Finite Group Databasebeta

← #96932

Subgroups — #96932

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2857C211yesmaximal, characteristic
2#10712C211yesmaximal, characteristic
3#10712C211yesmaximal, characteristic
4C2×#1271C214yesmaximal, complemented
5C22×C4⋊Q8C411yescharacteristic
6C22×C22⋊Q8—21no—
7C23×C42C2211yescharacteristic
8C22×C22⋊C8—21no—
9C22×C22⋊C8—21no—
10#2857—28no—
11C2×#391—21no—
12C2×#389—21no—
13C2×#389C2212yes—
14C2×#384C411yescharacteristic
15C2×#380—21no—
16C2×#380C412yes—
17C2×#336—21no—
18C2×#336—21no—
19C2×#336C2212yes—
20C2×#336C2212yes—
21#1271—28no—
22C23×Q8—21no—
23C24×C4C2×C411yescharacteristic
24C24×C4D411yescharacteristic
25C24×C4D411yescharacteristic
26C23×C8—22no—
27C23×C8—22no—
28C2×C4⋊Q8—24no—
29C2×C4⋊Q8—42no—
30C2×C4⋊Q8—42no—
31C2×C22⋊Q8—28no—
32C2×C22⋊Q8—44no—
33C2×C22⋊Q8—42no—
34C2×C22⋊Q8—42no—
35C22×C4⋊C4—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—22no—
39C22×C4⋊C4C2×C412yes—
40C22×C4⋊C4C2×C412yes—
41C22×C22⋊C4—22no—
42C22×C42—24no—
43C22×C42—42no—
44C22×C42—42no—
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42C2311yescharacteristic
49C2×C22⋊C8—24no—
50C2×C22⋊C8—24no—
51C2×C22⋊C8—42no—
52C2×C22⋊C8—42no—
53C2×C22⋊C8—42no—
54C2×C22⋊C8—42no—
55#391—28no—
56#389—28no—
57#389—24no—
58#389D418yes—
59#384—24no—
60#380—28no—
61#380—24no—
62#380—24no—
63C2×#95—22no—
64C2×#95—22no—
65#336—24no—
66#336—24no—
67#336—42no—
68#336—42no—
69C25C22⋊C411yescharacteristic
70C22×Q8—24no—
71C22×Q8—42no—
72C22×Q8—42no—
73C22×Q8—42no—
74C22×Q8—42no—
75C22×Q8—42no—
76C22×Q8—42no—
77C23×C4—22no—
78C23×C4—22no—
79C23×C4—22no—
80C23×C4—22no—
81C23×C4—22no—
82C23×C4—22no—
83C23×C4—22no—
84C23×C4—22no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—22no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—41no—
93C23×C4—41no—
94C23×C4—41no—
95C23×C4—41no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4D812yes—
102C23×C4SD1612yes—
103C23×C4SD1612yes—
104C23×C4Q1612yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C22×C8—24no—
109C22×C8—24no—
110C22×C8—42no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—42no—
114C22×C8—42no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8—42no—
121C22×C8—42no—
122C4⋊Q8—416no—
123C22⋊Q8—416no—
124C22⋊Q8—416no—
125C2×C4⋊C4—24no—
126C2×C4⋊C4—24no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—24no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—28no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4—42no—
137C2×C4⋊C4—42no—
138C2×C4⋊C4—42no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4C22⋊C414yes—
145C2×C4⋊C4C22⋊C414yes—
146C2×C22⋊C4—28no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4—44no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—22no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—416no—
157C2×C42—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42C2×D414yes—
162C22⋊C8—48no—
163C22⋊C8—48no—
164#95—28no—
165#95—28no—
166C24—21no—
167C24—21no—
168C24—21no—
169C24—21no—
170C24—21no—
171C24—21no—
172C24—21no—
173C24—21no—
174C24—21no—
175C24—21no—
176C24—21no—
177C24—21no—
178C24#10112yes—
179C24#10212yes—
180C24#10312yes—
181C24C2×C22⋊C411yescharacteristic
182C2×Q8—48no—
183C2×Q8—48no—
184C2×Q8—48no—
185C2×Q8—84no—
186C2×Q8—84no—
187C2×Q8—84no—
188C2×Q8—84no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—28no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—44no—
236C22×C4—44no—
237C22×C4—44no—
238C22×C4—44no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4#9912yes—
257C22×C4#9912yes—
258C22×C4#10112yes—
259C22×C4#10212yes—
260C22×C4#10312yes—
