Finite Group Databasebeta

← #127201

Subgroups — #127201

Order 512.

939 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20722C211yesmaximal, characteristic
2#17364C211yesmaximal, characteristic
3#13784C211yesmaximal, characteristic
4#7213C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6#2920—21no—
7C2×#386—21no—
8#2280—21no—
9C2×C4×C22⋊C4C411yescharacteristic, complemented
10C22×#95C411yescharacteristic, complemented
11#1819C2212yes—
12#1799—21no—
13#1799C412yescomplemented
14#1781—21no—
15C4×#99—21no—
16C4×#99C412yescomplemented
17#1466—21no—
18#1466C412yescomplemented
19#1462—21no—
20#1462C2212yes—
21#1460—21no—
22#1460C412yescomplemented
23#1437C2212yes—
24#1436C412yescomplemented
25#1434—21no—
26C22⋊#95—21no—
27C22⋊#95—21no—
28C23×D4D411yescharacteristic
29C24×C4—21no—
30C24×C4C2×C411yescharacteristic
31C24×C4D411yescharacteristic
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⋊C4C2×C411yescharacteristic
46C22×C22⋊C4—21no—
47C22×C22⋊C4—22no—
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#386—28no—
62#381—41no—
63#380—22no—
64#380—41no—
65#379—22no—
66C4×C22⋊C4—22no—
67C4×C22⋊C4—41no—
68C4×C22⋊C4—41no—
69C4×C22⋊C4—41no—
70C2×#95—22no—
71C2×#95—21no—
72C2×#95—21no—
73C2×#95—22no—
74C2×#95—22no—
75C2×#95—41no—
76C2×#95—41no—
77C2×#95C2×C412yes—
78#342—22no—
79#342C2×C414yescomplemented
80#323—22no—
81#323—22no—
82#323—22no—
83#323C2×C414yescomplemented
84C25C22⋊C411yescharacteristic
85C25C2×D411yescharacteristic
86C22×D4—24no—
87C22×D4—41no—
88C22×D4—41no—
89C22×D4—42no—
90C22×D4C22⋊C411yescharacteristic
91C22×D4C22⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C22×D4C4⋊C411yescharacteristic
94C23×C4—21no—
95C23×C4—21no—
96C23×C4—21no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—24no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—82no—
108C23×C4—82no—
109C23×C4—82no—
110C23×C4—82no—
111C23×C4C4211yescharacteristic
112C23×C4C4211yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C22⋊C411yescharacteristic
118C23×C4C4⋊C411yescharacteristic
119C23×C4C4⋊C411yescharacteristic
120C23×C4C4⋊C411yescharacteristic
121C23×C4C4⋊C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×M16—42no—
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—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141#117—84no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4C4211yescharacteristic
146C2×C4⋊C4C4211yescharacteristic
147C2×C4⋊C4C4⋊C411yescharacteristic
148C2×C4⋊C4C4⋊C411yescharacteristic
149C2×C22⋊C4—24no—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—42no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4—82no—
161C2×C22⋊C4—82no—
162C2×C22⋊C4C22⋊C411yescharacteristic
163C2×C22⋊C4C22⋊C411yescharacteristic
164C2×C22⋊C4C4⋊C411yescharacteristic
165C2×C22⋊C4C4⋊C411yescharacteristic
166C2×C22⋊C4C22×C411yescharacteristic
167C2×C22⋊C4C22×C411yescharacteristic
168C2×C22⋊C4C2×D411yescharacteristic
169C2×C22⋊C4C2×Q811yescharacteristic
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—82no—
177C2×C42C22⋊C411yescharacteristic
178C2×C42C22⋊C411yescharacteristic
179C2×C42C4⋊C411yescharacteristic
180C2×C42C4⋊C411yescharacteristic
181C4⋊C8—42no—
182C4⋊C8—42no—
183#99—44no—
184#99—44no—
185C22⋊C8—24no—
186C22⋊C8—42no—
187#97—42no—
188C4×C8—42no—
189#95—28no—
190#95—42no—
191#95—42no—
192#95—42no—
193#95—41no—
194#95—41no—
195#95—41no—
196#95—41no—
197#95—82no—
198#95—82no—
199C24—22no—
200C24—22no—
201C24—21no—
202C24—21no—
203C24—22no—
204C24—21no—
205C24—21no—
206C24—41no—
207C24—42no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24—42no—
212C24—41no—
213C24#9511yescharacteristic
214C24#9511yescharacteristic
215C24C2×C22⋊C411yescharacteristic
216C24C2×C22⋊C411yescharacteristic
217C24C2×C4⋊C411yescharacteristic
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—48no—
230C2×D4—48no—
231C2×D4#10414yes—
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—22no—
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—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—48no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—81no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—88no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—81no—
295C22×C4—81no—
296C22×C4—84no—
297C22×C4—88no—
298C22×C4—84no—
299C22×C4#9511yescharacteristic
300C22×C4#9511yescharacteristic
301C22×C4#9511yescharacteristic
302C22×C4#9511yescharacteristic
303C22×C4#9511yescharacteristic
304C22×C4#9511yescharacteristic
305C22×C4#9911yescharacteristic
306C22×C4#9911yescharacteristic
307C22×C4#9911yescharacteristic
308C22×C4#9911yescharacteristic
309C22×C4C2×C4211yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C4⋊C411yescharacteristic
312C22×C4C2×C4⋊C411yescharacteristic
313M16—82no—
314M16—82no—
315C2×C8—44no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—84no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C22⋊C4—41no—
329C22⋊C4—48no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—88no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—84no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—84no—
345C42—84no—
346C23—22no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—24no—
352C23—24no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—44no—
379C23—44no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—81no—
387C23—81no—
388C23—82no—
389C23—82no—
390C23H(3,ℤ/4)12yes—
391C23#33912yes—
392C23C2×#9511yescharacteristic
393C23C2×#9911yescharacteristic
394C23C2×#9911yescharacteristic
395C23C2×#10412yes—
396D4—44no—
397D4—44no—
398D4—44no—
399D4—44no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
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—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
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—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—816no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—84no—
464C2×C4—84no—
465C2×C4—84no—
466C2×C4#33912yes—
467C2×C4#34211yescharacteristic
468C2×C4#34211yescharacteristic
469C2×C4#34211yescharacteristic
470C2×C4#34211yescharacteristic
471C8—84no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—21no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—44no—
489C22—48no—
490C22—42no—
491C22—44no—
492C22—42no—
493C22—42no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—41no—
503C22—81no—
504C22—82no—
505C22—84no—
506C22—82no—
507C22—81no—
508C22C2×H(3,ℤ/4)11yescharacteristic
509C22C2×#33911yescharacteristic
510C22C2×#34211yescharacteristic
511C22#212112yes—
512C22#212811yescharacteristic
513C22#212911yescharacteristic
514C4—22no—
515C4—21no—
516C4—21no—
517C4—21no—
518C4—21no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—84no—
527C4—84no—
528C4—164no—
529C2—21no—
530C2—22no—
531C2—21no—
532C2—21no—
533C2—21no—
534C2—42no—
535C2—42no—
536C2—42no—
537C2—81no—
538C2#794211yescharacteristic
539C2#1257411yescharacteristic
540C2#1258311yescharacteristic
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
9. C2×C4×C22⋊C4C4semidirect
10. C22×#95C4semidirect
13. #1799C4semidirect
16. C4×#99C4semidirect
18. #1466C4semidirect
22. #1460C4semidirect
24. #1436C4semidirect
79. #342C2×C4semidirect
83. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
#207222
#173642
#137842
#72132