Finite Group Databasebeta

← #165085

Subgroups — #165085

Order 512.

830 conjugacy classes of subgroups.

The subgroup lattice has 485 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36534C211yesmaximal, characteristic
2#22830C212yesmaximal, complemented
3#7702C212yesmaximal, complemented
4#7696C212yesmaximal, complemented
5#7656C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic, complemented
7#7637C211yesmaximal, characteristic
8#7635C211yesmaximal, characteristic, complemented
9#7603C211yesmaximal, characteristic
10#7601C211yesmaximal, characteristic, complemented
11#7567C211yesmaximal, characteristic, complemented
12C2×#556C411yescharacteristic
13C2×#554C411yescharacteristic
14#2862C2211yescharacteristic
15#2861C2211yescharacteristic
16#2861C2211yescharacteristic
17#2565C411yescharacteristic
18#2565C411yescharacteristic
19#2563C411yescharacteristic
20#2560—22no—
21#2560—22no—
22#2547C411yescharacteristic
23#2544C411yescharacteristic
24#2544C411yescharacteristic
25#2536—22no—
26#2352—22no—
27C4×#127C2211yescharacteristic
28C4×#124C2211yescharacteristic
29C4×#124C2211yescharacteristic
30C42×Q8C2211yescharacteristic
31#1985—22no—
32#1917—22no—
33#1916—22no—
34#1839C2212yes—
35#1839C2212yes—
36#1831C2212yes—
37#1771—22no—
38#1767C2212yes—
39#1745C2212yes—
40#1744C2212yes—
41#1272C2212yes—
42#1272C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1270C2212yes—
46#1270C2212yes—
47#1270C2211yescharacteristic
48#1270C2211yescharacteristic
49#1270C2211yescharacteristic
50#1270C2211yescharacteristic
51#556—44no—
52#554—44no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#127C2×C411yescharacteristic
56C2×#126—21no—
57C2×#126—21no—
58C2×#125C2×C411yescharacteristic
59C2×#125C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8D412yes—
72C2×C4×D4—22no—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×#97—22no—
82C2×#97—22no—
83C2×C4×C8—22no—
84#398—22no—
85#398—22no—
86#397—22no—
87#394—22no—
88#394—22no—
89#393—22no—
90#391—22no—
91#391C2×C411yescharacteristic
92#391C2×C411yescharacteristic
93#391C2×C411yescharacteristic
94#391C2×C411yescharacteristic
95#391C2311yescharacteristic
96#391C2311yescharacteristic
97#390—22no—
98#390C2×C411yescharacteristic
99#390C2×C411yescharacteristic
100#390C2311yescharacteristic
101#390C2311yescharacteristic
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#387—22no—
109#387—22no—
110#386—22no—
111#386—22no—
112#384C2×C411yescharacteristic
113#382—22no—
114#382C2×C411yescharacteristic
115#382C2×C411yescharacteristic
116#380—22no—
117#380—22no—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D412yes—
123C4×C4⋊C4D412yes—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—22no—
126C43C2311yescharacteristic
127C43C2311yescharacteristic
128C43C2311yescharacteristic
129#336—22no—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2312yes—
139#336C2312yes—
140C22×Q8—21no—
141C22×Q8C22⋊C411yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×Q8C22×C411yescharacteristic
144C22×D4—22no—
145C22×D4C22×C411yescharacteristic
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157#127—22no—
158#127—41no—
159#126—44no—
160#126—44no—
161#125—42no—
162#125—42no—
163#124—22no—
164#124—22no—
165#124—41no—
166#124—41no—
167#124—42no—
168#124—42no—
169C22⋊Q8—44no—
170C22⋊Q8—44no—
171C22⋊Q8—44no—
172C22⋊Q8—44no—
173C4×Q8—24no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×Q8—42no—
178C4×D4—44no—
179C4×D4—44no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184#117—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4C22⋊C411yescharacteristic
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C4⋊C4C22⋊C412yes—
199C2×C4⋊C4C22⋊C412yes—
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2411yescharacteristic
235C4⋊C8—44no—
236C4⋊C8—44no—
237#97—44no—
238#97—44no—
239C4×C8—44no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250#95C22⋊C412yes—
251C24—21no—
252C24—21no—
253C2×Q8—22no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—82no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—44no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—44no—
274C2×D4—82no—
275C2×D4#13112yes—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C23×C411yescharacteristic
300C22×C4C22×D412yes—
301C2×C8—24no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4#10014yes—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—82no—
348C22⋊C4—82no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—41no—
368C42—42no—
369C42—42no—
370C23—22no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23C22×C22⋊C411yescharacteristic
381Q8—44no—
382Q8—81no—
383Q8—81no—
384Q8—81no—
385Q8—81no—
386Q8—162no—
387D4—42no—
388D4—42no—
389D4—81no—
390D4—81no—
391D4—88no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—84no—
429C2×C4—84no—
430C2×C4—162no—
431C2×C4—162no—
432C2×C4C2×#10012yes—
433C2×C4C2×#10012yes—
434C2×C4C2×#10012yes—
435C2×C4C2×#13111yescharacteristic
436C2×C4#56811yescharacteristic
437C2×C4#57512yes—
438C2×C4#57512yes—
439C8—44no—
440C8—82no—
441C8—82no—
442C8—82no—
443C22—42no—
444C22—44no—
445C22—81no—
446C22—81no—
447C22—82no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—82no—
452C22—82no—
453C22#288011yescharacteristic
454C22#288411yescharacteristic
455C22C22×#10011yescharacteristic
456C22#289612yes—
457C22#290112yes—
458C4—21no—
459C4—21no—
460C4—22no—
461C4—22no—
462C4—22no—
463C4—22no—
464C4—22no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—81no—
472C4—81no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—81no—
477C2—82no—
478C2—161no—
479C2#1742511yescharacteristic
480C2#1751312yes—
481C2#1762412yes—
482C2#1768511yescharacteristic
483C2#1773811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22830C2semidirect
3. #7702C2semidirect
4. #7696C2semidirect
5. #7656C2semidirect
6. #7649C2semidirect
8. #7635C2semidirect
10. #7601C2semidirect
11. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#365342
#228302
#77022
#76962
#76562
#76492
#76372
#76352
#76032
#76012
#75672