Finite Group Databasebeta

← #190038

Subgroups — #190038

Order 512.

1003 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34245C211yesmaximal, characteristic
2C2×#1472C211yesmaximal, characteristic
3C2×#1472C211yesmaximal, characteristic
4#8737C214yesmaximal, complemented
5#8735C214yesmaximal, complemented
6#7929C211yesmaximal, characteristic
7#7927C211yesmaximal, characteristic
8#7925C211yesmaximal, characteristic
9C2×#1282C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C2×#407C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2368—22no—
16#2362—22no—
17#2344C412yes—
18#2342C414yes—
19Q8×C22⋊C4C412yes—
20#2332—22no—
21#2313C2212yes—
22C4×C22⋊Q8—22no—
23#2280C2211yescharacteristic
24#2274C2212yes—
25C2×C4×C22⋊C4C2211yescharacteristic
26#1472—22no—
27#1472—22no—
28#1472—22no—
29#1472—22no—
30C2×#324C2214yes—
31C2×#324C2214yes—
32#1282C2212yes—
33#1282C2212yes—
34#1282C2214yes—
35#1282C2214yes—
36#1282C2212yes—
37#1282C2212yes—
38C23×Q8C2×C411yescharacteristic
39C24×C4C2311yescharacteristic
40C22×M16—21no—
41C22×M16—21no—
42C23×C8—21no—
43C23×C8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8D412yes—
53C2×C22⋊Q8D412yes—
54C2×C22⋊Q8D412yes—
55C2×C22⋊Q8D412yes—
56C2×C4×Q8—22no—
57C2×#117—21no—
58C2×#117C2×C414yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4C2×C412yes—
62C22×C22⋊C4C2×C412yes—
63C22×C42—21no—
64#407—21no—
65#407—21no—
66#407—21no—
67#407—21no—
68#407—21no—
69#407—21no—
70#407—21no—
71#407—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×C22⋊C8C2312yes—
87C2×C22⋊C8C2312yes—
88#391—24no—
89#391—22no—
90#389—22no—
91#388—24no—
92#388—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#385—24no—
97#381—21no—
98#381C2×C414yes—
99#380—21no—
100#380C2×C414yes—
101#380C2×C412yes—
102#379C2×C412yes—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C22⋊C4—21no—
107C4×C22⋊C4—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—24no—
110C4×C22⋊C4C2×C414yes—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4C2311yescharacteristic
113C4×C22⋊C4C2312yes—
114C4×C22⋊C4C2311yescharacteristic
115C2×#95—21no—
116#324—28no—
117#324—28no—
118C25C22×C411yescharacteristic
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—82no—
124C22×Q8C22⋊C412yes—
125C22×Q8C22⋊C412yes—
126C23×C4—22no—
127C23×C4—82no—
128C23×C4C22×C412yes—
129C23×C4C22×C412yes—
130C23×C4C22×C411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4#4111yescharacteristic
136C23×C4#4111yescharacteristic
137C23×C4#4111yescharacteristic
138C23×C4#4111yescharacteristic
139C23×C4#4111yescharacteristic
140C23×C4#4111yescharacteristic
141C23×C4#4111yescharacteristic
142C23×C4#4111yescharacteristic
143C23×C4C2411yescharacteristic
144C2×M16—41no—
145C2×M16—41no—
146C2×M16—41no—
147C2×M16—41no—
148C2×M16—82no—
149C2×M16—82no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—41no—
155C22×C8—41no—
156C22×C8—41no—
157C22×C8—41no—
158C22×C8—82no—
159C22×C8—82no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168C22⋊Q8—22no—
169C22⋊Q8—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C4×Q8—84no—
177#117—28no—
178#117—44no—
179#117—82no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2×D414yes—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4C22⋊C414yes—
205C2×C22⋊C4C22⋊C414yes—
206C2×C22⋊C4C22×C412yes—
207C2×C22⋊C4C22×C414yes—
208C2×C22⋊C4C22×C412yes—
209C2×C42—24no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—82no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—41no—
225C22⋊C8—41no—
226C22⋊C8—41no—
227C22⋊C8—41no—
228C22⋊C8—41no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232#95—22no—
233#95—22no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—21no—
239#95—21no—
240#95—22no—
241C24—22no—
242C24—22no—
243C24—42no—
244C24—81no—
