Finite Group Databasebeta

← #164978

Subgroups — #164978

Order 512.

915 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36261C211yesmaximal, characteristic
2#22835C211yesmaximal, characteristic
3#22700C211yesmaximal, characteristic
4#7728C212yesmaximal, complemented
5#7685C212yesmaximal, complemented
6#7663C211yesmaximal, characteristic, complemented
7#7660C211yesmaximal, characteristic, complemented
8#7652C211yesmaximal, characteristic, complemented
9#7610C212yesmaximal, complemented
10#7573C212yesmaximal, complemented
11#7570C211yesmaximal, characteristic, complemented
12C2×#554—21no—
13C2×#436—21no—
14#2859C2211yescharacteristic
15#2857C2212yes—
16#2584—21no—
17#2583—22no—
18#2552C411yescharacteristic
19#2551C411yescharacteristic
20#2550—21no—
21#2549—22no—
22#2440C412yes—
23#2440C412yes—
24#2349—21no—
25D4×C4⋊C4C411yescharacteristic
26D4×C4⋊C4C411yescharacteristic
27C4×C4⋊Q8C2211yescharacteristic
28C2×#389C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1923—22no—
33#1851—22no—
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1817C2212yes—
37#1797C2212yes—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42#1728C2212yes—
43#1271C2211yescharacteristic
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×M16—22no—
54#554—44no—
55#554—44no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C412yes—
59C2×#125—21no—
60C2×#125—21no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×D4—21no—
66C2×C4×D4—21no—
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C22×C4⋊C4—22no—
71C22×C4⋊C4C2×C412yes—
72C22×C4⋊C4C2×C412yes—
73C22×C42C2312yes—
74#436—44no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—22no—
86#398—22no—
87#398—22no—
88#398—22no—
89#395—22no—
90#395—22no—
91#395—22no—
92#391—21no—
93#391—22no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—22no—
101#389—22no—
102#389—21no—
103#389—22no—
104#389C2×C412yes—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2311yescharacteristic
108#389C2311yescharacteristic
109#389C2312yes—
110#385—22no—
111#385—22no—
112#385—22no—
113#385—22no—
114#384C2×C411yescharacteristic
115#384C2×C412yes—
116#384C2×C411yescharacteristic
117#384C2×C411yescharacteristic
118#382—21no—
119#380—22no—
120#380C2×C412yes—
121#380C2×C412yes—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131C2×#95—22no—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×Q8—22no—
145C22×D4C22⋊C411yescharacteristic
146C23×C4—22no—
147C23×C4—22no—
148C23×C4C22×C412yes—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C2×M16—44no—
152C2×M16—44no—
153C2×M16—82no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C4⋊Q8—22no—
161C4⋊Q8—41no—
162C4⋊Q8—44no—
163C4⋊Q8—82no—
164#125—44no—
165#125—42no—
166#125—42no—
167C22⋊Q8—44no—
168C22⋊Q8—44no—
169C22⋊Q8—48no—
170C22⋊Q8—48no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×D4—24no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—44no—
177C4×D4—44no—
178C4×D4D814yes—
179C4×D4SD1614yes—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—82no—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2411yescharacteristic
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8—82no—
235C22⋊C8—44no—
236C22⋊C8—44no—
237C22⋊C8—82no—
238C4×C8—42no—
239C4×C8—42no—
240C4×C8—82no—
241#95—22no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—21no—
249#95—22no—
250#95—21no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C24C2×C22⋊C412yes—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—44no—
258C2×Q8—81no—
259C2×Q8—82no—
260C2×Q8—82no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4#10214yes—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×D814yes—
294C22×C4C2×SD1614yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298M16—88no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—42no—
317C4⋊C4—21no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—42no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—41no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4—81no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4#10114yes—
348C4⋊C4#10214yes—
349C4⋊C4#13112yes—
350C22⋊C4—24no—
351C22⋊C4—24no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4#10218yes—
355C42—21no—
356C42—21no—
357C42—22no—
358C42—42no—
359C42—41no—
360C42—41no—
361C42—42no—
362C42—41no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42C2×D812yes—
367C42C2×SD1612yes—
368C42#13712yes—
369C42#13712yes—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23C2×#10214yes—
377C23C22×C22⋊C411yescharacteristic
378Q8—84no—
379Q8—164no—
380D4—28no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—88no—
426C2×C4—164no—
427C2×C4C2×#10114yes—
428C2×C4C2×#10112yes—
429C2×C4C2×#10212yes—
430C2×C4C2×#10214yes—
431C2×C4C2×#10212yes—
432C2×C4#41912yes—
433C2×C4#41912yes—
434C2×C4C2×#13111yescharacteristic
435C2×C4#56811yescharacteristic
436C2×C4C22×D811yescharacteristic
437C2×C4C22×SD1611yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4C2×#13711yescharacteristic
440C8—42no—
441C8—42no—
442C8—81no—
443C8—81no—
444C8—82no—
445C8—82no—
446C22—24no—
447C22—24no—
448C22—24no—
449C22—42no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22#288011yescharacteristic
454C22#288311yescharacteristic
455C22C22×#10111yescharacteristic
456C22C22×#10211yescharacteristic
457C22C2×#41911yescharacteristic
458C22#290111yescharacteristic
459C22#290111yescharacteristic
460C4—21no—
461C4—21no—
462C4—24no—
463C4—22no—
464C4—22no—
465C4—21no—
466C4—21no—
467C4—24no—
468C4—21no—
469C4—21no—
470C4—41no—
471C4—42no—
472C4—42no—
473C4—82no—
474C4—81no—
475C4—81no—
476C4—82no—
477C4—81no—
478C4#154614yes—
479C2—44no—
480C2#1742611yescharacteristic
481C2#1750711yescharacteristic
482C2#1751411yescharacteristic
483C2#1761311yescharacteristic
484C2#1762711yescharacteristic
485C2#1769211yescharacteristic
486C2C2×#154611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
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
4. #7728C2semidirect
5. #7685C2semidirect
6. #7663C2semidirect
7. #7660C2semidirect
8. #7652C2semidirect
9. #7610C2semidirect
10. #7573C2semidirect
11. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#362612
#228352
#227002
#77282
#76852
#76632
#76602
#76522
#76102
#75732
#75702