Finite Group Databasebeta

← #162424

Subgroups — #162424

Order 512.

909 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
1#35170C211yesmaximal, characteristic
2#22696C211yesmaximal, characteristic, complemented
3#22008C211yesmaximal, characteristic, complemented
4#7728C212yesmaximal, complemented
5#7714C212yesmaximal, complemented
6#7668C212yesmaximal, complemented
7#7654C212yesmaximal, complemented
8#7588C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12C2×#542—21no—
13C2×C8×D4—21no—
14#2857C2212yes—
15#2854C2212yes—
16#2663—22no—
17#2551C411yescharacteristic
18#2547C411yescharacteristic
19#2542C411yescharacteristic
20#2537C411yescharacteristic
21#2521—22no—
22#2496C412yes—
23#2492—21no—
24#2491C412yes—
25#2427—21no—
26#2323—21no—
27C4×C4⋊Q8C2211yescharacteristic
28#2293C2211yescharacteristic
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1939—21no—
33#1923—22no—
34#1850—22no—
35#1831C2211yescharacteristic
36#1817C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1744C2211yescharacteristic
42C2×#336C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45#1271C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C23×C8—22no—
54#542—44no—
55#542—44no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C412yes—
59C2×#126—22no—
60C2×#124—21no—
61C2×#124—22no—
62C2×#123—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×#121C2×C412yes—
68C2×C4×Q8—21no—
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×#117—21no—
74C22×C42C2312yes—
75C8×D4—44no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#400—22no—
88#399—22no—
89#397—22no—
90#397—22no—
91#395—22no—
92#394—22no—
93#394—22no—
94#393—22no—
95#391—21no—
96#391—21no—
97#391—22no—
98#390C2×C412yes—
99#390C2×C411yescharacteristic
100#390C2311yescharacteristic
101#389—21no—
102#389—21no—
103#389—22no—
104#389C2×C412yes—
105#389C2×C411yescharacteristic
106#389C2312yes—
107#389C2311yescharacteristic
108#388—22no—
109#388—22no—
110#388—21no—
111#388—21no—
112#388—22no—
113#387—22no—
114#386—22no—
115#384C2×C411yescharacteristic
116#384C2×C412yes—
117#380—22no—
118#380—21no—
119#380C2×C412yes—
120#380C2×C412yes—
121#380C2×C412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C412yes—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4C2×C412yes—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×Q8—22no—
145C22×D4—22no—
146C22×D4C22⋊C411yescharacteristic
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C23×C4C22×C412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—44no—
159C22×C8—44no—
160C22×C8—82no—
161C4⋊Q8—24no—
162C4⋊Q8—44no—
163C4⋊Q8—82no—
164#126—48no—
165#124—44no—
166#124—48no—
167#123—44no—
168#123—44no—
169C22⋊Q8—44no—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172C22⋊Q8—44no—
173#121—44no—
174#121—82no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—42no—
180C4×D4D814yes—
181C4×D4SD1614yes—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22⋊C411yescharacteristic
195C2×C4⋊C4C22⋊C411yescharacteristic
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×C22⋊C4—22no—
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—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—82no—
219C2×C42C22×C412yes—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2411yescharacteristic
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—82no—
233C4⋊C8—82no—
234C22⋊C8—44no—
235C22⋊C8—82no—
236C22⋊C8—82no—
237C4×C8—42no—
238C4×C8—42no—
239C4×C8—81no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—22no—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251#95C22⋊C412yes—
252C24—21no—
253C24C2×C22⋊C412yes—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—44no—
257C2×Q8—42no—
258C2×Q8—82no—
259C2×Q8—82no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—44no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4#10214yes—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×D814yes—
294C22×C4C2×SD1614yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—88no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—41no—
324C4⋊C4—44no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—81no—
335C4⋊C4—82no—
336C4⋊C4#10214yes—
337C22⋊C4—24no—
338C22⋊C4—42no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—44no—
350C22⋊C4—81no—
351C22⋊C4—82no—
352C22⋊C4#10218yes—
353C42—21no—
354C42—21no—
355C42—22no—
356C42—41no—
357C42—42no—
358C42—42no—
359C42—41no—
360C42—42no—
361C42—41no—
362C42—42no—
363C42—44no—
364C42#13114yes—
365C42C2×D812yes—
366C42C2×SD1612yes—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23C2×#10214yes—
381C23C22×C22⋊C411yescharacteristic
382Q8—84no—
383Q8—164no—
384D4—28no—
385D4—82no—
386D4—164no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4#40812yes—
430C2×C4#40812yes—
431C2×C4C2×#10212yes—
432C2×C4C2×#10214yes—
433C2×C4C2×#10212yes—
434C2×C4#42112yes—
435C2×C4C2×#13112yes—
436C2×C4C22×D811yescharacteristic
437C2×C4C22×SD1611yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57811yescharacteristic
440C8—42no—
441C8—81no—
442C8—82no—
443C8—81no—
444C8—81no—
445C8—82no—
446C22—24no—
447C22—24no—
448C22—24no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22C2×#40811yescharacteristic
461C22#288311yescharacteristic
462C22#288911yescharacteristic
463C22C22×#10211yescharacteristic
464C22#289811yescharacteristic
465C22C2×#42111yescharacteristic
466C22#290111yescharacteristic
467C4—22no—
468C4—22no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—24no—
474C4—22no—
475C4—41no—
476C4—42no—
477C4—42no—
478C4—81no—
479C4—82no—
480C4—81no—
481C4—82no—
482C4—81no—
483C4#148014yes—
484C2—44no—
485C2—161no—
486C2#1738811yescharacteristic
487C2C2×#148011yescharacteristic
488C2#1748411yescharacteristic
489C2#1760011yescharacteristic
490C2#1761311yescharacteristic
491C2#1768311yescharacteristic
492C2#1772111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2537C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22696C2semidirect
3. #22008C2semidirect
4. #7728C2semidirect
5. #7714C2semidirect
6. #7668C2semidirect
7. #7654C2semidirect
8. #7588C2semidirect
9. #7574C2semidirect
10. #7570C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#351702
#226962
#220082
#77282
#77142
#76682
#76542
#75882
#75742
#75702
#75602