Finite Group Databasebeta

← #164844

Subgroups — #164844

Order 512.

743 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36407C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic
3#22012C211yesmaximal, characteristic
4#7728C212yesmaximal, complemented
5#7725C212yesmaximal, complemented
6#7683C211yesmaximal, characteristic, complemented
7#7680C211yesmaximal, characteristic, complemented
8#7646C212yesmaximal, complemented
9#7636C211yesmaximal, characteristic, complemented
10#7598C212yesmaximal, complemented
11#7568C211yesmaximal, characteristic, complemented
12#2859C2211yescharacteristic
13#2857C2212yes—
14#2574—22no—
15#2556—22no—
16#2552C411yescharacteristic
17#2551C411yescharacteristic
18#2454—21no—
19#2452—21no—
20#2441C412yes—
21#2439C412yes—
22#2370—21no—
23#2369—21no—
24#2354C411yescharacteristic
25#2354C411yescharacteristic
26C4×#125C2211yescharacteristic
27#2296C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1939—21no—
33#1924—22no—
34#1924—22no—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38#1797C2212yes—
39#1767C2211yescharacteristic
40#1767C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×#125—21no—
56C2×#125—21no—
57C2×#125C2×C411yescharacteristic
58C2×C22⋊Q8—22no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×#117C2×C412yes—
63C2×#117C2×C412yes—
64C22×C42C2312yes—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#399—22no—
78#398—22no—
79#396—22no—
80#396—22no—
81#395—22no—
82#393—22no—
83#391—22no—
84#391—22no—
85#390—21no—
86#390—21no—
87#389—21no—
88#389—21no—
89#389—22no—
90#389—21no—
91#389—21no—
92#389—22no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2311yescharacteristic
97#389C2311yescharacteristic
98#389C2312yes—
99#388—22no—
100#388—22no—
101#387—21no—
102#387—21no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#384C2×C412yes—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#382—21no—
111#382C2×C411yescharacteristic
112#380—22no—
113#380C2×C412yes—
114#380C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129C43C2311yescharacteristic
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140C22×Q8—22no—
141C22×D4C22⋊C411yescharacteristic
142C23×C4C22×C412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C4⋊Q8—44no—
152C4⋊Q8—82no—
153#125—22no—
154#125—41no—
155#125—44no—
156#125—81no—
157#125—81no—
158C22⋊Q8—82no—
159C22⋊Q8—82no—
160C4×D4—24no—
161C4×D4—42no—
162C4×D4—42no—
163#117—24no—
164#117—42no—
165#117—82no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—82no—
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—42no—
213C4⋊C8—42no—
214C4⋊C8—82no—
215C4⋊C8—82no—
216C22⋊C8—82no—
217C22⋊C8—82no—
218C22⋊C8—82no—
219C22⋊C8—82no—
220#97—81no—
221#97—81no—
222C4×C8—42no—
223C4×C8—42no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—22no—
233#95—21no—
234#95—21no—
235#95—22no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239C24C2×C22⋊C412yes—
240C2×Q8—44no—
241C2×Q8—82no—
242C2×Q8—82no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—82no—
316C4⋊C4—81no—
317C4⋊C4—82no—
318C4⋊C4—81no—
319C4⋊C4—82no—
320C4⋊C4#10114yes—
321C4⋊C4#13112yes—
322C22⋊C4—24no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—82no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—42no—
339C42—42no—
340C42—41no—
341C42—42no—
342C42—44no—
343C42—42no—
344C42—42no—
345C42—82no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23C22×C22⋊C411yescharacteristic
354Q8—164no—
355D4—44no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4C2×#10114yes—
404C2×C4C2×#10112yes—
405C2×C4#42112yes—
406C2×C4#42112yes—
407C2×C4C2×#13111yescharacteristic
408C2×C4#56811yescharacteristic
409C2×C4#57611yescharacteristic
410C2×C4#57711yescharacteristic
411C2×C4#57711yescharacteristic
412C2×C4#57811yescharacteristic
413C8—81no—
414C8—82no—
415C8—81no—
416C8—81no—
417C8—82no—
418C8—81no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22#288011yescharacteristic
427C22#288311yescharacteristic
428C22C22×#10111yescharacteristic
429C22#289711yescharacteristic
430C22#289811yescharacteristic
431C22C2×#42111yescharacteristic
432C22C2×#42111yescharacteristic
433C4—21no—
434C4—21no—
435C4—24no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—41no—
441C4—41no—
442C4—42no—
443C4—41no—
444C4—42no—
445C4—42no—
446C4—82no—
447C4—81no—
448C4—82no—
449C4—81no—
450C4—81no—
451C4#153714yes—
452C2—82no—
453C2#1742611yescharacteristic
454C2#1752311yescharacteristic
455C2#1752611yescharacteristic
456C2#1759911yescharacteristic
457C2#1760011yescharacteristic
458C2C2×#153711yescharacteristic
459C2#1777711yescharacteristic
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. #7725C2semidirect
6. #7683C2semidirect
7. #7680C2semidirect
8. #7646C2semidirect
9. #7636C2semidirect
10. #7598C2semidirect
11. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#364072
#220122
#77282
#77252
#76832
#76802
#76462
#76362
#75982
#75682