Finite Group Databasebeta

← #99752

Subgroups — #99752

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1831C211yesmaximal, characteristic, complemented
2#14700C211yesmaximal, characteristic
3#10795C211yesmaximal, characteristic, complemented
4C2×#1270C212yesmaximal
5#7025C212yesmaximal, complemented
6C22×C4⋊C8—21no—
7C2×#390C411yescharacteristic
8C2×#390C411yescharacteristic
9C2×#390C2211yescharacteristic
10C2×#389C411yescharacteristic
11C2×#388—21no—
12C2×#388—21no—
13C2×C43C2211yescharacteristic
14#1831—24no—
15C2×#336—21no—
16C2×#336—21no—
17C2×#336—21no—
18C2×#336C2212yes—
19C2×#336C2212yes—
20#1437—21no—
21#1437—21no—
22C4⋊#95C411yescharacteristic
23C4⋊#95C2211yescharacteristic
24#1270—24no—
25C23×D4C2×C411yescharacteristic
26C23×C8—22no—
27C23×C8—22no—
28C22×C4⋊C4—21no—
29C22×C4⋊C4C2×C411yescharacteristic
30C22×C4⋊C4C2×C411yescharacteristic
31C22×C4⋊C4C2×C411yescharacteristic
32C22×C22⋊C4—21no—
33C22×C22⋊C4—21no—
34C22×C22⋊C4—21no—
35C22×C22⋊C4—21no—
36C22×C22⋊C4—21no—
37C22×C22⋊C4—21no—
38C22×C42C2×C411yescharacteristic
39C22×C42C2×C411yescharacteristic
40C22×C42D411yescharacteristic
41C22×C42D411yescharacteristic
42C22×C42D411yescharacteristic
43C22×C42D411yescharacteristic
44C22×C42C2311yescharacteristic
45C2×C4⋊C8—24no—
46C2×C4⋊C8—42no—
47C2×C4⋊C8—42no—
48#390—22no—
49#390—22no—
50#390—22no—
51#390—42no—
52#390C814yescomplemented
53#389—42no—
54#388—44no—
55#388—42no—
56#388—42no—
57C2×#95—21no—
58C2×#95—21no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—21no—
62C43—22no—
63C43—22no—
64#336—24no—
65#336—24no—
66#336—22no—
67#336—22no—
68#336—22no—
69#336—42no—
70#336D414yes—
71C25—21no—
72C22×D4—22no—
73C22×D4—21no—
74C22×D4—22no—
75C22×D4—41no—
76C22×D4—41no—
77C22×D4—41no—
78C22×D4—41no—
79C22×D4C2×C812yes—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—21no—
83C23×C4C22⋊C411yescharacteristic
84C23×C4C22⋊C411yescharacteristic
85C23×C4C22⋊C411yescharacteristic
86C23×C4C22⋊C411yescharacteristic
87C23×C4C22×C411yescharacteristic
88C23×C4C2×D411yescharacteristic
89C23×C4C2×D411yescharacteristic
90C22×C8—24no—
91C22×C8—24no—
92C22×C8—42no—
93C22×C8—42no—
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—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—41no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4—41no—
110C2×C4⋊C4—41no—
111C2×C4⋊C4—41no—
112C2×C4⋊C4—41no—
113C2×C4⋊C4—42no—
114C2×C4⋊C4—42no—
115C2×C4⋊C4—42no—
116C2×C4⋊C4C2×C812yes—
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4—42no—
121C2×C22⋊C4—42no—
122C2×C22⋊C4—42no—
123C2×C22⋊C4—42no—
124C2×C22⋊C4—42no—
125C2×C22⋊C4—42no—
126C2×C22⋊C4—42no—
127C2×C22⋊C4—42no—
128C2×C22⋊C4—42no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—41no—
132C2×C22⋊C4—41no—
133C2×C22⋊C4—41no—
134C2×C22⋊C4—41no—
135C2×C22⋊C4—41no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C42—22no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—21no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—21no—
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×C42C2×C812yes—
162C2×C42M1612yes—
163C2×C42D812yes—
164C2×C42SD1612yes—
165C2×C42C2×D412yes—
166C2×C42#4112yes—
167C4⋊C8—48no—
168#95—42no—
169#95—42no—
170#95—42no—
171#95—41no—
172#95—41no—
173#95—42no—
174C24—22no—
175C24—41no—
176C24—41no—
177C24—41no—
178C24—41no—
179C24—41no—
180C24—41no—
181C24—41no—
182C24—41no—
183C24—41no—
184C24—41no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—41no—
189C24C2×C22⋊C411yescharacteristic
190C2×D4—24no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—82no—
197C2×D4—82no—
198C2×D4—82no—
199C2×D4—82no—
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—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—22no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
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×C4C22⋊C812yes—
256C22×C4C22⋊C812yes—
257C22×C4#10012yes—
258C22×C4#10212yes—
259C22×C4#10212yes—
260C22×C4#10412yes—
261C22×C4C4×D412yes—
262C22×C4#12011yescharacteristic
263C22×C4#12112yes—
264C22×C4#12311yescharacteristic
265C22×C4C22×C811yescharacteristic
266C22×C4C2×M1611yescharacteristic
267C22×C4#13112yes—
268C22×C4C2×D811yescharacteristic
