Finite Group Databasebeta

← #99766

Subgroups — #99766

Order 512.

990 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1797C211yesmaximal, characteristic
2#14697C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1271C212yesmaximal, complemented
5#7025C212yesmaximal, complemented
6C23×C42C2211yescharacteristic
7C22×C4⋊C8—21no—
8C22×C22⋊C8—21no—
9C22×C22⋊C8—21no—
10C2×#389C411yescharacteristic
11C2×#389C2211yescharacteristic
12C2×#385—21no—
13C2×#380C411yescharacteristic
14C2×#380C411yescharacteristic
15#1797—24no—
16C2×#336—21no—
17C2×#336C2212yes—
18C2×#336C2212yes—
19#1436—21no—
20#1436—21no—
21C4⋊#95C411yescharacteristic
22C4⋊#95C2211yescharacteristic
23C22⋊#95—21no—
24#1271—24no—
25C24×C4C2×C411yescharacteristic
26C24×C4D411yescharacteristic
27C24×C4D411yescharacteristic
28C23×C8—22no—
29C23×C8—22no—
30C22×C4⋊C4—21no—
31C22×C4⋊C4—21no—
32C22×C4⋊C4C2×C411yescharacteristic
33C22×C4⋊C4C2×C411yescharacteristic
34C22×C4⋊C4C2×C411yescharacteristic
35C22×C4⋊C4C2×C411yescharacteristic
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C42—22no—
39C22×C42—22no—
40C22×C42—42no—
41C22×C42—42no—
42C22×C42C2×C411yescharacteristic
43C22×C42D411yescharacteristic
44C22×C42D411yescharacteristic
45C22×C42C2311yescharacteristic
46C2×C4⋊C8—24no—
47C2×C4⋊C8—42no—
48C2×C4⋊C8—42no—
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8—42no—
52C2×C22⋊C8—42no—
53C2×C22⋊C8—42no—
54C2×C22⋊C8—42no—
55#389—22no—
56#389—22no—
57#389—42no—
58#385—42no—
59#385—42no—
60#380—22no—
61#380—22no—
62#380—42no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—21no—
70C2×#95—21no—
71#336—22no—
72#336—22no—
73#336—22no—
74#336—42no—
75#336D414yes—
76C25C22⋊C411yescharacteristic
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—22no—
83C23×C4—21no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—22no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4—41no—
101C23×C4—41no—
102C23×C4—41no—
103C23×C4C22⋊C411yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4D812yes—
107C23×C4SD1612yes—
108C23×C4SD1612yes—
109C23×C4Q1612yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C22×C8—24no—
114C22×C8—24no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—42no—
121C22×C8—42no—
122C22×C8—42no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8—42no—
126C22×C8—42no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4M1612yes—
149C2×C4⋊C4M1612yes—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—42no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—48no—
165C2×C42—48no—
166C2×C42—41no—
167C2×C42—41no—
168C2×C42—41no—
169C2×C42—41no—
170C2×C42C2×D412yes—
171C2×C42#4112yes—
172C4⋊C8—48no—
173C22⋊C8—48no—
174C22⋊C8—48no—
175#95—24no—
176#95—42no—
177#95—42no—
178#95—42no—
179#95—42no—
180#95—42no—
181#95—41no—
182#95—41no—
183#95—42no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24—21no—
189C24—21no—
190C24—21no—
191C24—21no—
192C24—21no—
193C24—21no—
194C24—21no—
195C24—21no—
196C24#10112yes—
197C24#10212yes—
198C24#10312yes—
199C24C2×C22⋊C411yescharacteristic
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
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—24no—
217C22×C4—24no—
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—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
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—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4#9912yes—
293C22×C4#9912yes—
294C22×C4#10212yes—
295C22×C4#10312yes—
296C22×C4#10412yes—
297C22×C4C4×D412yes—
298C22×C4#12011yescharacteristic
299C22×C4#12112yes—
300C22×C4#12311yescharacteristic
301C22×C4C2×M1611yescharacteristic
302C22×C4C2×M1611yescharacteristic
303C22×C4#13112yes—
304C22×C4C2×D811yescharacteristic
