Finite Group Databasebeta

← #164794

Subgroups — #164794

Order 512.

876 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36473C211yesmaximal, characteristic
2#22632C211yesmaximal, characteristic, complemented
3#20754C211yesmaximal, characteristic, complemented
4#7732C212yesmaximal, complemented
5#7685C211yesmaximal, characteristic, complemented
6#7668C211yesmaximal, characteristic, complemented
7#7666C212yesmaximal, complemented
8#7652C212yesmaximal, complemented
9#7616C211yesmaximal, characteristic
10#7588C212yesmaximal, complemented
11#7565C211yesmaximal, characteristic
12C2×#556—21no—
13C2×#424—21no—
14#2860C2212yes—
15#2857C2211yescharacteristic
16#2604—22no—
17#2551C411yescharacteristic
18#2542C411yescharacteristic
19#2521—22no—
20#2496C412yes—
21#2492—21no—
22#2491C412yes—
23#2415—21no—
24#2365C411yescharacteristic
25#2362—21no—
26#2354C411yescharacteristic
27C4×C4⋊Q8C2212yes—
28C4×#121C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1939—21no—
31#1923—22no—
32#1832—22no—
33#1831C2212yes—
34#1831C2212yes—
35#1831C2212yes—
36#1797C2211yescharacteristic
37C8×C4⋊C4—21no—
38C4×C22⋊C8C2211yescharacteristic
39#1744C2212yes—
40#1729—21no—
41#1728C2211yescharacteristic
42C2×#336C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C22×M16—21no—
54C23×C8—21no—
55#556—44no—
56#556—44no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#126—22no—
61C2×#126—22no—
62C2×#124—22no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×#121C2×C411yescharacteristic
67C2×C4×Q8—21no—
68C2×C4×Q8—22no—
69C2×C4×D4—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C42C2311yescharacteristic
75#424—44no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C4×C8—22no—
83C2×C4×C8—21no—
84#400—22no—
85#399—22no—
86#397—22no—
87#395—22no—
88#394—22no—
89#393—22no—
90#391—21no—
91#391—22no—
92#391—21no—
93#390C2×C412yes—
94#390C2×C412yes—
95#390C2311yescharacteristic
96#389—21no—
97#389—22no—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2311yescharacteristic
102#389C2312yes—
103#388—21no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#387—22no—
110#387—22no—
111#386—21no—
112#384C2×C412yes—
113#384C2×C411yescharacteristic
114#380—21no—
115#380—21no—
116#380C2×C412yes—
117#380C2×C411yescharacteristic
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129#376D412yes—
130C43C2312yes—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—21no—
144C22×Q8—22no—
145C22×D4—21no—
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C2×M16—42no—
152C2×M16—42no—
153C2×M16—42no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—42no—
161C22×C8—42no—
162C22×C8—42no—
163C4⋊Q8—24no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—44no—
167#126—48no—
168#126—44no—
169#126—44no—
170#124—48no—
171C22⋊Q8—44no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C22⋊Q8—44no—
175#121—22no—
176#121—41no—
177C4×Q8—44no—
178C4×Q8—44no—
179C4×Q8—44no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183#117—42no—
184#117—44no—
185#117D814yes—
186#117SD1614yes—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C411yescharacteristic
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—22no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—22no—
221C2×C42—42no—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C412yes—
224C2×C42C22×C412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2411yescharacteristic
232C4⋊C8—42no—
233C4⋊C8—44no—
234C4⋊C8—44no—
235C4⋊C8—82no—
236C22⋊C8—42no—
237C22⋊C8—81no—
238C4×C8—42no—
239C4×C8—44no—
240#95—21no—
241#95—22no—
242#95—21no—
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—
252#95C22⋊C412yes—
253C24—21no—
254C24C2×C22⋊C411yescharacteristic
255C2×Q8—44no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—44no—
259C2×Q8—42no—
260C2×Q8—82no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—42no—
267C2×D4—81no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×D812yes—
294C22×C4C2×SD1612yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298M16—84no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—84no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—44no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—41no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—81no—
338C4⋊C4#10214yes—
339C4⋊C4#10214yes—
340C4⋊C4#13112yes—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—42no—
346C22⋊C4—44no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—82no—
354C22⋊C4#10218yes—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—42no—
362C42—41no—
363C42—41no—
364C42—42no—
365C42—42no—
366C42—44no—
367C42—42no—
368C42—44no—
369C42C2×D814yes—
370C42C2×SD1614yes—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23C2×#10212yes—
385C23C22×C22⋊C411yescharacteristic
386Q8—88no—
387Q8—84no—
388D4—81no—
389D4—81no—
390D4—84no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—84no—
435C2×C4#41012yes—
436C2×C4C2×#10214yes—
437C2×C4C2×#10214yes—
438C2×C4C2×#10212yes—
439C2×C4#42112yes—
440C2×C4C2×#13111yescharacteristic
441C2×C4#56811yescharacteristic
442C2×C4C22×D811yescharacteristic
443C2×C4C22×SD1611yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57811yescharacteristic
446C8—42no—
447C8—81no—
448C8—82no—
449C8—81no—
450C8—81no—
451C8—82no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22#288011yescharacteristic
467C22C2×#41011yescharacteristic
468C22#288911yescharacteristic
469C22C22×#10211yescharacteristic
470C22#289811yescharacteristic
471C22C2×#42111yescharacteristic
472C22#290111yescharacteristic
473C4—21no—
474C4—21no—
475C4—24no—
476C4—24no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—41no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—41no—
486C4—81no—
487C4—82no—
488C4—81no—
489C4—82no—
490C4#151714yes—
491C2—42no—
492C2—161no—
493C2#1742011yescharacteristic
494C2#1750711yescharacteristic
495C2#1752511yescharacteristic
496C2C2×#151711yescharacteristic
497C2#1760511yescharacteristic
498C2#1768311yescharacteristic
499C2#1772111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22632C2semidirect
3. #20754C2semidirect
4. #7732C2semidirect
5. #7685C2semidirect
6. #7668C2semidirect
7. #7666C2semidirect
8. #7652C2semidirect
10. #7588C2semidirect

Maximal subgroups

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

subgroupindex
#364732
#226322
#207542
#77322
#76852
#76682
#76662
#76522
#76162
#75882
#75652