Finite Group Databasebeta

← #214589

Subgroups — #214589

Order 512.

986 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35279C211yesmaximal, characteristic
2#21853C211yesmaximal, characteristic
3#20721C211yesmaximal, characteristic
4#9136C212yesmaximal, complemented
5#9136C212yesmaximal, complemented
6#9135C212yesmaximal, complemented
7#9128C212yesmaximal, complemented
8#8727C212yesmaximal, complemented
9#8342C212yesmaximal, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2857C2212yes—
17#2857C2212yes—
18#2628—21no—
19#2628—21no—
20#2568C412yes—
21#2567—21no—
22#2563—21no—
23#2542C412yes—
24#2487—21no—
25#2487—21no—
26#2409C412yes—
27#2409C412yes—
28#2317—21no—
29#2311—21no—
30#2285C2211yescharacteristic
31#1985—21no—
32#1955—21no—
33#1870—21no—
34#1870—21no—
35#1818—21no—
36#1797C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1797C2212yes—
40#1780—21no—
41#1750—21no—
42#1750—21no—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#333C2212yes—
46C2×#333C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1296C2212yes—
51C22×#41D411yescharacteristic
52C23×Q8C2×C412yes—
53C22×M16—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—41no—
57C2×C4⋊Q8—41no—
58C2×C4⋊Q8—41no—
59C2×C4⋊Q8—41no—
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2312yes—
62C2×C22⋊Q8C2×C414yes—
63C2×C22⋊Q8C2×C414yes—
64C4×#41—41no—
65C4×#41—41no—
66C4×#41—41no—
67C4×#41—41no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×#117—21no—
77C2×#117D411yescharacteristic
78C2×#117D411yescharacteristic
79C2×#117D411yescharacteristic
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85#432—41no—
86#432—41no—
87#432—41no—
88#432—41no—
89#431—41no—
90#431—41no—
91#423—41no—
92#423—41no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×C22⋊C8—22no—
99C2×C22⋊C8—22no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×#97—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108#400—22no—
109#400—22no—
110#399—22no—
111#399—22no—
112#395—22no—
113#395—22no—
114#393—22no—
115#393—22no—
116#391—22no—
117#391C2×C414yes—
118#391C2×C414yes—
119#390—22no—
120#389—22no—
121#389—22no—
122#389—22no—
123#389—22no—
124#386—22no—
125#386—22no—
126#385—22no—
127#385—22no—
128#384—22no—
129#384C2312yes—
130#382—22no—
131#382—21no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—22no—
136#380—21no—
137#380C2×C412yes—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2×C412yes—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#333—42no—
145#333—42no—
146#333—42no—
147#333—42no—
148C2×#41—24no—
149C2×#41—41no—
150C2×#41—41no—
151C2×#41—42no—
152C22×Q8—22no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—84no—
156C22×Q8C22⋊C411yescharacteristic
157C22×Q8C22×C414yes—
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22⋊C411yescharacteristic
160C22×D4C22⋊C411yescharacteristic
161C23×C4—22no—
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C2×M16—42no—
172C2×M16—82no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187C22⋊Q8—84no—
188C22⋊Q8—84no—
189C4×Q8—41no—
190C4×Q8—41no—
191C4×Q8—41no—
192C4×Q8—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4—41no—
196C4×D4—41no—
197C4×D4—41no—
198C4×D4—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205#117—41no—
206#117—41no—
207#117—41no—
208#117—41no—
209#117—41no—
210#117—41no—
211#117—44no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4C22⋊C411yescharacteristic
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22⋊C411yescharacteristic
228C2×C4⋊C4C22⋊C411yescharacteristic
229C2×C4⋊C4C22⋊C411yescharacteristic
230C2×C4⋊C4C22×C414yes—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4C22⋊C411yescharacteristic
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22⋊C411yescharacteristic
239C2×C22⋊C4C22⋊C411yescharacteristic
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—22no—
253C2×C42C22×C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C22×C411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2411yescharacteristic
263C4⋊C8—22no—
264C4⋊C8—22no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—22no—
271C4⋊C8—42no—
272C22⋊C8—44no—
273C22⋊C8—42no—
274C22⋊C8—42no—
275C22⋊C8—42no—
276C22⋊C8—44no—
277C22⋊C8—82no—
278C22⋊C8—82no—
279#97—42no—
280#97—42no—
281C4×C8—42no—
282#95—22no—
283#95—22no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95—24no—
288C24C2×C22⋊C411yescharacteristic
289C24C2×C22⋊C411yescharacteristic
290C24C2×C22⋊C411yescharacteristic
291#41—48no—
292#41—48no—
293C2×Q8—22no—
294C2×Q8—24no—
295C2×Q8—24no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—84no—
302C2×Q8—88no—
303C2×Q8—88no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—84no—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4#13714yes—
359C22×C4C23×C411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362M16—84no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—84no—
392C4⋊C4—84no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—41no—
398C22⋊C4—41no—
399C22⋊C4—44no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—21no—
417C42—21no—
418C42—21no—
419C42—21no—
420C42—44no—
421C42H(5,𝔽2)11yescharacteristic
422C42H(5,𝔽2)11yescharacteristic
423C42H(5,𝔽2)11yescharacteristic
424C42H(5,𝔽2)11yescharacteristic
425C23—21no—
426C23—22no—
427C23—21no—
428C23—22no—
429C23—21no—
430C23—21no—
431C23—22no—
432C23—22no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—22no—
439C23#41912yes—
440C23C22×C22⋊C411yescharacteristic
441Q8—44no—
442Q8—84no—
443Q8—84no—
444Q8—88no—
445D4—44no—
446D4—44no—
447D4—44no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—88no—
497C2×C4#41912yes—
498C2×C4#51811yescharacteristic
499C2×C4#51811yescharacteristic
500C2×C4#53411yescharacteristic
501C2×C4#53411yescharacteristic
502C2×C4#53411yescharacteristic
503C2×C4#53411yescharacteristic
504C2×C4C2×#13712yes—
505C2×C4#57512yes—
506C2×C4#57712yes—
507C2×C4#57812yes—
508C8—82no—
509C8—82no—
510C8—82no—
511C8—82no—
512C22—22no—
513C22—24no—
514C22—41no—
515C22—42no—
516C22—41no—
517C22—41no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—41no—
526C22—41no—
527C22#230811yescharacteristic
528C22#288812yes—
529C22C2×#41911yescharacteristic
530C22#289611yescharacteristic
531C22#289812yes—
532C4—22no—
533C4—21no—
534C4—21no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—41no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—41no—
543C4—84no—
544C4—84no—
545C4#319612yes—
546C4#319612yes—
547C4#320011yescharacteristic
548C4#320112yes—
549C4#320211yescharacteristic
550C2—42no—
551C2—81no—
552C2—81no—
553C2#1776012yes—
554C2#1776812yes—
555C2#2175511yescharacteristic
556C2#2176811yescharacteristic
557C2#2177111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9136C2semidirect
5. #9136C2semidirect
6. #9135C2semidirect
7. #9128C2semidirect
8. #8727C2semidirect
9. #8342C2semidirect

Maximal subgroups

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

subgroupindex
#352792
#218532
#207212
#91362
#91352
#91282
#87272
#83422