245C24C2×C22⋊C411yescharacteristic
246C24C2×C22⋊C411yescharacteristic
247C24#11711yescharacteristic
248C24#11711yescharacteristic
249C24#11711yescharacteristic
250C24#11711yescharacteristic
251C24C23×C411yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—44no—
257C2×Q8—82no—
258C2×Q8—82no—
259C2×Q8—82no—
260C2×Q8—82no—
261C2×Q8—82no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4C2×C22⋊C414yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C414yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4#11711yescharacteristic
294C22×C4#11711yescharacteristic
295C22×C4#11711yescharacteristic
296C22×C4#11711yescharacteristic
297C22×C4#12311yescharacteristic
298C22×C4#12311yescharacteristic
299C22×C4#12311yescharacteristic
300C22×C4#12311yescharacteristic
301C22×C4#12311yescharacteristic
302C22×C4#12311yescharacteristic
303C22×C4#12311yescharacteristic
304C22×C4#12311yescharacteristic
305C22×C4#12311yescharacteristic
306C22×C4#12311yescharacteristic
307C22×C4#12311yescharacteristic
308C22×C4#12311yescharacteristic
309C22×C4#12311yescharacteristic
310C22×C4#12311yescharacteristic
311C22×C4#12311yescharacteristic
312C22×C4#12311yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4C2×#4111yescharacteristic
319M16—82no—
320M16—82no—
321M16—82no—
322M16—82no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—82no—
328C2×C8—82no—
329C2×C8—82no—
330C2×C8—82no—
331C2×C8—82no—
332C2×C8—82no—
333C2×C8—82no—
334C2×C8—82no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—48no—
344C4⋊C4—44no—
345C4⋊C4—84no—
346C22⋊C4—22no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—22no—
352C22⋊C4—24no—
353C22⋊C4—22no—
354C22⋊C4—24no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—82no—
358C22⋊C4#13114yes—
359C42—48no—
360C42—44no—
361C42—81no—
362C42—81no—
363C42—82no—
364C42—84no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—21no—
370C23—21no—
371C23—22no—
372C23—22no—
373C23—21no—
374C23—21no—
375C23—22no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—84no—
388C23—84no—
389C23#38111yescharacteristic
390C23#38111yescharacteristic
391C23#38111yescharacteristic
392C23#38111yescharacteristic
393C23C22×C22⋊C411yescharacteristic
394C23C2×#11711yescharacteristic
395C23C2×#11711yescharacteristic
396C23C2×#13112yes—
397C23#56812yes—
398Q8—44no—
399Q8—44no—
400Q8—82no—
401Q8—82no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
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—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—42no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—88no—
450C2×C4#38111yescharacteristic
451C2×C4#38111yescharacteristic
452C2×C4#38111yescharacteristic
453C2×C4#38111yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C2×C4C2×#12311yescharacteristic
456C2×C4C2×#12311yescharacteristic
457C2×C4C2×#12311yescharacteristic
458C8—82no—
459C8—82no—
460C8—82no—
461C8—82no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—42no—
480C22—42no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—44no—
485C22—82no—
486C22—81no—
487C22—81no—
488C22—82no—
489C22#183012yes—
490C22C2×#38111yescharacteristic
491C22#288012yes—
492C22#292211yescharacteristic
493C22#292311yescharacteristic
494C22#292512yes—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—21no—
502C4—44no—
503C4—44no—
504C4—44no—
505C4—84no—
506C4—82no—
507C4#182812yes—
508C2—22no—
509C2—22no—
510C2—21no—
511C2—21no—
512C2—22no—
513C2—41no—
514C2—41no—
515C2—41no—
516C2#2068411yescharacteristic
517C2C2×#182811yescharacteristic
518C2C2×#183011yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8737C2semidirect
5. #8735C2semidirect

Maximal subgroups

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

subgroupindex
#342452
C2×#14722
#87372
#87352
#79292
#79272
#79252
C2×#12822