269C22×C4C2×SD1611yescharacteristic
270C22×C4AGL(1,ℤ/8)11yescharacteristic
271C22×C4AGL(1,ℤ/8)12yes—
272C22×C4AGL(1,ℤ/8)11yescharacteristic
273C22×C4#13711yescharacteristic
274C22×C4#13711yescharacteristic
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—82no—
282C2×C8—82no—
283C2×C8—82no—
284C2×C8—82no—
285C2×C8—82no—
286C2×C8—82no—
287C2×C8—82no—
288C2×C8—82no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—84no—
294C22⋊C4—48no—
295C22⋊C4—48no—
296C22⋊C4—84no—
297C22⋊C4—84no—
298C22⋊C4—81no—
299C22⋊C4—81no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—81no—
303C22⋊C4—81no—
304C42—24no—
305C42—24no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—42no—
311C42—42no—
312C42—44no—
313C42—44no—
314C42—42no—
315C42—42no—
316C42—42no—
317C42—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—81no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23—81no—
333C23—81no—
334C23—81no—
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#38611yescharacteristic
354C23C2×C22⋊C811yescharacteristic
355C23#40811yescharacteristic
356C23#41011yescharacteristic
357C23#41011yescharacteristic
358C23C2×#10011yescharacteristic
359C23#41311yescharacteristic
360C23C2×#10211yescharacteristic
361C23#41711yescharacteristic
362C23#41711yescharacteristic
363C23#41811yescharacteristic
364C23C2×#10411yescharacteristic
365C23#42111yescharacteristic
366C23#42111yescharacteristic
367C23#42111yescharacteristic
368D4—84no—
369D4—84no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—41no—
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—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
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×C4D4⋊C814yes—
447C2×C4C8×D412yes—
448C2×C4#43512yes—
449C2×C4#44711yescharacteristic
450C2×C4#44811yescharacteristic
451C2×C4#46812yes—
452C2×C4#46912yes—
453C2×C4#48011yescharacteristic
454C2×C4#48111yescharacteristic
455C8—84no—
456C8—84no—
457C22—82no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22—82no—
464C22—161no—
465C22—161no—
466C22—161no—
467C22—161no—
468C22—161no—
469C22—161no—
470C22—161no—
471C22—161no—
472C22#145811yescharacteristic
473C22#146011yescharacteristic
474C22#146911yescharacteristic
475C22#147311yescharacteristic
476C22C2×D4⋊C811yescharacteristic
477C22#147711yescharacteristic
478C22#147711yescharacteristic
479C22#147811yescharacteristic
480C22#148011yescharacteristic
481C22#148311yescharacteristic
482C22#149511yescharacteristic
483C22#149611yescharacteristic
484C22#150511yescharacteristic
485C22#150911yescharacteristic
486C22#151211yescharacteristic
487C22#151211yescharacteristic
488C22#151211yescharacteristic
489C22#151411yescharacteristic
490C22#151711yescharacteristic
491C22#151711yescharacteristic
492C22#152911yescharacteristic
493C22#153011yescharacteristic
494C22#153311yescharacteristic
495C22#153511yescharacteristic
496C22#154511yescharacteristic
497C22#154711yescharacteristic
498C22#155711yescharacteristic
499C22#156011yescharacteristic
500C22#186911yescharacteristic
501C22#187111yescharacteristic
502C22#187411yescharacteristic
503C22#189511yescharacteristic
504C22#189711yescharacteristic
505C22#189911yescharacteristic
506C22#190011yescharacteristic
507C4—22no—
508C4—22no—
509C4—22no—
510C4—22no—
511C4—21no—
512C4—21no—
513C4—21no—
514C4—21no—
515C4—24no—
516C4—42no—
517C4—42no—
518C4—42no—
519C4—82no—
520C4—161no—
521C4—161no—
522C2—162no—
523C2#757411yescharacteristic
524C2#758711yescharacteristic
525C2#759411yescharacteristic
526C2#760111yescharacteristic
527C2#769611yescharacteristic
528C2#770611yescharacteristic
529C2#771911yescharacteristic
530C2#773211yescharacteristic
531C2#893911yescharacteristic
532C2#907411yescharacteristic
533C2#908011yescharacteristic
534C2#911811yescharacteristic
535C2#912411yescharacteristic
536C2#914111yescharacteristic
537C2#915611yescharacteristic
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
1. C2×#1831C2semidirect
3. #10795C2semidirect
5. #7025C2semidirect
52. #390C8semidirect

Maximal subgroups

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

subgroupindex
C2×#18312
#147002
#107952
C2×#12702
#70252