305C22×C4C2×SD1611yescharacteristic
306C22×C4C2×SD1611yescharacteristic
307C22×C4C2×Q1611yescharacteristic
308C22×C4#13512yes—
309C22×C4AGL(1,ℤ/8)11yescharacteristic
310C22×C4#13711yescharacteristic
311C2×C8—44no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—82no—
318C2×C8—82no—
319C2×C8—82no—
320C2×C8—82no—
321C2×C8—82no—
322C2×C8—82no—
323C2×C8—82no—
324C2×C8—82no—
325C4⋊C4—24no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—82no—
330C4⋊C4—82no—
331C4⋊C4—84no—
332C4⋊C4—81no—
333C4⋊C4—81no—
334C4⋊C4—81no—
335C4⋊C4—81no—
336C22⋊C4—48no—
337C22⋊C4—84no—
338C42—24no—
339C42—24no—
340C42—42no—
341C42—42no—
342C42—44no—
343C42—44no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23#33214yes—
379C23#38611yescharacteristic
380C23#40711yescharacteristic
381C23#40811yescharacteristic
382C23C2×#9911yescharacteristic
383C23C2×#9911yescharacteristic
384C23C2×#10111yescharacteristic
385C23#41311yescharacteristic
386C23C2×#10211yescharacteristic
387C23C2×#10311yescharacteristic
388C23#41611yescharacteristic
389C23#41711yescharacteristic
390C23C2×#10411yescharacteristic
391C23#42111yescharacteristic
392C23#42111yescharacteristic
393C23#42111yescharacteristic
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—48no—
435C2×C4—48no—
436C2×C4—48no—
437C2×C4—48no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
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#32814yes—
479C2×C4#43512yes—
480C2×C4#43612yes—
481C2×C4#44711yescharacteristic
482C2×C4#44811yescharacteristic
483C2×C4#46512yes—
484C2×C4#46712yes—
485C2×C4#48011yescharacteristic
486C2×C4#48111yescharacteristic
487C8—84no—
488C8—84no—
489C22—24no—
490C22—24no—
491C22—24no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22#146311yescharacteristic
505C22#146511yescharacteristic
506C22#146811yescharacteristic
507C22#147211yescharacteristic
508C22#148111yescharacteristic
509C22#148211yescharacteristic
510C22#148611yescharacteristic
511C22#148711yescharacteristic
512C22#148811yescharacteristic
513C22#148911yescharacteristic
514C22#149211yescharacteristic
515C22#149511yescharacteristic
516C22C2×#32811yescharacteristic
517C22#150111yescharacteristic
518C22#151311yescharacteristic
519C22#151311yescharacteristic
520C22#151611yescharacteristic
521C22#151911yescharacteristic
522C22#151911yescharacteristic
523C22#152011yescharacteristic
524C22#152911yescharacteristic
525C22#153111yescharacteristic
526C22#153411yescharacteristic
527C22#153511yescharacteristic
528C22C2×#33211yescharacteristic
529C22#154011yescharacteristic
530C22#155011yescharacteristic
531C22#155211yescharacteristic
532C22#187011yescharacteristic
533C22#187111yescharacteristic
534C22#187411yescharacteristic
535C22#189211yescharacteristic
536C22#189711yescharacteristic
537C22#189811yescharacteristic
538C22#190111yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—24no—
543C4—22no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—21no—
548C4—42no—
549C4—42no—
550C4—82no—
551C4—82no—
552C4—161no—
553C4—161no—
554C2—44no—
555C2#761311yescharacteristic
556C2#761711yescharacteristic
557C2#764111yescharacteristic
558C2#764711yescharacteristic
559C2#765811yescharacteristic
560C2#766511yescharacteristic
561C2#771511yescharacteristic
562C2#773111yescharacteristic
563C2#894411yescharacteristic
564C2#907511yescharacteristic
565C2#908511yescharacteristic
566C2#911111yescharacteristic
567C2#912611yescharacteristic
568C2#913111yescharacteristic
569C2#915511yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1271C2semidirect
5. #7025C2semidirect

Maximal subgroups

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

subgroupindex
C2×#17972
#146972
#107122
C2×#12712
#70252