Finite Group Databasebeta

← #98812

Subgroups — #98812

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 557 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14700C211yesmaximal, characteristic
2#10832C211yesmaximal, characteristic, complemented
3#10832C211yesmaximal, characteristic, complemented
4C2×#1270C212yesmaximal
5#7025C212yesmaximal, complemented
6C22×C4⋊C8—21no—
7C22×C4⋊C8—21no—
8C2×#390C411yescharacteristic
9C2×#390C411yescharacteristic
10C2×#390C2211yescharacteristic
11C2×#389C411yescharacteristic
12C2×#388—21no—
13C2×#388—21no—
14C2×C43C2211yescharacteristic
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
25C23×C8—22no—
26C23×C8—22no—
27C22×C4⋊C4—21no—
28C22×C4⋊C4C2×C411yescharacteristic
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C22⋊C4—21no—
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C42C2×C411yescharacteristic
38C22×C42C2×C411yescharacteristic
39C22×C42D411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42C22×C42D411yescharacteristic
43C22×C42C2311yescharacteristic
44C2×C4⋊C8—24no—
45C2×C4⋊C8—24no—
46C2×C4⋊C8—42no—
47C2×C4⋊C8—42no—
48C2×C4⋊C8—42no—
49C2×C4⋊C8—42no—
50#390—21no—
51#390—21no—
52#390—42no—
53#390—42no—
54#390D412yes—
55#390Q812yes—
56#389—42no—
57#388—42no—
58#388—42no—
59#388—42no—
60#388—42no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C43—42no—
67#336—22no—
68#336—22no—
69#336—22no—
70#336—22no—
71#336—42no—
72#336—42no—
73C25—21no—
74C22×D4—21no—
75C22×D4—22no—
76C22×D4—22no—
77C22×D4—41no—
78C22×D4—41no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4C4⋊C412yes—
82C23×C4—21no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22⋊C411yescharacteristic
88C23×C4C22⋊C411yescharacteristic
89C23×C4C22×C411yescharacteristic
90C23×C4C2×D411yescharacteristic
91C23×C4C2×D411yescharacteristic
92C22×C8—24no—
93C22×C8—24no—
94C22×C8—42no—
95C22×C8—42no—
96C22×C8—42no—
97C22×C8—42no—
98C22×C8—42no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8—42no—
102C22×C8—42no—
103C22×C8—42no—
104C22×C8—42no—
105C22×C8—42no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—41no—
114C2×C4⋊C4—41no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4—42no—
117C2×C4⋊C4—42no—
118C2×C4⋊C4C4⋊C412yes—
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—22no—
122C2×C22⋊C4—22no—
123C2×C22⋊C4—41no—
124C2×C22⋊C4—41no—
125C2×C22⋊C4—41no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—41no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—42no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
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×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42D812yes—
164C2×C42D812yes—
165C2×C42SD1612yes—
166C2×C42SD1612yes—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×Q811yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C4⋊C8—48no—
172C4⋊C8—48no—
173#95—42no—
174#95—42no—
175#95—42no—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—41no—
180C24—22no—
181C24—41no—
182C24—41no—
183C24—41no—
184C24—41no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24—41no—
190C24—41no—
191C24—41no—
192C24—41no—
193C24—41no—
194C24—41no—
195C24C2×C22⋊C411yescharacteristic
196C2×D4—24no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—82no—
203C2×D4—82no—
204C2×D4—82no—
205C2×D4—82no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
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—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4#9912yes—
260C22×C4#10012yes—
261C22×C4#10212yes—
262C22×C4#10212yes—
263C22×C4#10212yes—
264C22×C4#10212yes—
265C22×C4#10412yes—
266C22×C4#10412yes—
267C22×C4C2×C4⋊C411yescharacteristic
268C22×C4#11711yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4C22⋊Q811yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4#13112yes—
274C22×C4#13112yes—
275C22×C4C2×D811yescharacteristic
276C22×C4C2×D811yescharacteristic
277C22×C4C2×SD1611yescharacteristic
278C22×C4C2×SD1611yescharacteristic
279C22×C4AGL(1,ℤ/8)11yescharacteristic
280C22×C4AGL(1,ℤ/8)11yescharacteristic
