Finite Group Databasebeta

← #161109

Subgroups — #161109

Order 512.

846 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34324C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic
3#22008C211yesmaximal, characteristic
4#7648C212yesmaximal, complemented
5#7646C212yesmaximal, complemented
6#7623C212yesmaximal, complemented
7#7613C212yesmaximal, complemented
8#7588C211yesmaximal, characteristic, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12C2×#556C411yescharacteristic
13#2373—22no—
14#2367—21no—
15#2365C411yescharacteristic
16#2362C412yes—
17#2355—21no—
18#2354C411yescharacteristic
19Q8×C4⋊C4C411yescharacteristic
20#2346C412yes—
21#2334—22no—
22#2326—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C22⋊Q8C2212yes—
25#2296C2212yes—
26#2295C2211yescharacteristic
27#2276—21no—
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1923—22no—
34#1831C2211yescharacteristic
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38#1770—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1749C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2211yescharacteristic
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2212yes—
50#1271C2212yes—
51#1271C2212yes—
52#1271C2212yes—
53#556—24no—
54#556—24no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—22no—
57C2×#124—21no—
58C2×C22⋊Q8C2×C412yes—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8C2×C411yescharacteristic
63C2×C4×Q8C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×#117C2×C412yes—
68C2×#117C2×C412yes—
69C22×C42C2312yes—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×#97—22no—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82#391—21no—
83#391—21no—
84#391C2×C411yescharacteristic
85#391C2311yescharacteristic
86#390—21no—
87#389—21no—
88#389—22no—
89#389C2×C411yescharacteristic
90#389C2×C412yes—
91#389C2×C412yes—
92#389C2312yes—
93#389C2311yescharacteristic
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#387—21no—
100#387—22no—
101#387—21no—
102#387—22no—
103#387—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#384—21no—
108#384C2×C411yescharacteristic
109#382—22no—
110#380C2×C412yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—22no—
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129#376—21no—
130#376—22no—
131C43C2311yescharacteristic
132#336—22no—
133#336—22no—
134#336—21no—
135#336—21no—
136#336C2311yescharacteristic
137#336C2312yes—
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×Q8C22×C411yescharacteristic
144C22×D4C22⋊C411yescharacteristic
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C4⋊Q8—22no—
155C4⋊Q8M1614yes—
156#126—216no—
157#124—28no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C4×Q8—28no—
161C4×Q8—22no—
162C4×Q8—44no—
163C4×Q8—42no—
164C4×Q8—42no—
165C4×Q8—82no—
166C4×Q8M1614yes—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—42no—
172#117—24no—
173#117—24no—
174#117—82no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—82no—
209C2×C42C22×C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2411yescharacteristic
219C4⋊C8—42no—
220C4⋊C8—42no—
221C4⋊C8—42no—
222C4⋊C8—82no—
223C22⋊C8—44no—
224C22⋊C8—82no—
225C22⋊C8—82no—
226C22⋊C8—82no—
227#97—82no—
228C4×C8—42no—
229C4×C8—42no—
230C4×C8—81no—
231#95—21no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C24C2×C22⋊C412yes—
248C2×Q8—21no—
249C2×Q8—24no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8C2×M1612yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C412yes—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C2×C8—22no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—41no—
303C2×C8—41no—
304C4⋊C4—28no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—81no—
322C4⋊C4—81no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4C2×M1612yes—
326C4⋊C4C2×M1614yes—
327C22⋊C4—28no—
328C22⋊C4—28no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C22⋊C4—82no—
333C22⋊C4—82no—
334C42—28no—
335C42—21no—
336C42—21no—
337C42—24no—
338C42—22no—
339C42—42no—
340C42—41no—
341C42—42no—
342C42—41no—
343C42—42no—
344C42—41no—
345C42—44no—
346C42—44no—
347C42—42no—
348C42—82no—
349C42—82no—
350C42C2×M1612yes—
351C42C2×M1612yes—
352C42#13512yes—
353C42#13512yes—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23C22×C22⋊C411yescharacteristic
362Q8—42no—
363Q8—42no—
364Q8—48no—
365D4—44no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#40712yes—
413C2×C4#40712yes—
414C2×C4#41612yes—
415C2×C4#41612yes—
416C2×C4#42112yes—
417C2×C4C22×M1611yescharacteristic
418C2×C4C22×M1611yescharacteristic
419C2×C4C2×#13511yescharacteristic
420C2×C4C2×#13511yescharacteristic
421C2×C4#57711yescharacteristic
422C2×C4#57811yescharacteristic
423C8—42no—
424C8—81no—
425C8—82no—
426C8—81no—
427C8—81no—
428C8—82no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22C2×#40711yescharacteristic
437C22#288311yescharacteristic
438C22#288911yescharacteristic
439C22C2×#41611yescharacteristic
440C22#289811yescharacteristic
441C22C2×#42111yescharacteristic
442C22#290111yescharacteristic
443C4—22no—
444C4—22no—
445C4—21no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—21no—
451C4—41no—
452C4—44no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—81no—
459C4—81no—
460C4—82no—
461C4#148814yes—
462C2—82no—
463C2#1737011yescharacteristic
464C2C2×#148811yescharacteristic
465C2#1745411yescharacteristic
466C2#1760011yescharacteristic
467C2#1761711yescharacteristic
468C2#1768311yescharacteristic
469C2#1772511yescharacteristic
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. #7648C2semidirect
5. #7646C2semidirect
6. #7623C2semidirect
7. #7613C2semidirect
8. #7588C2semidirect
9. #7577C2semidirect
10. #7570C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#343242
#228392
#220082
#76482
#76462
#76232
#76132
#75882
#75772
#75702
#75622