Finite Group Databasebeta

← #99746

Subgroups — #99746

Order 512.

964 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1817C211yesmaximal, characteristic
2#14697C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1271C212yesmaximal, complemented
5#7025C212yesmaximal, complemented
6C23×C42C2211yescharacteristic
7C22×C22⋊C8—21no—
8C22×C22⋊C8—21no—
9C2×#389C411yescharacteristic
10C2×#389C2211yescharacteristic
11C2×#385—21no—
12C2×#380C411yescharacteristic
13C2×#380C411yescharacteristic
14#1817—24no—
15C2×#336—21no—
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×C22⋊C8—24no—
47C2×C22⋊C8—24no—
48C2×C22⋊C8—42no—
49C2×C22⋊C8—42no—
50C2×C22⋊C8—42no—
51C2×C22⋊C8—42no—
52#389—22no—
53#389—22no—
54#389—42no—
55#385—44no—
56#380—22no—
57#380—42no—
58#380C814yescomplemented
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C2×#95—21no—
63C2×#95—21no—
64C2×#95—21no—
65C2×#95—21no—
66C2×#95—21no—
67#336—24no—
68#336—22no—
69#336—22no—
70#336—22no—
71#336—42no—
72#336D414yes—
73C25C22⋊C411yescharacteristic
74C23×C4—22no—
75C23×C4—22no—
76C23×C4—22no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4—22no—
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—21no—
93C23×C4—21no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4—41no—
99C23×C4—41no—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C411yescharacteristic
103C23×C4C2×C812yes—
104C23×C4M1612yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C22×C8—24no—
109C22×C8—24no—
110C22×C8—42no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—42no—
114C22×C8—42no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—42no—
121C22×C8—42no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4C2×C812yes—
142C2×C4⋊C4C2×C812yes—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—42no—
148C2×C22⋊C4—42no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—21no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—48no—
162C2×C42—48no—
163C2×C42C2×D412yes—
164C2×C42#4112yes—
165C22⋊C8—48no—
166C22⋊C8—48no—
167#95—24no—
168#95—42no—
169#95—42no—
170#95—42no—
171#95—42no—
172#95—41no—
173#95—41no—
174#95—42no—
175#95—42no—
176C24—21no—
177C24—21no—
178C24—21no—
179C24—21no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24C22⋊C812yes—
189C24#9912yes—
190C24#10112yes—
191C24C2×C22⋊C411yescharacteristic
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
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—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—41no—
255C22×C4—41no—
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—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4C22⋊C812yes—
281C22×C4#9912yes—
282C22×C4#10412yes—
283C22×C4C4×D412yes—
284C22×C4#12011yescharacteristic
285C22×C4#12112yes—
286C22×C4#12311yescharacteristic
287C22×C4C22×C811yescharacteristic
288C22×C4C2×M1611yescharacteristic
289C22×C4#13112yes—
290C22×C4#13512yes—
291C22×C4AGL(1,ℤ/8)12yes—
292C22×C4AGL(1,ℤ/8)11yescharacteristic
293C22×C4#13712yes—
294C22×C4#13711yescharacteristic
295C2×C8—44no—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8—82no—
302C2×C8—82no—
303C2×C8—82no—
304C2×C8—82no—
305C2×C8—82no—
306C2×C8—82no—
307C2×C8—82no—
308C2×C8—82no—
309C4⋊C4—24no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—82no—
314C4⋊C4—82no—
315C4⋊C4—84no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C22⋊C4—48no—
319C22⋊C4—84no—
320C42—42no—
321C42—42no—
322C42—44no—
323C42—44no—
324C42—42no—
325C42—42no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23#32414yes—
361C23#38611yescharacteristic
362C23C2×C22⋊C811yescharacteristic
363C23#40811yescharacteristic
364C23C2×#9911yescharacteristic
365C23C2×#9911yescharacteristic
366C23C2×#10111yescharacteristic
367C23#41311yescharacteristic
368C23#41611yescharacteristic
369C23#41711yescharacteristic
370C23#41811yescharacteristic
371C23#41911yescharacteristic
372C23C2×#10411yescharacteristic
373C23#42111yescharacteristic
374C23#42111yescharacteristic
375C23#42111yescharacteristic
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—48no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
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×C4C8×D412yes—
456C2×C4#43512yes—
457C2×C4#45212yes—
458C2×C4#46812yes—
459C2×C4#46912yes—
460C2×C4#48512yes—
461C8—84no—
462C8—84no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22C2×#32411yescharacteristic
479C22#145911yescharacteristic
480C22#146811yescharacteristic
481C22#147211yescharacteristic
482C22#147611yescharacteristic
483C22#147711yescharacteristic
484C22#147811yescharacteristic
485C22#147911yescharacteristic
486C22#148211yescharacteristic
487C22#148511yescharacteristic
488C22#149511yescharacteristic
489C22#149611yescharacteristic
490C22#150411yescharacteristic
491C22#150711yescharacteristic
492C22#151011yescharacteristic
493C22#151111yescharacteristic
494C22#151111yescharacteristic
495C22#151311yescharacteristic
496C22#151611yescharacteristic
497C22#151611yescharacteristic
498C22#152911yescharacteristic
499C22#153111yescharacteristic
500C22#153411yescharacteristic
501C22#153511yescharacteristic
502C22#155211yescharacteristic
503C22#155311yescharacteristic
504C22#155611yescharacteristic
505C22#155911yescharacteristic
506C22#186911yescharacteristic
507C22#187111yescharacteristic
508C22#187311yescharacteristic
509C22#189511yescharacteristic
510C22#189711yescharacteristic
511C22#189811yescharacteristic
512C22#190111yescharacteristic
513C4—22no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—22no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—82no—
523C4—82no—
524C4—161no—
525C4—161no—
526C2—44no—
527C2#757911yescharacteristic
528C2#758211yescharacteristic
529C2#759311yescharacteristic
530C2#760011yescharacteristic
531C2#769511yescharacteristic
532C2#770511yescharacteristic
533C2#771811yescharacteristic
534C2#773111yescharacteristic
535C2#893811yescharacteristic
536C2#907411yescharacteristic
537C2#908111yescharacteristic
538C2#911711yescharacteristic
539C2#912611yescharacteristic
540C2#913811yescharacteristic
541C2#915411yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC22×C4C22×C8
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×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
58. #380C8semidirect

Maximal subgroups

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

subgroupindex
C2×#18172
#146972
#107122
C2×#12712
#70252