Finite Group Databasebeta

← #164953

Subgroups — #164953

Order 512.

870 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36534C211yesmaximal, characteristic
2#22832C211yesmaximal, characteristic, complemented
3#22734C211yesmaximal, characteristic, complemented
4#7716C212yesmaximal, complemented
5#7676C212yesmaximal, complemented
6#7662C211yesmaximal, characteristic, complemented
7#7662C211yesmaximal, characteristic, complemented
8#7651C211yesmaximal, characteristic
9#7618C212yesmaximal, complemented
10#7584C212yesmaximal, complemented
11#7569C211yesmaximal, characteristic
12C2×#556C411yescharacteristic
13C2×#554C411yescharacteristic
14C2×C8×Q8—21no—
15#2862C2211yescharacteristic
16#2861C2212yes—
17#2565C412yes—
18#2563C411yescharacteristic
19#2560—22no—
20#2560—22no—
21#2547C411yescharacteristic
22#2544C412yes—
23#2536—21no—
24#2536—21no—
25#2352—21no—
26#2352—21no—
27C4×#127C2211yescharacteristic
28C4×#124C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—22no—
32#1939—21no—
33#1917—22no—
34#1839C2212yes—
35#1831C2212yes—
36#1831C2211yescharacteristic
37#1831C2211yescharacteristic
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2212yes—
42#1745C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1270C2211yescharacteristic
48#1270C2212yes—
49#1270C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53#556—44no—
54#554—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#127C2×C411yescharacteristic
58C2×#126—22no—
59C2×#125C2×C412yes—
60C2×#124C2×C412yes—
61C2×#124C2×C412yes—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—22no—
65C2×C4×Q8C2×C412yes—
66C2×C4×Q8C2×C411yescharacteristic
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C412yes—
74C2×#117—21no—
75C22×C4⋊C4—21no—
76C8×Q8—44no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—21no—
83C2×#97—22no—
84C2×C4×C8—22no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#398—22no—
88#398—22no—
89#397—22no—
90#394—22no—
91#394—22no—
92#393—22no—
93#391—21no—
94#391—21no—
95#391C2×C412yes—
96#391C2×C412yes—
97#391C2312yes—
98#390—21no—
99#390—21no—
100#390C2×C412yes—
101#390C2311yescharacteristic
102#390C2311yescharacteristic
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#387—22no—
110#387—22no—
111#386—22no—
112#386—22no—
113#384C2×C411yescharacteristic
114#382—21no—
115#382—21no—
116#382C2×C412yes—
117#380—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C412yes—
123C4×C4⋊C4D412yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C43C2311yescharacteristic
130C43C2312yes—
131#336—21no—
132#336—22no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×Q8C22⋊C411yescharacteristic
145C22×Q8C22×C412yes—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22×C411yescharacteristic
149C22×D4C22×C411yescharacteristic
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C4⋊Q8—42no—
160C4⋊Q8—42no—
161#127—22no—
162#127—41no—
163#126—48no—
164#125—44no—
165#124—24no—
166#124—42no—
167#124—44no—
168C22⋊Q8—48no—
169C22⋊Q8—44no—
170C22⋊Q8—44no—
171C4×Q8—24no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—44no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—44no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184#117—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—22no—
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C412yes—
219C2×C42C22×C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2411yescharacteristic
230C4⋊C8—44no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233C4⋊C8—44no—
234C4⋊C8—42no—
235C4⋊C8—42no—
236#97—44no—
237C4×C8—44no—
238C4×C8—42no—
239C4×C8—42no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24—21no—
252C24—21no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—81no—
261C2×Q8—81no—
262C2×Q8#10014yes—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—81no—
275C2×D4—81no—
276C2×D4#13112yes—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—41no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C42—22no—
354C42—21no—
355C42—22no—
356C42—21no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—24no—
363C42—42no—
364C42—41no—
365C42—41no—
366C42—42no—
367C42—41no—
368C42—44no—
369C42#13512yes—
370C42#13512yes—
371C42AGL(1,ℤ/8)12yes—
372C42#13712yes—
373C23—22no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C22×C22⋊C411yescharacteristic
388Q8—28no—
389Q8—82no—
390Q8—82no—
391Q8—162no—
392D4—42no—
393D4—42no—
394D4—84no—
395D4—84no—
396D4—81no—
397D4—81no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—84no—
435C2×C4—84no—
436C2×C4—164no—
437C2×C4C2×#10014yes—
438C2×C4C2×#10012yes—
439C2×C4#41612yes—
440C2×C4#41612yes—
441C2×C4#41712yes—
442C2×C4#41712yes—
443C2×C4C2×#13111yescharacteristic
444C2×C4#56811yescharacteristic
445C2×C4C2×#13511yescharacteristic
446C2×C4C2×#13511yescharacteristic
447C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
448C2×C4C2×#13711yescharacteristic
449C8—42no—
450C8—42no—
451C8—81no—
452C8—82no—
453C8—82no—
454C8—81no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22#288011yescharacteristic
470C22#288411yescharacteristic
471C22C22×#10011yescharacteristic
472C22C2×#41611yescharacteristic
473C22C2×#41711yescharacteristic
474C22#290111yescharacteristic
475C22#290111yescharacteristic
476C4—21no—
477C4—21no—
478C4—24no—
479C4—22no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—42no—
486C4—41no—
487C4—42no—
488C4—42no—
489C4—82no—
490C4—82no—
491C4—81no—
492C4—81no—
493C4#155114yes—
494C2—82no—
495C2—161no—
496C2#1742511yescharacteristic
497C2#1750811yescharacteristic
498C2#1751211yescharacteristic
499C2#1761611yescharacteristic
500C2#1762811yescharacteristic
501C2#1768811yescharacteristic
502C2C2×#155111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22832C2semidirect
3. #22734C2semidirect
4. #7716C2semidirect
5. #7676C2semidirect
6. #7662C2semidirect
7. #7662C2semidirect
9. #7618C2semidirect
10. #7584C2semidirect

Maximal subgroups

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

subgroupindex
#365342
#228322
#227342
#77162
#76762
#76622
#76512
#76182
#75842
#75692