Finite Group Databasebeta

← #161102

Subgroups — #161102

Order 512.

922 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34316C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic, complemented
3#22008C211yesmaximal, characteristic, complemented
4#7648C212yesmaximal, complemented
5#7645C212yesmaximal, complemented
6#7623C212yesmaximal, complemented
7#7612C212yesmaximal, complemented
8#7590C211yesmaximal, characteristic, complemented
9#7574C211yesmaximal, characteristic, complemented
10#7571C211yesmaximal, characteristic, complemented
11#7560C211yesmaximal, characteristic, complemented
12C2×#547C411yescharacteristic
13#2374—22no—
14#2365C411yescharacteristic
15#2363—21no—
16#2362C412yes—
17#2354C411yescharacteristic
18#2354C411yescharacteristic
19#2345—21no—
20#2341C412yes—
21#2335—22no—
22#2324—21no—
23C4×C4⋊Q8C2211yescharacteristic
24C4×C22⋊Q8C2212yes—
25#2297C2212yes—
26#2293C2211yescharacteristic
27#2274—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#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×#41C2×C412yes—
54#547—24no—
55#547—24no—
56C2×C4⋊Q8C2×C411yescharacteristic
57C2×#123—22no—
58C2×C22⋊Q8C2×C412yes—
59C2×#121—21no—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C412yes—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4C2×C411yescharacteristic
64C2×C4×D4D411yescharacteristic
65C2×C4×D4D411yescharacteristic
66C2×#117—21no—
67C2×#117—21no—
68C2×#117C2×C412yes—
69C22×C4⋊C4—21no—
70C22×C42C2312yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97—22no—
80C2×C4×C8—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83#391C2×C412yes—
84#391C2312yes—
85#390—21no—
86#390—22no—
87#390—22no—
88#390C2×C411yescharacteristic
89#390C2×C412yes—
90#390C2311yescharacteristic
91#389—21no—
92#389—21no—
93#389—21no—
94#389C2×C411yescharacteristic
95#389C2311yescharacteristic
96#388—22no—
97#388—22no—
98#388—22no—
99#388—21no—
100#388—21no—
101#388—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#386—22no—
106#386—22no—
107#385—22no—
108#384C2×C411yescharacteristic
109#380—21no—
110#380—22no—
111#380C2×C412yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C412yes—
119C4×C4⋊C4C2×C411yescharacteristic
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
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
142C2×#41—24no—
143C2×#41—24no—
144C2×#41—82no—
145C2×#41—82no—
146C22×Q8C22×C412yes—
147C22×D4—22no—
148C22×D4C22⋊C411yescharacteristic
149C22×D4C22×C411yescharacteristic
150C23×C4—21no—
151C23×C4—22no—
152C23×C4C22×C412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—21no—
160C22×C8—22no—
161C4⋊Q8—22no—
162C4⋊Q8M1614yes—
163#123—216no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166#121—28no—
167C4×Q8—82no—
168C4×Q8—82no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—42no—
175C4×D4M1614yes—
176#117—44no—
177#117—42no—
178#117—42no—
179#117—82no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C412yes—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4C22⋊C412yes—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—82no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C412yes—
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2411yescharacteristic
226C4⋊C8—42no—
227C4⋊C8—42no—
228C4⋊C8—42no—
229C4⋊C8—82no—
230C22⋊C8—44no—
231C22⋊C8—82no—
232C22⋊C8—82no—
233C22⋊C8—82no—
234#97—82no—
235C4×C8—42no—
236C4×C8—42no—
237C4×C8—81no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—22no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95C22⋊C412yes—
248#95C22⋊C412yes—
249#95C22⋊C412yes—
250C24—21no—
251C24C2×C22⋊C412yes—
252#41—88no—
253#41—88no—
254C2×Q8—22no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8C2×M1614yes—
258C2×D4—28no—
259C2×D4—21no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4C2×M1612yes—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—28no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C2×C8—22no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C4⋊C4—28no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—41no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—81no—
334C4⋊C4—82no—
335C4⋊C4—82no—
336C4⋊C4C2×M1612yes—
337C22⋊C4—28no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—42no—
342C22⋊C4—44no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—81no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C42—21no—
353C42—21no—
354C42—22no—
355C42—42no—
356C42—41no—
357C42—42no—
358C42—41no—
359C42—42no—
360C42—41no—
361C42—44no—
362C42—42no—
363C42—82no—
364C42—82no—
365C42—82no—
366C42C2×M1612yes—
367C42C2×M1612yes—
368C42#13512yes—
369C42#13512yes—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23C22×C22⋊C411yescharacteristic
385Q8—44no—
386Q8—44no—
387D4—42no—
388D4—42no—
389D4—44no—
390D4—88no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—48no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—88no—
436C2×C4#40712yes—
437C2×C4#40712yes—
438C2×C4#41612yes—
439C2×C4#41612yes—
440C2×C4#42112yes—
441C2×C4C22×M1611yescharacteristic
442C2×C4C22×M1611yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4C2×#13511yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57811yescharacteristic
447C8—42no—
448C8—81no—
449C8—82no—
450C8—81no—
451C8—81no—
452C8—82no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22C2×#40711yescharacteristic
468C22#288311yescharacteristic
469C22#288811yescharacteristic
470C22C2×#41611yescharacteristic
471C22#289811yescharacteristic
472C22C2×#42111yescharacteristic
473C22#290111yescharacteristic
474C4—22no—
475C4—22no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—41no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—44no—
488C4—82no—
489C4—81no—
490C4—81no—
491C4#148814yes—
492C2—82no—
493C2—82no—
494C2#1736911yescharacteristic
495C2C2×#148811yescharacteristic
496C2#1745411yescharacteristic
497C2#1760011yescharacteristic
498C2#1761711yescharacteristic
499C2#1768411yescharacteristic
500C2#1772611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#547C4
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. #22839C2semidirect
3. #22008C2semidirect
4. #7648C2semidirect
5. #7645C2semidirect
6. #7623C2semidirect
7. #7612C2semidirect
8. #7590C2semidirect
9. #7574C2semidirect
10. #7571C2semidirect
11. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#343162
#228392
#220082
#76482
#76452
#76232
#76122
#75902
#75742
#75712
#75602