Finite Group Databasebeta

← #496322

Subgroups — #496322

Order 512.

1061 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic, complemented
3#20727C211yesmaximal, characteristic, complemented
4#13228C214yesmaximal, complemented
5#13227C212yesmaximal, complemented
6#13226C212yesmaximal, complemented
7#12702C214yesmaximal, complemented
8C2×#536C2212yes—
9C2×#532C411yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#2920—21no—
12#2920—21no—
13#2854—21no—
14#2854—21no—
15#2618—21no—
16#2618—21no—
17#2548C411yescharacteristic
18#2548C2211yescharacteristic
19#2546C411yescharacteristic
20#2546C2211yescharacteristic
21#2545C412yes—
22#2545C2212yes—
23#2538—21no—
24#2538—21no—
25#2536—21no—
26#2536—21no—
27#2311C412yes—
28#2285C411yescharacteristic
29C2×#356C2218yes—
30C2×#355C2218yes—
31#1821—21no—
32#1821—21no—
33#1801—21no—
34#1801—21no—
35#1801C2212yes—
36#1801C2212yes—
37#1784C2212yes—
38#1784C2212yes—
39C4×#101—21no—
40C4×#101—21no—
41C4×#101C2212yes—
42C4×#101C2212yes—
43C22×#41C2×C412yes—
44C23×Q8C2311yescharacteristic
45#536—48no—
46#532—41no—
47#532—41no—
48#532—41no—
49#532—41no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C412yes—
52C2×C4⋊Q8C2312yes—
53C2×#127C2×C411yescharacteristic
54C2×#124C2×C412yes—
55C2×#124C2×C412yes—
56C2×#124C2312yes—
57C2×#123—22no—
58C2×#123—22no—
59C2×C22⋊Q8—22no—
60C2×C22⋊Q8—22no—
61C2×#121—21no—
62C2×#121—21no—
63C2×C4×Q8—41no—
64C2×C4×Q8—41no—
65C2×C4×Q8—41no—
66C2×C4×Q8—41no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8D411yescharacteristic
71C2×C4×Q8D411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77C2×#117C2×C412yes—
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4D411yescharacteristic
80C22×C4⋊C4D411yescharacteristic
81C22×C42C2×C411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42D411yescharacteristic
84#432—41no—
85#432—41no—
86#432—41no—
87#432—41no—
88#431—41no—
89#431—41no—
90#431—41no—
91#431—41no—
92#423—22no—
93#423—22no—
94C2×#101—22no—
95C2×#101—22no—
96C2×#101C2314yes—
97C2×#101C2314yes—
98#407—22no—
99#407—22no—
100C4×M16—22no—
101C4×M16—22no—
102#398—21no—
103#398—21no—
104#398—21no—
105#398—21no—
106#396—22no—
107#396—22no—
108#396—22no—
109#396—22no—
110#394—21no—
111#394—21no—
112#394—21no—
113#394—21no—
114#391C2×C412yes—
115#391C2×C412yes—
116#390—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#390C2×C412yes—
121#390C2×C412yes—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#388—22no—
127#388—22no—
128#388—22no—
129#388—22no—
130#386—22no—
131#386—22no—
132#384—21no—
133#382—21no—
134#380—21no—
135#380—21no—
136#380—22no—
137#380—22no—
138C4×C4⋊C4C2×C412yes—
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141#356—48no—
142#355—48no—
143#343—22no—
144#343—22no—
145#343—22no—
146#343—22no—
147C2×#41—44no—
148C2×#41—42no—
149C2×#41—42no—
150C2×#41—44no—
151C22×Q8—24no—
152C22×Q8—41no—
153C22×Q8—41no—
154C22×Q8—42no—
155C22×Q8C22×C412yes—
156C22×Q8C22×C411yescharacteristic
157C22×Q8C2×D411yescharacteristic
158C22×Q8C2×D411yescharacteristic
159C22×Q8C2411yescharacteristic
160C22×D4—22no—
161C22×D4—21no—
162C22×D4—21no—
163C22×D4C22×C412yes—
164C23×C4—22no—
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22×C411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C2×M16—24no—
173C2×M16—24no—
174C4⋊Q8—28no—
175C4⋊Q8—44no—
176C4⋊Q8—41no—
177C4⋊Q8—41no—
178#127—41no—
179#127—41no—
180#124—28no—
181#124—44no—
182#124—42no—
183#124—42no—
184#123—48no—
185#123—48no—
186C22⋊Q8—48no—
187C22⋊Q8—48no—
188#121—42no—
189#121—42no—
190#121—41no—
191#121—41no—
192#121—41no—
193#121—41no—
194C4×Q8—41no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×Q8—41no—