261C22×C4#10412yes—
262C22×C4#10412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4#12012yes—
265C22×C4#12012yes—
266C22×C4#13112yes—
267C22×C4#13112yes—
268C22×C4C2×D811yescharacteristic
269C22×C4C2×SD1611yescharacteristic
270C22×C4C2×SD1611yescharacteristic
271C22×C4C2×Q1611yescharacteristic
272C22×C4AGL(1,ℤ/8)12yes—
273C22×C4#13712yes—
274C2×C8—44no—
275C2×C8—44no—
276C2×C8—44no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C2×C8—82no—
281C2×C8—82no—
282C2×C8—82no—
283C2×C8—82no—
284C2×C8—82no—
285C2×C8—82no—
286C2×C8—82no—
287C2×C8—82no—
288C4⋊C4—28no—
289C4⋊C4—44no—
290C4⋊C4—48no—
291C4⋊C4—48no—
292C4⋊C4—48no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—48no—
296C4⋊C4—48no—
297C22⋊C4—416no—
298C42—28no—
299C42—44no—
300C42—44no—
301C42—44no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23#33214yes—
337C23#37911yescharacteristic
338C23#40811yescharacteristic
339C23#40811yescharacteristic
340C23C2×#9911yescharacteristic
341C23C2×#9911yescharacteristic
342C23C2×#10111yescharacteristic
343C23C2×#10111yescharacteristic
344C23C2×#10211yescharacteristic
345C23C2×#10311yescharacteristic
346C23#41811yescharacteristic
347C23#41911yescharacteristic
348C23C2×#10411yescharacteristic
349C23C2×#10411yescharacteristic
350C23#42111yescharacteristic
351C23#42111yescharacteristic
352Q8—816no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—416no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4#32714yes—
414C2×C4#32814yes—
415C2×C4#40814yes—
416C2×C4#44712yes—
417C2×C4#44812yes—
418C2×C4#45012yes—
419C2×C4#45112yes—
420C8—84no—
421C8—84no—
422C22—24no—
423C22—24no—
424C22—24no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—42no—
437C22#147211yescharacteristic
438C22#147211yescharacteristic
439C22#147211yescharacteristic
440C22#147211yescharacteristic
441C22#148011yescharacteristic
442C22#148111yescharacteristic
443C22#148411yescharacteristic
444C22#148511yescharacteristic
445C22#149211yescharacteristic
446C22#149311yescharacteristic
447C22#149611yescharacteristic
448C22#149711yescharacteristic
449C22C2×#32711yescharacteristic
450C22C2×#32811yescharacteristic
451C22#150711yescharacteristic
452C22#150811yescharacteristic
453C22#151011yescharacteristic
454C22#151111yescharacteristic
455C22#151911yescharacteristic
456C22#152011yescharacteristic
457C22#152611yescharacteristic
458C22#153111yescharacteristic
459C22#153111yescharacteristic
460C22#153711yescharacteristic
461C22C2×#33211yescharacteristic
462C22#154611yescharacteristic
463C22#154911yescharacteristic
464C22#155611yescharacteristic
465C22#178011yescharacteristic
466C22#178111yescharacteristic
467C22#178411yescharacteristic
468C22#178611yescharacteristic
469C22#178711yescharacteristic
470C22#178911yescharacteristic
471C22#178911yescharacteristic
472C4—24no—
473C4—22no—
474C4—24no—
475C4—22no—
476C4—22no—
477C4—24no—
478C4—22no—
479C4—42no—
480C4—84no—
481C4—84no—
482C2—44no—
483C2#766511yescharacteristic
484C2#766811yescharacteristic
485C2#768511yescharacteristic
486C2#768811yescharacteristic
487C2#770511yescharacteristic
488C2#770811yescharacteristic
489C2#772511yescharacteristic
490C2#772811yescharacteristic
491C2#830811yescharacteristic
492C2#831311yescharacteristic
493C2#831411yescharacteristic
494C2#832811yescharacteristic
495C2#832911yescharacteristic
496C2#833811yescharacteristic
497C2#834011yescharacteristic
subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23×C4C22×C4
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1271C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28572
#107122
C2×#12712