Finite Group Databasebeta

← #164988

Subgroups — #164988

Order 512.

743 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36476C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic
3#22835C211yesmaximal, characteristic
4#7721C212yesmaximal, complemented
5#7675C212yesmaximal, complemented
6#7662C211yesmaximal, characteristic, complemented
7#7662C211yesmaximal, characteristic, complemented
8#7652C211yesmaximal, characteristic, complemented
9#7623C212yesmaximal, complemented
10#7583C212yesmaximal, complemented
11#7570C211yesmaximal, characteristic, complemented
12#2862C2211yescharacteristic
13#2855C2212yes—
14#2607—22no—
15#2589—22no—
16#2580—21no—
17#2570—21no—
18#2563C411yescharacteristic
19#2542C411yescharacteristic
20#2492C412yes—
21#2423C412yes—
22#2369C411yescharacteristic
23#2365C411yescharacteristic
24#2357—21no—
25#2355—21no—
26C4×C4⋊Q8C2211yescharacteristic
27C4×C22⋊Q8C2212yes—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1924—22no—
32#1923—22no—
33#1831C2211yescharacteristic
34#1831C2211yescharacteristic
35#1817C2212yes—
36#1817C2212yes—
37#1770—21no—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1749C2212yes—
42C4×C22⋊C8C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×C22⋊Q8C2×C412yes—
57C2×C22⋊Q8C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4D411yescharacteristic
63C22×C42C2312yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×#97—22no—
73C2×C4×C8—21no—
74C2×C4×C8—21no—
75C2×C4×C8—22no—
76#400—22no—
77#399—22no—
78#399—22no—
79#398—22no—
80#393—22no—
81#393—22no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—22no—
86#391—22no—
87#391C2×C412yes—
88#391C2×C412yes—
89#391C2312yes—
90#390—21no—
91#390—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389C2×C412yes—
96#389C2311yescharacteristic
97#389C2311yescharacteristic
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#384—21no—
108#384C2×C411yescharacteristic
109#382—21no—
110#382—21no—
111#382—21no—
112#381—22no—
113#380C2×C412yes—
114#380C2×C412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—22no—
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—21no—
141C22×Q8—21no—
142C22×Q8C22×C412yes—
143C22×D4C22⋊C411yescharacteristic
144C23×C4C22×C412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—41no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C22⋊Q8—24no—
158C22⋊Q8—42no—
159C22⋊Q8—82no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—82no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—42no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22×C411yescharacteristic
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C412yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4C22⋊C412yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—82no—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2411yescharacteristic
210C4⋊C8—42no—
211C4⋊C8—42no—
212C4⋊C8—42no—
213C4⋊C8—42no—
214C22⋊C8—44no—
215C22⋊C8—44no—
216C22⋊C8—82no—
217C22⋊C8—82no—
218#97—82no—
219C4×C8—42no—
220C4×C8—42no—
221C4×C8—82no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239C24C2×C22⋊C412yes—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—81no—
248C2×Q8—81no—
249C2×Q8—81no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—42no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—81no—
319C4⋊C4—82no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4#13112yes—
323C22⋊C4—24no—
324C22⋊C4—24no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—82no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C42—21no—
331C42—21no—
332C42—22no—
333C42—42no—
334C42—42no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—82no—
342C42—82no—
343C42#13512yes—
344C42#13512yes—
345C42#13712yes—
346C42#13712yes—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23C22×C22⋊C411yescharacteristic
355Q8—84no—
356Q8—82no—
357Q8—82no—
358Q8—162no—
359D4—44no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#41312yes—
407C2×C4#41612yes—
408C2×C4#41612yes—
409C2×C4#41912yes—
410C2×C4#41912yes—
411C2×C4C2×#13111yescharacteristic
412C2×C4#56811yescharacteristic
413C2×C4C2×#13511yescharacteristic
414C2×C4C2×#13511yescharacteristic
415C2×C4C2×#13711yescharacteristic
416C2×C4C2×#13711yescharacteristic
417C8—42no—
418C8—42no—
419C8—81no—
420C8—82no—
421C8—81no—
422C8—82no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22#288011yescharacteristic
431C22#288311yescharacteristic
432C22C2×#41311yescharacteristic
433C22C2×#41611yescharacteristic
434C22C2×#41911yescharacteristic
435C22#290111yescharacteristic
436C22#290111yescharacteristic
437C4—21no—
438C4—21no—
439C4—22no—
440C4—22no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—42no—
446C4—41no—
447C4—42no—
448C4—42no—
449C4—42no—
450C4—82no—
451C4—82no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4#155414yes—
456C2—82no—
457C2#1742911yescharacteristic
458C2#1750811yescharacteristic
459C2#1751411yescharacteristic
460C2#1761711yescharacteristic
461C2#1762711yescharacteristic
462C2#1769111yescharacteristic
463C2C2×#155411yescharacteristic
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. #7721C2semidirect
5. #7675C2semidirect
6. #7662C2semidirect
7. #7662C2semidirect
8. #7652C2semidirect
9. #7623C2semidirect
10. #7583C2semidirect
11. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#364762
#228392
#228352
#77212
#76752
#76622
#76522
#76232
#75832
#75702