200C4×Q8—41no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8—41no—
204C4×Q8—41no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—41no—
208C4×Q8—41no—
209C4×Q8—41no—
210C4×D4—42no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—42no—
214#117—44no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22⋊C411yescharacteristic
233C2×C4⋊C4C22⋊C411yescharacteristic
234C2×C4⋊C4C22⋊C411yescharacteristic
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22×C411yescharacteristic
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C22⋊C411yescharacteristic
267C2×C42C22⋊C411yescharacteristic
268C2×C42C22⋊C411yescharacteristic
269C2×C42C22⋊C411yescharacteristic
270C2×C42C22⋊C411yescharacteristic
271C2×C42C22⋊C411yescharacteristic
272C2×C42C22×C412yes—
273C2×C42C22×C412yes—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C4⋊C8—42no—
281C4⋊C8—42no—
282C4⋊C8—42no—
283C4⋊C8—42no—
284#101—24no—
285#101—28no—
286#101—24no—
287#101—44no—
288#101—44no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293#97—42no—
294#97—42no—
295C4×C8—42no—
296C4×C8—42no—
297#95—22no—
298#95—21no—
299#95—21no—
300#95—22no—
301#95—22no—
302#95—22no—
303#95—22no—
304C24—22no—
305C24C2×C22⋊C411yescharacteristic
306#41—88no—
307#41—88no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—41no—
319C2×Q8—41no—
320C2×Q8—48no—
321C2×Q8—48no—
322C2×Q8—42no—
323C2×Q8—42no—
324C2×Q8H(5,𝔽2)14yes—
325C2×D4—24no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—41no—
336C2×D4—41no—
337C2×D4—82no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—21no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—44no—
363C22×C4—81no—
364C22×C4—81no—
365C22×C4—81no—
366C22×C4#9911yescharacteristic
367C22×C4#9911yescharacteristic
368C22×C4#9911yescharacteristic
369C22×C4#9911yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C23×C411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384M16—44no—
385M16—44no—
386M16—44no—
387M16—44no—
388C2×C8—44no—
389C2×C8—44no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—82no—
413C22⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—42no—
420C22⋊C4—42no—
421C22⋊C4—82no—
422C22⋊C4—82no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—24no—
450C42—24no—
451C23—21no—
452C23—21no—
453C23—41no—
454C23—41no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23C2×#9911yescharacteristic
463C23C2×#9911yescharacteristic
464C23C22×C22⋊C411yescharacteristic
465Q8—44no—
466Q8—44no—
467Q8—44no—
468Q8—44no—
469Q8—84no—
470D4—84no—
471D4—82no—
472D4—84no—
473D4—84no—
474D4—84no—
475D4—82no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—88no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4C2×#9911yescharacteristic
531C2×C4C2×#9911yescharacteristic
532C2×C4C2×#9911yescharacteristic
533C2×C4C2×#9911yescharacteristic
534C2×C4#41012yes—
535C2×C4#51812yes—
536C2×C4#53512yes—
537C2×C4#53512yes—
538C8—84no—
539C8—84no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—42no—
549C22—42no—
550C22—82no—
551C22—82no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22#230911yescharacteristic
558C22C22×#9911yescharacteristic
559C22C2×#41011yescharacteristic
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4#213312yes—
579C2—21no—
580C2—21no—
581C2—41no—
582C2—41no—
583C2—162no—
584C2#2179311yescharacteristic
585C2C2×#213311yescharacteristic
586C2#3336911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#532C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20727C2semidirect
3. #20727C2semidirect
4. #13228C2semidirect
5. #13227C2semidirect
6. #13226C2semidirect
7. #12702C2semidirect

Maximal subgroups

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

subgroupindex
#351782
#207272
#132282
#132272
#132262
#127022