Finite Group Databasebeta

← #98783

Subgroups — #98783

Order 512.

998 conjugacy classes of subgroups.

The subgroup lattice has 579 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#390C211yesmaximal, characteristic
2#10832C211yesmaximal, characteristic, complemented
3#10795C211yesmaximal, characteristic, complemented
4C2×#1270C212yesmaximal
5#7025C212yesmaximal, complemented
6C22×C4×D4C411yescharacteristic
7C22×C4⋊C8—21no—
8C2×#390C411yescharacteristic
9C2×#390C2211yescharacteristic
10C2×C4×C4⋊C4C411yescharacteristic
11C2×C4×C22⋊C4—21no—
12C2×C4×C22⋊C4—21no—
13C2×C43C2211yescharacteristic
14C2×#336—21no—
15C2×#336—21no—
16C2×#336—21no—
17C2×#336C2212yes—
18C2×#336C2212yes—
19#1437—21no—
20#1437—21no—
21C4⋊#95C411yescharacteristic
22C4⋊#95C2211yescharacteristic
23#1270—24no—
24C23×D4C2×C411yescharacteristic
25C24×C4—21no—
26C23×C8—22no—
27C23×C8—22no—
28C2×C4×D4—22no—
29C2×C4×D4—42no—
30C2×C4×D4—42no—
31C2×C4×D4C814yescomplemented
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C22⋊C4—21no—
39C22×C22⋊C4—21no—
40C22×C42—21no—
41C22×C42C2×C411yescharacteristic
42C22×C42C2×C411yescharacteristic
43C22×C42D411yescharacteristic
44C22×C42D411yescharacteristic
45C22×C42D411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42C2311yescharacteristic
48C2×C4⋊C8—24no—
49C2×C4⋊C8—42no—
50C2×C4⋊C8—42no—
51#390—22no—
52#390—22no—
53#390—21no—
54#390—21no—
55#390D412yes—
56#390Q812yes—
57C4×C4⋊C4—42no—
58C4×C22⋊C4—44no—
59C4×C22⋊C4—42no—
60C4×C22⋊C4—42no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C43—42no—
67#336—24no—
68#336—22no—
69#336—22no—
70#336—22no—
71#336—22no—
72#336—42no—
73#336—42no—
74C25—21no—
75C22×D4—21no—
76C22×D4—21no—
77C22×D4—21no—
78C22×D4—41no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4C4⋊C412yes—
83C22×D4C2×C812yes—
84C22×D4M1612yes—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—24no—
89C23×C4—42no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C22×C8—24no—
103C22×C8—24no—
104C22×C8—42no—
105C22×C8—42no—
106C22×C8—42no—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8—42no—
110C22×C8—42no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—42no—
114C22×C8—42no—
115C22×C8—42no—
116C4×D4—48no—
117C4×D4—48no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4C4⋊C412yes—
128C2×C4⋊C4C2×C812yes—
129C2×C4⋊C4M1612yes—
130C2×C22⋊C4—24no—
131C2×C22⋊C4—24no—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—41no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—22no—
160C2×C42—22no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42—41no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—42no—
172C2×C42C2×C812yes—
173C2×C42M1612yes—
174C2×C42D812yes—
175C2×C42SD1612yes—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×Q811yescharacteristic
178C2×C42#4111yescharacteristic
179C2×C42#4111yescharacteristic
180C4⋊C8—48no—
181#95—42no—
182#95—42no—
183#95—42no—
184#95—41no—
185#95—41no—
186#95—42no—
187C24—22no—
188C24—22no—
189C24—22no—
190C24—41no—
191C24—41no—
192C24—41no—
193C24—41no—
194C24—41no—
195C24—41no—
196C24—41no—
197C24—41no—
198C24—41no—
199C24—41no—
200C24—41no—
201C24—41no—
202C24C2×C22⋊C411yescharacteristic
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—22no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4C4⋊C814yescomplemented
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—48no—
255C22×C4—48no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—48no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4C22⋊C812yes—
280C22×C4C22⋊C812yes—
281C22×C4#9912yes—
282C22×C4#10012yes—
283C22×C4#10212yes—
284C22×C4#10212yes—
285C22×C4#10412yes—
286C22×C4#10412yes—
287C22×C4C2×C4⋊C411yescharacteristic
288C22×C4#11711yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4C22⋊Q811yescharacteristic
292C22×C4C22⋊Q811yescharacteristic
293C22×C4C22×C811yescharacteristic
294C22×C4C2×M1611yescharacteristic
295C22×C4C2×M1611yescharacteristic
