Finite Group Databasebeta

← #127232

Subgroups — #127232

Order 512.

848 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20722C211yesmaximal, characteristic
2#17362C211yesmaximal, characteristic
3#13959C211yesmaximal, characteristic
4#7213C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#407—21no—
7C2×#383C411yescharacteristic, complemented
8#2280—21no—
9C2×#376C411yescharacteristic, complemented
10#1819C2212yes—
11#1799—21no—
12#1799C412yescomplemented
13#1781—21no—
14C4×#99—21no—
15C4×#99C412yescomplemented
16#1465—21no—
17#1465C412yescomplemented
18#1463—21no—
19#1462C2212yes—
20#1459—21no—
21#1459C412yescomplemented
22#1445—21no—
23#1438—21no—
24#1437C412yescomplemented
25#1437C2212yes—
26C22⋊#95—21no—
27C22⋊#95—21no—
28C23×D4D411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—41no—
32C2×C4×D4—41no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×C4×D4Q811yescharacteristic
44C2×#117—41no—
45C22×C4⋊C4D411yescharacteristic
46C22×C22⋊C4—22no—
47C22×C22⋊C4—22no—
48C22×C22⋊C4C2×C411yescharacteristic
49C22×C22⋊C4C2311yescharacteristic
50C22×C42C2×C411yescharacteristic
51C22×C42C2×C412yes—
52C2×#99—22no—
53C2×#99—22no—
54#407—22no—
55#407—22no—
56#407—41no—
57#407—41no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—41no—
60#383—41no—
61#383—41no—
62#381—41no—
63#380—22no—
64#380—41no—
65#379—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—41no—
68#376—41no—
69#376—41no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95C2×C412yes—
77#342—22no—
78#342C2×C414yes—
79#324—22no—
80#324—22no—
81#323—22no—
82#323C2×C414yes—
83C25C22⋊C411yescharacteristic
84C25C2×D411yescharacteristic
85C22×D4—41no—
86C22×D4—41no—
87C22×D4—42no—
88C22×D4—42no—
89C22×D4C22⋊C411yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C4⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C23×C4—21no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4C22⋊C411yescharacteristic
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C4⋊C411yescharacteristic
113C23×C4C4⋊C411yescharacteristic
114C23×C4C4⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—42no—
119C2×M16—42no—
120C2×M16—84no—
121C22×C8—42no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130C4×D4—41no—
131C4×D4—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
137C4×D4—41no—
138#117—84no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C4⋊C411yescharacteristic
145C2×C4⋊C4C4⋊C411yescharacteristic
146C2×C22⋊C4—24no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—44no—
153C2×C22⋊C4—44no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—82no—
158C2×C22⋊C4—82no—
159C2×C22⋊C4C4211yescharacteristic
160C2×C22⋊C4C4211yescharacteristic
161C2×C22⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4C4⋊C411yescharacteristic
163C2×C22⋊C4C22×C411yescharacteristic
164C2×C22⋊C4C22×C411yescharacteristic
165C2×C22⋊C4C2×D411yescharacteristic
166C2×C22⋊C4C2×Q811yescharacteristic
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42—44no—
173C2×C42—42no—
174C2×C42C4211yescharacteristic
175C2×C42C4211yescharacteristic
176C2×C42C4⋊C411yescharacteristic
177C2×C42C4⋊C411yescharacteristic
178#99—44no—
179#99—44no—
180C22⋊C8—24no—
181C22⋊C8—42no—
182C22⋊C8—42no—
183C22⋊C8—42no—
184#95—42no—
185#95—42no—
186#95—42no—
187#95—42no—
188#95—42no—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—82no—
194#95—82no—
195C24—22no—
196C24—22no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—41no—
202C24—41no—
203C24—42no—
204C24—42no—
205C24—42no—
206C24—42no—
207C24#9511yescharacteristic
208C24#9511yescharacteristic
209C24C2×C22⋊C411yescharacteristic
210C24C2×C22⋊C411yescharacteristic
211C24C2×C4⋊C411yescharacteristic
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—22no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4—48no—
225C2×D4—48no—
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—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—44no—
255C22×C4—44no—
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—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4#9511yescharacteristic
288C22×C4#9511yescharacteristic
289C22×C4#9511yescharacteristic
290C22×C4#9511yescharacteristic
291C22×C4#9511yescharacteristic
292C22×C4#9511yescharacteristic
293C22×C4#9911yescharacteristic
294C22×C4#9911yescharacteristic
295C22×C4#9911yescharacteristic
296C22×C4#9911yescharacteristic
297C22×C4C2×C4211yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300C22×C4C2×C4⋊C411yescharacteristic
301M16—84no—
302M16—84no—
303C2×C8—44no—
304C2×C8—84no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—84no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—88no—
321C22⋊C4—88no—
322C22⋊C4—84no—
323C22⋊C4—84no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—44no—
333C42—44no—
334C42—84no—
335C23—22no—
336C23—22no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—44no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23H(3,ℤ/4)12yes—
378C23C2×#9511yescharacteristic
379C23C2×#9911yescharacteristic
380C23C2×#9911yescharacteristic
381C23#42112yes—
382D4—44no—
383D4—44no—
384D4—44no—
385D4—44no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
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—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—81no—
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—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—84no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—82no—
451C2×C4#34211yescharacteristic
452C2×C4#34211yescharacteristic
453C2×C4#34211yescharacteristic
454C2×C4#34211yescharacteristic
455C8—84no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—22no—
467C22—22no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—81no—
483C22—81no—
484C22—84no—
485C22—84no—
486C22—84no—
487C22—84no—
488C22—81no—
489C22—81no—
490C22C2×H(3,ℤ/4)11yescharacteristic
491C22#173311yescharacteristic
492C22C2×#34211yescharacteristic
493C22#212012yes—
494C22#212712yes—
495C4—21no—
496C4—21no—
497C4—21no—
498C4—21no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—84no—
508C4—162no—
509C4—164no—
510C2—22no—
511C2—22no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—42no—
516C2—81no—
517C2—81no—
518C2#794611yescharacteristic
519C2#1257311yescharacteristic
520C2#1258211yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7213C2semidirect
7. C2×#383C4semidirect
9. C2×#376C4semidirect
12. #1799C4semidirect
15. C4×#99C4semidirect
17. #1465C4semidirect
21. #1459C4semidirect
24. #1437C4semidirect

Maximal subgroups

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

subgroupindex
#207222
#173622
#139592
#72132