Finite Group Databasebeta

← #163990

Subgroups — #163990

Order 512.

973 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35255C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic, complemented
3#22530C211yesmaximal, characteristic, complemented
4#7715C212yesmaximal, complemented
5#7698C212yesmaximal, complemented
6#7675C212yesmaximal, complemented
7#7658C212yesmaximal, complemented
8#7597C211yesmaximal, characteristic, complemented
9#7576C211yesmaximal, characteristic, complemented
10#7573C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic, complemented
12C2×#537C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#446—21no—
15C2×C8×Q8—21no—
16#2855C2212yes—
17#2854C2212yes—
18#2625C412yes—
19#2624C412yes—
20#2444—22no—
21#2443—22no—
22#2423—21no—
23#2422—21no—
24#2421—21no—
25#2420—21no—
26#2317C411yescharacteristic
27#2311C411yescharacteristic
28#2311C411yescharacteristic
29#2297C2211yescharacteristic
30C2×#389C2211yescharacteristic
31#1955—22no—
32#1954—21no—
33#1954—22no—
34#1954—21no—
35#1797C2212yes—
36#1797C2211yescharacteristic
37#1797C2212yes—
38#1797C2211yescharacteristic
39#1749C2212yes—
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42C4×C22⋊C8C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×#41C2×C411yescharacteristic
54C22×#41C2×C411yescharacteristic
55#537—44no—
56C2×#123—22no—
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8C2×C412yes—
59C2×#121—21no—
60C2×#121—22no—
61C2×#121C2×C412yes—
62C4×#41—44no—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D412yes—
68C2×C4×D4D412yes—
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C412yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#446—44no—
78C8×Q8—44no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—21no—
91C2×#97—22no—
92C2×C4×C8—21no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95#400—22no—
96#399—22no—
97#398—22no—
98#397—22no—
99#397—22no—
100#397—22no—
101#394—22no—
102#394—22no—
103#391—21no—
104#391C2×C412yes—
105#391C2×C411yescharacteristic
106#391C2311yescharacteristic
107#390—21no—
108#390C2×C412yes—
109#390C2×C411yescharacteristic
110#390C2311yescharacteristic
111#389—21no—
112#389—21no—
113#389C2×C412yes—
114#389C2×C412yes—
115#389C2×C412yes—
116#389C2312yes—
117#388—22no—
118#388—22no—
119#387—22no—
120#387—22no—
121#386—22no—
122#386—22no—
123#386—21no—
124#386—21no—
125#386—22no—
126#385—22no—
127#385—22no—
128#384—21no—
129#382—21no—
130#381—22no—
131#380—21no—
132#380—21no—
133#380C2×C412yes—
134#380C2×C412yes—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137C2×#95—21no—
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—24no—
146C2×#41—24no—
147C2×#41—42no—
148C2×#41—42no—
149C2×#41—42no—
150C22×Q8C22⋊C411yescharacteristic
151C22×Q8C22×C411yescharacteristic
152C22×D4—21no—
153C22×D4—22no—
154C22×D4C22⋊C411yescharacteristic
155C22×D4C22⋊C412yes—
156C22×D4C22×C411yescharacteristic
157C23×C4—21no—
158C23×C4—21no—
159C23×C4—21no—
160C23×C4C22×C412yes—
161C23×C4C22×C411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172#123—82no—
173#123—82no—
174C22⋊Q8—44no—
175C22⋊Q8—44no—
176#121—44no—
177#121—48no—
178#121—44no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—44no—
184C4×D4—44no—
185#117—42no—
186#117—44no—
187#117—42no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C42—21no—
222C2×C42—42no—
223C2×C42—44no—
224C2×C42C22×C412yes—
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2411yescharacteristic
231C4⋊C8—44no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—44no—
235C22⋊C8—42no—
236C22⋊C8—44no—
237C22⋊C8—44no—
238C22⋊C8—42no—
239C22⋊C8—81no—
240C22⋊C8—82no—
241C22⋊C8—81no—
242C22⋊C8—82no—
243#97—44no—
244C4×C8—44no—
245C4×C8—42no—
246C4×C8—42no—
247#95—22no—
248#95—21no—
249#95—21no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—21no—
258#95—21no—
259C24—21no—
260C24C2×C22⋊C412yes—
261C24C2×C22⋊C411yescharacteristic
262#41—416no—
263#41—88no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8#9914yes—
270C2×D4—24no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—44no—
283C2×D4—82no—
284C2×D4—82no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4#13114yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—41no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—44no—
369C22⋊C4—41no—
370C22⋊C4—41no—
371C22⋊C4—41no—
372C22⋊C4—41no—
373C22⋊C4—44no—
374C22⋊C4—81no—
375C22⋊C4—82no—
376C22⋊C4—81no—
377C22⋊C4—82no—
378C22⋊C4—82no—
379C22⋊C4—82no—
380C42—24no—
381C42—22no—
382C42—22no—
383C42—24no—
384C42—41no—
385C42—41no—
386C42—42no—
387C42—42no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23#40812yes—
409C23C22×C22⋊C411yescharacteristic
410Q8—28no—
411Q8—82no—
412D4—48no—
413D4—44no—
414D4—82no—
415D4—84no—
416D4—164no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—28no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—48no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4—162no—
465C2×C4#40812yes—
466C2×C4C2×#9914yes—
467C2×C4C2×#9912yes—
468C2×C4#41312yes—
469C2×C4C2×#13112yes—
470C2×C4#57611yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57711yescharacteristic
473C2×C4#57811yescharacteristic
474C8—42no—
475C8—42no—
476C8—82no—
477C8—81no—
478C8—81no—
479C8—82no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—42no—
493C22—42no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22C2×#40811yescharacteristic
502C22C22×#9911yescharacteristic
503C22C2×#41311yescharacteristic
504C22#289711yescharacteristic
505C22#289811yescharacteristic
506C22#290111yescharacteristic
507C22#290111yescharacteristic
508C4—21no—
509C4—21no—
510C4—21no—
511C4—21no—
512C4—22no—
513C4—24no—
514C4—41no—
515C4—42no—
516C4—42no—
517C4—41no—
518C4—81no—
519C4—82no—
520C4—81no—
521C4—82no—
522C4—81no—
523C4#146814yes—
524C2—42no—
525C2—82no—
526C2—161no—
527C2C2×#146811yescharacteristic
528C2#1746911yescharacteristic
529C2#1748311yescharacteristic
530C2#1760611yescharacteristic
531C2#1761111yescharacteristic
532C2#1768711yescharacteristic
533C2#1777211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22532C2semidirect
3. #22530C2semidirect
4. #7715C2semidirect
5. #7698C2semidirect
6. #7675C2semidirect
7. #7658C2semidirect
8. #7597C2semidirect
9. #7576C2semidirect
10. #7573C2semidirect
11. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#352552
#225322
#225302
#77152
#76982
#76752
#76582
#75972
#75762
#75732
#75632