281C22×C4#13711yescharacteristic
282C22×C4#13711yescharacteristic
283C2×C8—44no—
284C2×C8—44no—
285C2×C8—44no—
286C2×C8—44no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—82no—
290C2×C8—82no—
291C2×C8—82no—
292C2×C8—82no—
293C2×C8—82no—
294C2×C8—82no—
295C2×C8—82no—
296C2×C8—82no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—84no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—84no—
311C22⋊C4—84no—
312C22⋊C4—84no—
313C22⋊C4—84no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—42no—
319C42—42no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—44no—
325C42—44no—
326C42—42no—
327C42—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—81no—
338C23—81no—
339C23—81no—
340C23—81no—
341C23—81no—
342C23—81no—
343C23—81no—
344C23—81no—
345C23—81no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23#38011yescharacteristic
364C23#40811yescharacteristic
365C23#40811yescharacteristic
366C23C2×#9911yescharacteristic
367C23#41011yescharacteristic
368C23C2×#10011yescharacteristic
369C23#41311yescharacteristic
370C23C2×#10211yescharacteristic
371C23C2×#10211yescharacteristic
372C23#41711yescharacteristic
373C23#41711yescharacteristic
374C23C2×#10411yescharacteristic
375C23C2×#10411yescharacteristic
376C23#42111yescharacteristic
377C23#42111yescharacteristic
378D4—84no—
379D4—84no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—22no—
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—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4#32714yes—
460C2×C4#33114yes—
461C2×C4#42412yes—
462C2×C4#43312yes—
463C2×C4#45911yescharacteristic
464C2×C4#45911yescharacteristic
465C2×C4#46111yescharacteristic
466C2×C4#46111yescharacteristic
467C2×C4#47411yescharacteristic
468C2×C4#47411yescharacteristic
469C2×C4#47611yescharacteristic
470C2×C4#47611yescharacteristic
471C8—84no—
472C8—84no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—161no—
481C22—161no—
482C22—161no—
483C22—161no—
484C22—161no—
485C22—161no—
486C22—161no—
487C22—161no—
488C22#146811yescharacteristic
489C22#146911yescharacteristic
490C22#147111yescharacteristic
491C22#147311yescharacteristic
492C22#148011yescharacteristic
493C22#148011yescharacteristic
494C22#148311yescharacteristic
495C22#148311yescharacteristic
496C22#149211yescharacteristic
497C22#149211yescharacteristic
498C22#149511yescharacteristic
499C22#149511yescharacteristic
500C22C2×#32711yescharacteristic
501C22#150211yescharacteristic
502C22#150411yescharacteristic
503C22#150911yescharacteristic
504C22#151211yescharacteristic
505C22#151311yescharacteristic
506C22#151711yescharacteristic
507C22#151911yescharacteristic
508C22#152411yescharacteristic
509C22#152711yescharacteristic
510C22#153011yescharacteristic
511C22#153511yescharacteristic
512C22C2×#33111yescharacteristic
513C22#154711yescharacteristic
514C22#154711yescharacteristic
515C22#156011yescharacteristic
516C22#179811yescharacteristic
517C22#179911yescharacteristic
518C22#180011yescharacteristic
519C22#180211yescharacteristic
520C22#180211yescharacteristic
521C22#180611yescharacteristic
522C22#180611yescharacteristic
523C4—22no—
524C4—22no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—21no—
530C4—21no—
531C4—21no—
532C4—21no—
533C4—24no—
534C4—24no—
535C4—42no—
536C4—42no—
537C4—82no—
538C4—161no—
539C4—161no—
540C2—162no—
541C2#765911yescharacteristic
542C2#766411yescharacteristic
543C2#767811yescharacteristic
544C2#768611yescharacteristic
545C2#769611yescharacteristic
546C2#771011yescharacteristic
547C2#771411yescharacteristic
548C2#773211yescharacteristic
549C2#856111yescharacteristic
550C2#856711yescharacteristic
551C2#856711yescharacteristic
552C2#857111yescharacteristic
553C2#857711yescharacteristic
554C2#858111yescharacteristic
555C2#858511yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC23×D4C2×C4
LCSG ▷ C23×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. #10832C2semidirect
5. #7025C2semidirect

Maximal subgroups

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

subgroupindex
#147002
#108322
C2×#12702
#70252