Finite Group Databasebeta

← #127617

Subgroups — #127617

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1799C211yesmaximal, characteristic
2C2×#1460C211yesmaximal, characteristic
3#13960C211yesmaximal, characteristic
4#7250C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#431—21no—
7C22×#99—21no—
8C2×#386—21no—
9C2×#380—21no—
10C2×#380C411yescharacteristic, complemented
11C22×#95C411yescharacteristic, complemented
12#1799—22no—
13#1799C414yescomplemented
14C2×#342C2212yes—
15#1460—22no—
16#1460C414yescomplemented
17C2×#323—21no—
18C2×#323C2212yes—
19#1441C412yescomplemented
20#1439C2212yes—
21#1434—21no—
22C22⋊#95—21no—
23C22⋊#95—21no—
24C23×D4D411yescharacteristic
25C24×C4—21no—
26C24×C4C2×C411yescharacteristic
27C24×C4D411yescharacteristic
28C2×C4×D4—41no—
29C2×C4×D4—41no—
30C2×C4×D4—42no—
31C2×C4×D4C2×C412yes—
32C2×C4×D4C2×C412yes—
33C2×C4×D4D412yes—
34C2×C4×D4Q812yes—
35C22×C4⋊C4—41no—
36C22×C4⋊C4C2×C411yescharacteristic
37C22×C4⋊C4C2×C412yes—
38C22×C22⋊C4—22no—
39C22×C22⋊C4—21no—
40C22×C22⋊C4—22no—
41C22×C22⋊C4D411yescharacteristic
42C22×C22⋊C4C2311yescharacteristic
43C22×C42D411yescharacteristic
44#431—24no—
45#431—42no—
46C2×C4⋊C8—41no—
47C2×#99—24no—
48C2×#99—42no—
49C2×#99—42no—
50C2×C22⋊C8—22no—
51C2×C4×C8—41no—
52#386—42no—
53#386—42no—
54#380—24no—
55#380—42no—
56#380—41no—
57#380—41no—
58C2×#95—22no—
59C2×#95—21no—
60C2×#95—21no—
61C2×#95—22no—
62C2×#95—22no—
63C2×#95—41no—
64C2×#95—42no—
65C2×#95C2×C412yes—
66#342—22no—
67#342C2×C414yescomplemented
68#323—24no—
69#323—22no—
70#323C2×C414yescomplemented
71C25C22⋊C411yescharacteristic
72C25C2×D411yescharacteristic
73C22×D4—41no—
74C22×D4—41no—
75C22×D4—41no—
76C22×D4—41no—
77C22×D4C22⋊C412yes—
78C22×D4C4⋊C412yes—
79C22×D4D812yes—
80C22×D4SD1612yes—
81C22×D4SD1612yes—
82C22×D4Q1612yes—
83C23×C4—21no—
84C23×C4—22no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—44no—
92C23×C4—42no—
93C23×C4—42no—
94C23×C4—42no—
95C23×C4—42no—
96C23×C4—82no—
97C23×C4—82no—
98C23×C4—82no—
99C23×C4C4212yes—
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C22⋊C412yes—
103C23×C4C22⋊C411yescharacteristic
104C23×C4C4⋊C412yes—
105C23×C4C4⋊C412yes—
106C23×C4C22×C411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C22×C8—42no—
109C4×D4—42no—
110C4×D4—42no—
111C4×D4—42no—
112C4×D4—42no—
113C4×D4—42no—
114C4×D4—42no—
115C4×D4—42no—
116C4×D4—42no—
117C2×C4⋊C4—41no—
118C2×C4⋊C4—41no—
119C2×C4⋊C4—42no—
120C2×C4⋊C4—44no—
121C2×C4⋊C4—84no—
122C2×C4⋊C4—82no—
123C2×C4⋊C4C4212yes—
124C2×C4⋊C4C4⋊C412yes—
125C2×C22⋊C4—24no—
126C2×C22⋊C4—41no—
127C2×C22⋊C4—41no—
128C2×C22⋊C4—41no—
129C2×C22⋊C4—41no—
130C2×C22⋊C4—42no—
131C2×C22⋊C4—42no—
132C2×C22⋊C4—42no—
133C2×C22⋊C4—42no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—42no—
136C2×C22⋊C4—42no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—42no—
141C2×C22⋊C4—82no—
142C2×C22⋊C4C22⋊C412yes—
143C2×C22⋊C4C4⋊C412yes—
144C2×C22⋊C4C22×C411yescharacteristic
145C2×C22⋊C4C22×C411yescharacteristic
146C2×C22⋊C4C2×D411yescharacteristic
147C2×C22⋊C4C2×Q811yescharacteristic
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—22no—
151C2×C42C22⋊C412yes—
152C2×C42C4⋊C412yes—
153C4⋊C8—44no—
154#99—48no—
155C22⋊C8—24no—
156C22⋊C8—42no—
157C4×C8—44no—
158#95—42no—
159#95—44no—
160#95—44no—
161#95—42no—
162#95—41no—
163#95—41no—
164#95—44no—
165#95—82no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—21no—
170C24—21no—
171C24—21no—
172C24—21no—
173C24—21no—
174C24—42no—
175C24—42no—
176C24—42no—
177C24—42no—
178C24#9512yes—
179C24#10212yes—
180C24#10312yes—
181C24C2×C22⋊C411yescharacteristic
182C24C2×C22⋊C411yescharacteristic
183C24C2×C4⋊C411yescharacteristic
184C24C2×D811yescharacteristic
