Finite Group Databasebeta

← #164471

Subgroups — #164471

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36508C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic, complemented
3#20572C211yesmaximal, characteristic, complemented
4#7696C212yesmaximal
5#7682C211yesmaximal, characteristic, complemented
6#7682C211yesmaximal, characteristic, complemented
7#7656C212yesmaximal
8#7635C211yesmaximal, characteristic
9#7624C212yesmaximal, complemented
10#7590C212yesmaximal, complemented
11#7567C211yesmaximal, characteristic
12C2×#557C411yescharacteristic
13C2×#556C411yescharacteristic
14C2×C8×Q8—21no—
15#2862C2211yescharacteristic
16#2861C2212yes—
17#2565C412yes—
18#2546C411yescharacteristic
19#2545C412yes—
20#2542C411yescharacteristic
21#2519—22no—
22#2519—22no—
23#2429—21no—
24#2429—21no—
25#2362—21no—
26#2362—21no—
27C4×C4⋊Q8C2212yes—
28C4×#124C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1839C2212yes—
34#1831C2212yes—
35#1770—21no—
36#1770—22no—
37#1767C2211yescharacteristic
38#1767C2211yescharacteristic
39#1767C2212yes—
40#1744C2211yescharacteristic
41#1744C2211yescharacteristic
42#1744C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2211yescharacteristic
48#1272C2211yescharacteristic
49#1272C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53#557—44no—
54#556—44no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8C2×C412yes—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×#126—22no—
60C2×#125—22no—
61C2×#125C2×C412yes—
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C412yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8C2×C412yes—
69C2×C4×Q8C2×C412yes—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8D411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×#117—21no—
76C8×Q8—28no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—22no—
81C2×#97—21no—
82C2×#97—22no—
83C2×#97—21no—
84C2×C4×C8—21no—
85C2×C4×C8—22no—
86C2×C4×C8—21no—
87#399—22no—
88#398—22no—
89#398—22no—
90#398—22no—
91#396—22no—
92#396—22no—
93#391—21no—
94#391—21no—
95#391C2×C412yes—
96#391C2×C412yes—
97#391C2311yescharacteristic
98#391C2312yes—
99#390C2×C412yes—
100#390C2311yescharacteristic
101#389—21no—
102#389—21no—
103#389—22no—
104#389—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—21no—
110#388—21no—
111#388—22no—
112#388—22no—
113#387—22no—
114#387—22no—
115#384C2×C412yes—
116#382—21no—
117#382—21no—
118#382C2×C411yescharacteristic
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4C2×C412yes—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C43C2312yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2312yes—
143C22×Q8—21no—
144C22×Q8C22⋊C411yescharacteristic
145C22×Q8C22×C412yes—
146C22×Q8C22×C411yescharacteristic
147C22×D4C22×C411yescharacteristic
148C23×C4—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C4⋊Q8—24no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159C4⋊Q8—44no—
160#126—48no—
161#125—44no—
162#125—48no—
163#124—22no—
164#124—41no—
165#124—44no—
166C22⋊Q8—44no—
167C22⋊Q8—44no—
168C22⋊Q8—44no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—42no—
172C4×Q8—44no—
173C4×Q8—44no—
174C4×Q8—42no—
175C4×Q8—44no—
176C4×D4—42no—
177#117—44no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C412yes—
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C22×C411yescharacteristic
214C2×C42C22×C412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2411yescharacteristic
224C4⋊C8—28no—
225C4⋊C8—24no—
226C4⋊C8—42no—
227C4⋊C8—82no—
228#97—82no—
229#97—81no—
230#97—81no—
231C4×C8—28no—
232C4×C8—24no—
233C4×C8—42no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C24—21no—
248C2×Q8—21no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8#10014yes—
261C2×Q8#13112yes—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—41no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—41no—
289C2×C8—42no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—41no—
302C2×C8—41no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4#10414yes—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—41no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C42—21no—
336C42—22no—
337C42—21no—
338C42—22no—
339C42—22no—
340C42—21no—
341C42—21no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—21no—
346C42—21no—
347C42—22no—
348C42—24no—
349C42—24no—
350C42—22no—
351C42—22no—
352C42—42no—
353C42—41no—
354C42—44no—
355C42—44no—
356C42—42no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23C22×C22⋊C411yescharacteristic
365Q8—28no—
366Q8—42no—
367Q8—42no—
368Q8—82no—
369Q8—82no—
370Q8—84no—
371D4—81no—
372D4—81no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—84no—
414C2×C4C2×#10014yes—
415C2×C4C2×#10012yes—
416C2×C4C2×#10414yes—
417C2×C4C2×#10412yes—
418C2×C4#42112yes—
419C2×C4C2×#13111yescharacteristic
420C2×C4#56811yescharacteristic
421C2×C4#57711yescharacteristic
422C2×C4#57711yescharacteristic
423C2×C4#57811yescharacteristic
424C2×C4#57811yescharacteristic
425C8—44no—
426C8—42no—
427C8—42no—
428C8—81no—
429C8—82no—
430C8—81no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22—81no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22#288011yescharacteristic
439C22#288411yescharacteristic
440C22C22×#10011yescharacteristic
441C22#289811yescharacteristic
442C22#289811yescharacteristic
443C22C22×#10411yescharacteristic
444C22C2×#42111yescharacteristic
445C4—21no—
446C4—21no—
447C4—24no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—21no—
452C4—24no—
453C4—42no—
454C4—42no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—42no—
459C4—82no—
460C4—81no—
461C4—81no—
462C4—82no—
463C4#153014yes—
464C2—161no—
465C2#1742511yescharacteristic
466C2#1751911yescharacteristic
467C2#1752511yescharacteristic
468C2#1754911yescharacteristic
469C2#1760111yescharacteristic
470C2C2×#153011yescharacteristic
471C2#1777311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×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. #22022C2semidirect
3. #20572C2semidirect
5. #7682C2semidirect
6. #7682C2semidirect
9. #7624C2semidirect
10. #7590C2semidirect

Maximal subgroups

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

subgroupindex
#365082
#220222
#205722
#76962
#76822
#76562
#76352
#76242
#75902
#75672