296C22×C4C2×M1611yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4C2×SD1611yescharacteristic
299C22×C4#13512yes—
300C22×C4AGL(1,ℤ/8)11yescharacteristic
301C22×C4AGL(1,ℤ/8)12yes—
302C22×C4#13711yescharacteristic
303C2×C8—44no—
304C2×C8—44no—
305C2×C8—44no—
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—82no—
310C2×C8—82no—
311C2×C8—82no—
312C2×C8—82no—
313C2×C8—82no—
314C2×C8—82no—
315C2×C8—82no—
316C2×C8—82no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C22⋊C4—48no—
324C22⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—24no—
336C42—42no—
337C42—42no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—44no—
343C42—44no—
344C42—42no—
345C42—42no—
346C42—84no—
347C23—24no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23—81no—
382C23#38011yescharacteristic
383C23C2×C22⋊C811yescharacteristic
384C23#40711yescharacteristic
385C23C2×#9911yescharacteristic
386C23#41011yescharacteristic
387C23C2×#10011yescharacteristic
388C23#41311yescharacteristic
389C23C2×#10211yescharacteristic
390C23#41611yescharacteristic
391C23#41711yescharacteristic
392C23#41811yescharacteristic
393C23C2×#10411yescharacteristic
394C23C2×#10411yescharacteristic
395C23#42111yescharacteristic
396C23#42111yescharacteristic
397D4—44no—
398D4—44no—
399D4—44no—
400D4—44no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—816no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4#32314yes—
482C2×C4D4⋊C814yes—
483C2×C4C2×C4⋊C811yescharacteristic
484C2×C4#42311yescharacteristic
485C2×C4#43111yescharacteristic
486C2×C4#43211yescharacteristic
487C2×C4#45911yescharacteristic
488C2×C4#46111yescharacteristic
489C2×C4#46312yes—
490C2×C4#47411yescharacteristic
491C2×C4#47611yescharacteristic
492C2×C4#47812yes—
493C8—84no—
494C8—84no—
495C22—44no—
496C22—44no—
497C22—44no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—82no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—82no—
509C22—82no—
510C22C2×#32311yescharacteristic
511C22#145811yescharacteristic
512C22#146511yescharacteristic
513C22#146611yescharacteristic
514C22C2×D4⋊C811yescharacteristic
515C22#147611yescharacteristic
516C22#147711yescharacteristic
517C22#147811yescharacteristic
518C22#148611yescharacteristic
519C22#148811yescharacteristic
520C22#148911yescharacteristic
521C22#149011yescharacteristic
522C22#150011yescharacteristic
523C22#150211yescharacteristic
524C22#150411yescharacteristic
525C22#150711yescharacteristic
526C22#151211yescharacteristic
527C22#151411yescharacteristic
528C22#151611yescharacteristic
529C22#151911yescharacteristic
530C22#152411yescharacteristic
531C22#152711yescharacteristic
532C22#153011yescharacteristic
533C22#153511yescharacteristic
534C22#154011yescharacteristic
535C22#154511yescharacteristic
536C22#155111yescharacteristic
537C22#155711yescharacteristic
538C22#179711yescharacteristic
539C22#179911yescharacteristic
540C22#180011yescharacteristic
541C22#180211yescharacteristic
542C22#180411yescharacteristic
543C22#180611yescharacteristic
544C22#180611yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—22no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—24no—
556C4—24no—
557C4—42no—
558C4—42no—
559C4—82no—
560C4—82no—
561C4—82no—
562C2—84no—
563C2#756011yescharacteristic
564C2#757211yescharacteristic
565C2#757811yescharacteristic
566C2#759411yescharacteristic
567C2#761811yescharacteristic
568C2#762111yescharacteristic
569C2#763311yescharacteristic
570C2#764711yescharacteristic
571C2#855711yescharacteristic
572C2#856311yescharacteristic
573C2#856511yescharacteristic
574C2#857311yescharacteristic
575C2#857711yescharacteristic
576C2#858111yescharacteristic
577C2#858811yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #10832C2semidirect
3. #10795C2semidirect
5. #7025C2semidirect
31. C2×C4×D4C8semidirect
218. C2×D4C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#3902
#108322
#107952
C2×#12702
#70252