Finite Group Databasebeta

← #206923

Subgroups — #206923

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2295C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic
3C4×#432C211yesmaximal, characteristic
4#9881C214yesmaximal
5#9879C212yesmaximal
6#9878C212yesmaximal
7#7910C214yesmaximal
8C22×C4×Q8C411yescharacteristic
9C2×#432C2212yes—
10C2×#432C2212yes—
11#2295—21no—
12#2295—21no—
13#2295—21no—
14#2295—21no—
15#2295—21no—
16#2295—21no—
17#2295—21no—
18#2295—21no—
19C2×#391C411yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#389C411yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#387C412yes—
24C2×#387C2212yes—
25C2×C4×C4⋊C4C412yes—
26C2×C4×C4⋊C4C411yescharacteristic
27C2×#376C2212yes—
28C2×C43C2211yescharacteristic
29#1842—21no—
30#1831—21no—
31#1797C2212yes—
32#1797C2212yes—
33#1767—21no—
34#1767—21no—
35#1767—21no—
36C4×C4⋊C8—21no—
37#1749C2212yes—
38C4×C22⋊C8C2212yes—
39#1745—21no—
40C4×#97—21no—
41#1280C2218yes—
42#1280C2218yes—
43C23×Q8C2×C411yescharacteristic
44C2×C4×Q8—24no—
45C2×C4×Q8—24no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C22×C4⋊C4C2×C412yes—
51C22×C4⋊C4C2×C411yescharacteristic
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C42C2×C412yes—
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C412yes—
59C22×C42C2×C411yescharacteristic
60C22×C42C2×C411yescharacteristic
61C22×C42C2×C411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65#432—24no—
66#432—28no—
67#432Q818yescomplemented
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C22⋊C8C2314yes—
73C2×C22⋊C8C2314yes—
74C2×#97—22no—
75C2×#97—22no—
76C2×#97—22no—
77C2×C4×C8—22no—
78#391—21no—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110#376—22no—
111#376—22no—
112#376—22no—
113#376—22no—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2×C412yes—
117C2×#95C2×C412yes—
118C2×#95C2×C412yes—
119C2×#95C2312yes—
120C2×#95C2312yes—
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43D411yescharacteristic
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8M1614yes—
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C411yescharacteristic
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C412yes—
147C23×C4C22×C412yes—
148C23×C4C22×C411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—24no—
151C22×C8—24no—
152C4×Q8—28no—
153C4×Q8—28no—
154C4×Q8—28no—
155C4×Q8—28no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4M1614yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—44no—
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q814yes—
219C2×C42#4114yes—
220C4⋊C8—28no—
221C4⋊C8—44no—
222C4⋊C8—44no—
223C4⋊C8—44no—
224C22⋊C8—24no—
225C22⋊C8—28no—
226C22⋊C8C2×Q818yes—
227#97—28no—
228#97—44no—
229#97—44no—
230C4×C8—44no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245C24C23×C411yescharacteristic
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—48no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—48no—
252C2×Q8—42no—
253C2×Q8—42no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C4×Q814yes—
302C22×C4C4×Q814yes—
303C22×C4C2×M1612yes—
304C22×C4C2×M1612yes—
305C22×C4C2×M1614yes—
306C22×C4C2×M1614yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×Q812yes—
310C22×C4C2×#4112yes—
311C2×C8—28no—
312C2×C8—44no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C4⋊C4—28no—
318C4⋊C4—28no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C42—28no—
330C42—28no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—44no—
356C42—44no—
357C42C22⋊Q814yes—
358C42C22⋊Q814yes—
359C42#12412yes—
360C42#12412yes—
361C42C4⋊Q812yes—
362C42C4⋊Q812yes—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23C22×C22⋊C411yescharacteristic
368C23C2×C4×Q812yes—
369C23C22×M1611yescharacteristic
370C23C22×M1611yescharacteristic
371C23#56812yes—
372Q8—48no—
373Q8—48no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—21no—
401C2×C4—21no—
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—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—84no—
432C2×C4—84no—
433C2×C4#39112yes—
434C2×C4#39112yes—
435C2×C4#39112yes—
436C2×C4#39112yes—
437C2×C4#40712yes—
438C2×C4#40712yes—
439C2×C4#41012yes—
440C2×C4#41012yes—
441C2×C4#41312yes—
442C2×C4#41312yes—
443C2×C4C2×C22⋊Q812yes—
444C2×C4C2×#12411yescharacteristic
445C2×C4C2×C4⋊Q811yescharacteristic
446C8—48no—
447C8—84no—
448C22—22no—
449C22—22no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—22no—
456C22C2×#39111yescharacteristic
457C22C2×#40711yescharacteristic
458C22#287911yescharacteristic
459C22C2×#41011yescharacteristic
460C22C2×#41011yescharacteristic
461C22C2×#41311yescharacteristic
462C22C2×#41311yescharacteristic
463C22Q8×M1612yes—
464C22Q8×M1612yes—
465C4—22no—
466C4—22no—
467C4—22no—
468C4—21no—
469C4—21no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—44no—
476C4—41no—
477C4—41no—
478C4—44no—
479C4—41no—
480C4—41no—
481C4—82no—
482C4—82no—
483C4#146414yes—
484C2—22no—
485C2—22no—
486C2C2×#146411yescharacteristic
487C2#1736611yescharacteristic
488C2#1739511yescharacteristic
489C2#1739911yescharacteristic
490C2#2130311yescharacteristic
491C2#2133211yescharacteristic
492C2#2133711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
67. #432Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22952
#208102
C4×#4322
#98812
#98792
#98782
#79102