Finite Group Databasebeta

← #164878

Subgroups — #164878

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36464C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic, complemented
3#22012C211yesmaximal, characteristic, complemented
4#7718C212yesmaximal, complemented
5#7714C212yesmaximal, complemented
6#7683C211yesmaximal, characteristic, complemented
7#7680C211yesmaximal, characteristic, complemented
8#7648C212yesmaximal, complemented
9#7635C211yesmaximal, characteristic, complemented
10#7600C212yesmaximal, complemented
11#7567C211yesmaximal, characteristic, complemented
12#2859C2211yescharacteristic
13#2854C2212yes—
14#2558—22no—
15#2538C411yescharacteristic
16#2537C411yescharacteristic
17#2518—22no—
18#2497—21no—
19#2495C412yes—
20#2426C412yes—
21#2422—21no—
22#2363—21no—
23#2354C411yescharacteristic
24Q8×C4⋊C4C411yescharacteristic
25#2347—21no—
26C4×#124C2211yescharacteristic
27C4×C22⋊Q8C2212yes—
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#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C2×#125—21no—
54C2×#124C2×C411yescharacteristic
55C2×#123—22no—
56C2×C22⋊Q8C2×C412yes—
57C2×#121—21no—
58C2×#121C2×C412yes—
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8C2×C412yes—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×#117—21no—
66C22×C42C2312yes—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×#97—21no—
76C2×#97—21no—
77C2×C4×C8—21no—
78C2×C4×C8—21no—
79#400—22no—
80#397—22no—
81#396—22no—
82#396—22no—
83#394—22no—
84#394—22no—
85#391C2311yescharacteristic
86#390—21no—
87#390—22no—
88#390C2×C412yes—
89#390C2311yescharacteristic
90#389—21no—
91#389—21no—
92#389—21no—
93#389—22no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2312yes—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—21no—
101#388—21no—
102#388—22no—
103#388—22no—
104#387—22no—
105#386—22no—
106#385—22no—
107#384—21no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#383—22no—
111#382—21no—
112#382C2×C411yescharacteristic
113#380—21no—
114#380—21no—
115#380C2×C412yes—
116#380C2×C412yes—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—22no—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
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×Q8C22×C411yescharacteristic
141C22×D4—22no—
142C22×D4C22⋊C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154#125—81no—
155#125—81no—
156#124—22no—
157#124—41no—
158#123—82no—
159#123—82no—
160C22⋊Q8—24no—
161C22⋊Q8—42no—
162#121—44no—
163#121—82no—
164C4×Q8—42no—
165C4×Q8—82no—
166C4×D4—42no—
167C4×D4—42no—
168C4×D4—42no—
169C4×D4—42no—
170#117—44no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
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×C412yes—
187C2×C4⋊C4C22×C411yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—82no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—42no—
217C4⋊C8—42no—
218C4⋊C8—82no—
219C4⋊C8—82no—
220C22⋊C8—82no—
221C22⋊C8—82no—
222C22⋊C8—82no—
223C22⋊C8—82no—
224#97—81no—
225#97—81no—
226C4×C8—42no—
227C4×C8—42no—
228#95—21no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—21no—
236#95—21no—
237#95—22no—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241C24—21no—
242C24C2×C22⋊C412yes—
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8#13112yes—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—44no—
255C2×D4—82no—
256C2×D4—82no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—44no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—81no—
322C4⋊C4—82no—
323C4⋊C4—81no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—82no—
338C22⋊C4—82no—
339C22⋊C4—82no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—42no—
350C42—42no—
351C42—41no—
352C42—42no—
353C42—44no—
354C42—42no—
355C42—82no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23C22×C22⋊C411yescharacteristic
371Q8—42no—
372Q8—42no—
373D4—44no—
374D4—81no—
375D4—81no—
376D4—164no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—84no—
423C2×C4#41312yes—
424C2×C4#42112yes—
425C2×C4#42112yes—
426C2×C4C2×#13111yescharacteristic
427C2×C4#56811yescharacteristic
428C2×C4#57611yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57811yescharacteristic
432C8—81no—
433C8—82no—
434C8—81no—
435C8—81no—
436C8—82no—
437C8—81no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—42no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22#288011yescharacteristic
453C22#288311yescharacteristic
454C22C2×#41311yescharacteristic
455C22#289711yescharacteristic
456C22#289811yescharacteristic
457C22C2×#42111yescharacteristic
458C22C2×#42111yescharacteristic
459C4—21no—
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—41no—
468C4—41no—
469C4—41no—
470C4—42no—
471C4—42no—
472C4—81no—
473C4—82no—
474C4—82no—
475C4—81no—
476C4#153514yes—
477C2—82no—
478C2—161no—
479C2#1742411yescharacteristic
480C2#1752511yescharacteristic
481C2#1752611yescharacteristic
482C2#1759911yescharacteristic
483C2#1760011yescharacteristic
484C2C2×#153511yescharacteristic
485C2#1778111yescharacteristic
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. #22012C2semidirect
3. #22012C2semidirect
4. #7718C2semidirect
5. #7714C2semidirect
6. #7683C2semidirect
7. #7680C2semidirect
8. #7648C2semidirect
9. #7635C2semidirect
10. #7600C2semidirect
11. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#364642
#220122
#77182
#77142
#76832
#76802
#76482
#76352
#76002
#75672