363C22⋊C4—82no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—42no—
374C42—42no—
375C42—41no—
376C42—42no—
377C42—42no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23C2×#10414yes—
392C23C22×C22⋊C411yescharacteristic
393Q8—42no—
394Q8—42no—
395Q8—88no—
396Q8—82no—
397Q8—82no—
398Q8—84no—
399D4—28no—
400D4—81no—
401D4—81no—
402D4—88no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—88no—
448C2×C4—88no—
449C2×C4—84no—
450C2×C4C2×#10014yes—
451C2×C4C2×#10012yes—
452C2×C4C2×#10412yes—
453C2×C4#42112yes—
454C2×C4C2×#13111yescharacteristic
455C2×C4#56811yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57711yescharacteristic
458C2×C4#57811yescharacteristic
459C2×C4#57811yescharacteristic
460C8—42no—
461C8—44no—
462C8—42no—
463C8—81no—
464C8—82no—
465C8—81no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22—81no—
479C22—81no—
480C22#288011yescharacteristic
481C22#288311yescharacteristic
482C22C22×#10011yescharacteristic
483C22#289811yescharacteristic
484C22#289811yescharacteristic
485C22C22×#10411yescharacteristic
486C22C2×#42111yescharacteristic
487C4—21no—
488C4—21no—
489C4—24no—
490C4—21no—
491C4—21no—
492C4—21no—
493C4—21no—
494C4—42no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—42no—
499C4—42no—
500C4—82no—
501C4—81no—
502C4—81no—
503C4—82no—
504C4#152714yes—
505C2—44no—
506C2—161no—
507C2#1742111yescharacteristic
508C2#1751911yescharacteristic
509C2#1752511yescharacteristic
510C2#1754811yescharacteristic
511C2#1760011yescharacteristic
512C2C2×#152711yescharacteristic
513C2#1777311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C4
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. #22012C2semidirect
3. #20567C2semidirect
4. #7710C2semidirect
5. #7682C2semidirect
6. #7682C2semidirect
7. #7670C2semidirect
8. #7635C2semidirect
9. #7609C2semidirect
10. #7575C2semidirect
11. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#363032
#220122
#205672
#77102
#76822
#76702
#76352
#76092
#75752
#75672