Finite Group Databasebeta

← #127233

Subgroups — #127233

Order 512.

825 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20722C211yesmaximal, characteristic
2#17364C211yesmaximal, characteristic
3#13940C211yesmaximal, characteristic
4#7213C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6#2920—21no—
7#2280—21no—
8C2×#381C411yescharacteristic, complemented
9C2×C4×C22⋊C4C411yescharacteristic, complemented
10#1819C2212yes—
11#1799—21no—
12#1799C412yescomplemented
13#1781—21no—
14C4×#99—21no—
15C4×#99C412yescomplemented
16#1466—21no—
17#1466C412yescomplemented
18#1462—21no—
19#1462C2212yes—
20#1460—21no—
21#1460C412yescomplemented
22#1440—21no—
23#1438—21no—
24#1437C2212yes—
25#1436C412yescomplemented
26#1434—21no—
27C22⋊#95—21no—
28C23×D4D411yescharacteristic
29C24×C4C2×C411yescharacteristic
30C24×C4D411yescharacteristic
31C2×C4×D4—41no—
32C2×C4×D4—41no—
33C2×C4×D4—41no—
34C2×C4×D4—41no—
35C2×C4×D4C2×C411yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4Q811yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×#117—41no—
44C22×C4⋊C4C2×C411yescharacteristic
45C22×C22⋊C4—21no—
46C22×C22⋊C4—22no—
47C22×C22⋊C4—21no—
48C22×C22⋊C4D411yescharacteristic
49C22×C22⋊C4C2311yescharacteristic
50C22×C42C2×C412yes—
51C22×C42D411yescharacteristic
52#432—22no—
53#432—41no—
54#431—22no—
55#431—41no—
56#423—41no—
57C2×#99—22no—
58C2×#99—22no—
59#407—22no—
60C4×M16—41no—
61#381—41no—
62#381—41no—
63#381—41no—
64#380—22no—
65#380—41no—
66#379—22no—
67C4×C22⋊C4—22no—
68C4×C22⋊C4—41no—
69C4×C22⋊C4—41no—
70C4×C22⋊C4—41no—
71C2×#95—22no—
72C2×#95—21no—
73C2×#95—21no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95C2×C412yes—
77#342—22no—
78#342C2×C414yes—
79#323—22no—
80#323—22no—
81#323—22no—
82#323C2×C414yes—
83C25C22⋊C411yescharacteristic
84C25C2×D411yescharacteristic
85C22×D4—42no—
86C22×D4—41no—
87C22×D4—41no—
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×C4C4211yescharacteristic
106C23×C4C4211yescharacteristic
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—
119C4×D4—41no—
120C4×D4—41no—
121C4×D4—41no—
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—
135#117—84no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4C4211yescharacteristic
140C2×C4⋊C4C4211yescharacteristic
141C2×C4⋊C4C4⋊C411yescharacteristic
142C2×C4⋊C4C4⋊C411yescharacteristic
143C2×C22⋊C4—24no—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—44no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4—42no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—82no—
156C2×C22⋊C4—82no—
157C2×C22⋊C4C22⋊C411yescharacteristic
158C2×C22⋊C4C22⋊C411yescharacteristic
159C2×C22⋊C4C4⋊C411yescharacteristic
160C2×C22⋊C4C4⋊C411yescharacteristic
161C2×C22⋊C4C22×C411yescharacteristic
162C2×C22⋊C4C22×C411yescharacteristic
163C2×C22⋊C4C2×D411yescharacteristic
164C2×C22⋊C4C2×Q811yescharacteristic
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—82no—
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C4⋊C411yescharacteristic
175C2×C42C4⋊C411yescharacteristic
176C4⋊C8—42no—
177C4⋊C8—42no—
178#99—44no—
179#99—44no—
180C22⋊C8—24no—
181C22⋊C8—42no—
182#97—42no—
183C4×C8—42no—
184#95—42no—
185#95—44no—
186#95—44no—
187#95—42no—
188#95—42no—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—82no—
194C24—21no—
195C24—21no—
196C24—22no—
197C24—21no—
198C24—21no—
199C24—41no—
200C24—41no—
201C24—41no—
202C24—42no—
203C24—42no—
204C24—42no—
205C24—42no—
206C24#9511yescharacteristic
207C24#9511yescharacteristic
208C24C2×C22⋊C411yescharacteristic
209C24C2×C22⋊C411yescharacteristic
210C24C2×C4⋊C411yescharacteristic
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—22no—
214C2×D4—22no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—41no—
223C2×D4—48no—
224C2×D4—48no—
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—22no—
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—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—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4—81no—
278C22×C4—82no—
279C22×C4—82no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4#9511yescharacteristic
287C22×C4#9511yescharacteristic
288C22×C4#9511yescharacteristic
289C22×C4#9511yescharacteristic
290C22×C4#9511yescharacteristic
291C22×C4#9511yescharacteristic
292C22×C4#9911yescharacteristic
293C22×C4#9911yescharacteristic
294C22×C4#9911yescharacteristic
295C22×C4#9911yescharacteristic
296C22×C4C2×C4211yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C4⋊C411yescharacteristic
299C22×C4C2×C4⋊C411yescharacteristic
300M16—84no—
301C2×C8—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—84no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—48no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—88no—
319C22⋊C4—84no—
320C22⋊C4—84no—
321C22⋊C4—84no—
322C42—21no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—84no—
331C42—84no—
332C23—22no—
333C23—22no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—42no—
354C23—42no—
355C23—44no—
356C23—42no—
357C23—42no—
358C23—44no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—81no—
368C23—81no—
369C23—82no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23C2×#9511yescharacteristic
376C23C2×#9911yescharacteristic
377C23C2×#9911yescharacteristic
378C23#42112yes—
379D4—44no—
380D4—44no—
381D4—44no—
382D4—44no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
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—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—84no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—84no—
445C2×C4—84no—
446C2×C4—82no—
447C2×C4H(3,ℤ/4)12yes—
448C2×C4#34211yescharacteristic
449C2×C4#34211yescharacteristic
450C2×C4#34211yescharacteristic
451C2×C4#34211yescharacteristic
452C8—84no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—22no—
464C22—42no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—81no—
478C22—84no—
479C22—84no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—84no—
486C22—84no—
487C22C2×H(3,ℤ/4)11yescharacteristic
488C22#173311yescharacteristic
489C22C2×#34211yescharacteristic
490C22#212112yes—
491C22#212612yes—
492C4—22no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—21no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—84no—
505C4—162no—
506C4—164no—
507C2—22no—
508C2—22no—
509C2—21no—
510C2—21no—
511C2—42no—
512C2—81no—
513C2—81no—
514C2—81no—
515C2#794611yescharacteristic
516C2#1257411yescharacteristic
517C2#1258111yescharacteristic
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
8. C2×#381C4semidirect
9. C2×C4×C22⋊C4C4semidirect
12. #1799C4semidirect
15. C4×#99C4semidirect
17. #1466C4semidirect
21. #1460C4semidirect
25. #1436C4semidirect

Maximal subgroups

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

subgroupindex
#207222
#173642
#139402
#72132