185C24C2×SD1611yescharacteristic
186C24C2×SD1611yescharacteristic
187C24C2×Q1611yescharacteristic
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4#10214yes—
201C2×D4#10314yes—
202C2×D4#10614yes—
203C2×D4#10714yes—
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—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—48no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4—82no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—84no—
249C22×C4—84no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—88no—
253C22×C4—88no—
254C22×C4—84no—
255C22×C4—84no—
256C22×C4—82no—
257C22×C4—82no—
258C22×C4—84no—
259C22×C4—84no—
260C22×C4#9512yes—
261C22×C4#9512yes—
262C22×C4#9512yes—
263C22×C4#9911yescharacteristic
264C22×C4#9911yescharacteristic
265C22×C4#9912yes—
266C22×C4#10212yes—
267C22×C4#10312yes—
268C22×C4C2×C4211yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C4⋊C411yescharacteristic
271C22×C4C2×C4⋊C411yescharacteristic
272C2×C8—44no—
273C2×C8—84no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—84no—
277C4⋊C4—88no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—84no—
287C22⋊C4—84no—
288C22⋊C4—82no—
289C22⋊C4—82no—
290C22⋊C4—84no—
291C42—22no—
292C42—22no—
293C42—22no—
294C42—22no—
295C23—22no—
296C23—22no—
297C23—22no—
298C23—21no—
299C23—22no—
300C23—21no—
301C23—22no—
302C23—22no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—22no—
307C23—22no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—42no—
316C23—44no—
317C23—42no—
318C23—44no—
319C23—42no—
320C23—42no—
321C23—44no—
322C23—44no—
323C23—42no—
324C23—42no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—82no—
338C23—82no—
339C23#34014yes—
340C23C2×#9511yescharacteristic
341C23C2×#9911yescharacteristic
342C23C2×#9911yescharacteristic
343C23C2×#10211yescharacteristic
344C23C2×#10211yescharacteristic
345C23C2×#10311yescharacteristic
346C23C2×#10311yescharacteristic
347C23#42111yescharacteristic
348C23#42111yescharacteristic
349C23C2×#10611yescharacteristic
350C23C2×#10711yescharacteristic
351D4—44no—
352D4—44no—
353D4—44no—
354D4—44no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—816no—
396C2×C4—82no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—164no—
408C2×C4—164no—
409C2×C4#34014yes—
410C2×C4#34212yes—
411C2×C4#34212yes—
412C8—84no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—21no—
422C22—24no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—44no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—44no—
432C22—44no—
433C22—44no—
434C22—41no—
435C22—41no—
436C22—42no—
437C22—41no—
438C22—42no—
439C22—41no—
440C22—42no—
441C22—44no—
442C22—42no—
443C22—42no—
444C22—44no—
445C22—42no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#151911yescharacteristic
453C22#151911yescharacteristic
454C22#152011yescharacteristic
455C22#152011yescharacteristic
456C22#173311yescharacteristic
457C22C2×#34011yescharacteristic
458C22C2×#34211yescharacteristic
459C22#212112yes—
460C22#212612yes—
461C4—22no—
462C4—22no—
463C4—24no—
464C4—42no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—84no—
470C4—164no—
471C4—162no—
472C2—21no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2—41no—
477C2—44no—
478C2—41no—
479C2—41no—
480C2—41no—
481C2#802311yescharacteristic
482C2#1187612yes—
483C2#1190212yes—
484C2#1257411yescharacteristic
485C2#1258111yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ 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. #7250C2semidirect
10. C2×#380C4semidirect
11. C22×#95C4semidirect
13. #1799C4semidirect
16. #1460C4semidirect
19. #1441C4semidirect
67. #342C2×C4semidirect
70. #323C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17992
C2×#14602
#139602
#72502