Finite Group Databasebeta

← #164023

Subgroups — #164023

Order 512.

962 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35307C211yesmaximal, characteristic
2#22473C211yesmaximal, characteristic
3#22469C211yesmaximal, characteristic
4#7721C212yesmaximal, complemented
5#7695C212yesmaximal, complemented
6#7681C212yesmaximal, complemented
7#7655C212yesmaximal, complemented
8#7598C211yesmaximal, characteristic, complemented
9#7575C211yesmaximal, characteristic, complemented
10#7575C211yesmaximal, characteristic, complemented
11#7564C211yesmaximal, characteristic, complemented
12C2×#536C411yescharacteristic
13C2×C4×#41C2211yescharacteristic
14C2×#436—21no—
15C2×C8×D4—21no—
16#2855C2212yes—
17#2855C2212yes—
18#2625C412yes—
19#2625C412yes—
20#2447—21no—
21#2447—21no—
22#2446—21no—
23#2446—21no—
24#2419—22no—
25#2419—22no—
26#2316C411yescharacteristic
27#2311C411yescharacteristic
28#2310C411yescharacteristic
29#2296C2211yescharacteristic
30C2×#391C2211yescharacteristic
31#1984—21no—
32#1984—21no—
33#1870—22no—
34#1869—22no—
35#1817C2212yes—
36#1817C2212yes—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40C4×C22⋊C8C2211yescharacteristic
41#1728C2212yes—
42C2×#336C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2211yescharacteristic
53C22×#41C2×C411yescharacteristic
54C23×Q8C2×C411yescharacteristic
55C22×M16—22no—
56C23×C8—22no—
57#536—44no—
58C2×C4⋊Q8—21no—
59C2×#124—21no—
60C2×#123—22no—
61C2×C22⋊Q8C2×C412yes—
62C2×C22⋊Q8C2×C412yes—
63C4×#41—44no—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8D411yescharacteristic
66C2×C4×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D412yes—
69C2×C4×D4D412yes—
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C412yes—
74C22×C4⋊C4C2×C411yescharacteristic
75C22×C42C2311yescharacteristic
76C22×C42C2312yes—
77#436—44no—
78C8×D4—44no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89#400—22no—
90#400—22no—
91#398—22no—
92#398—22no—
93#397—22no—
94#397—22no—
95#397—22no—
96#397—22no—
97#391—21no—
98#391—21no—
99#391—21no—
100#391—21no—
101#391—21no—
102#391C2×C412yes—
103#391C2×C412yes—
104#391C2×C412yes—
105#391C2312yes—
106#390—21no—
107#389—21no—
108#389—21no—
109#389C2×C412yes—
110#389C2×C412yes—
111#389C2×C411yescharacteristic
112#389C2×C411yescharacteristic
113#389C2311yescharacteristic
114#389C2311yescharacteristic
115#387—22no—
116#387—22no—
117#387—22no—
118#387—22no—
119#386—22no—
120#385—22no—
121#385—22no—
122#385—21no—
123#385—21no—
124#381—22no—
125#381—22no—
126#381—22no—
127#380—22no—
128#380C2×C412yes—
129#380C2×C412yes—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C2×#95—21no—
133C2×#95—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137#336—22no—
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144C2×#41—24no—
145C2×#41—24no—
146C2×#41—42no—
147C22×Q8—21no—
148C22×Q8—24no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8C22⋊C411yescharacteristic
152C22×Q8C22×C412yes—
153C22×D4C22⋊C412yes—
154C22×D4C22⋊C411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C22×C412yes—
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C2×M16—44no—
164C2×M16—44no—
165C2×M16—82no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—44no—
173C22×C8—44no—
174C22×C8—82no—
175C4⋊Q8—44no—
176#124—44no—
177#123—48no—
178C22⋊Q8—44no—
179C22⋊Q8—44no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—44no—
185C4×D4—44no—
186#117—24no—
187#117—44no—
188#117—42no—
189#117—42no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22⋊C411yescharacteristic
202C2×C4⋊C4C22⋊C412yes—
203C2×C4⋊C4C22⋊C412yes—
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C412yes—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—42no—
221C2×C42—44no—
222C2×C42C22×C412yes—
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D412yes—
227C2×C42C2×D412yes—
228C2×C42C2411yescharacteristic
229C4⋊C8—42no—
230C4⋊C8—42no—
231C22⋊C8—42no—
232C22⋊C8—42no—
233C22⋊C8—44no—
234C22⋊C8—44no—
235C22⋊C8—81no—
236C22⋊C8—81no—
237C4×C8—42no—
238C4×C8—42no—
239#95—22no—
240#95—22no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—21no—
254#95—21no—
255C24C2×C22⋊C411yescharacteristic
256C24C2×C22⋊C412yes—
257#41—416no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—88no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—42no—
273C2×D4#9914yes—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C412yes—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4#13114yes—
314C22×C4C23×C411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317M16—88no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—88no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—42no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C42—22no—
367C42—22no—
368C42—41no—
369C42—41no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C23—22no—
376C23—21no—
377C23—21no—
378C23—22no—
379C23—21no—
380C23—21no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—21no—
386C23—21no—
387C23#40812yes—
388C23C2×#9914yes—
389C23C22×C22⋊C411yescharacteristic
390Q8—44no—
391Q8—84no—
392Q8—84no—
393D4—28no—
394D4—48no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—28no—
402C2×C4—22no—
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—21no—
412C2×C4—21no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—48no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—84no—
445C2×C4—162no—
446C2×C4#40812yes—
447C2×C4C2×#9912yes—
448C2×C4#41312yes—
449C2×C4C2×#13112yes—
450C2×C4#57711yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57811yescharacteristic
453C2×C4#57811yescharacteristic
454C8—42no—
455C8—42no—
456C8—81no—
457C8—81no—
458C8—82no—
459C8—82no—
460C22—22no—
461C22—22no—
462C22—24no—
463C22—24no—
464C22—22no—
465C22—24no—
466C22—41no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—41no—
471C22—42no—
472C22—41no—
473C22—41no—
474C22C2×#40811yescharacteristic
475C22C22×#9911yescharacteristic
476C22C2×#41311yescharacteristic
477C22#289811yescharacteristic
478C22#289811yescharacteristic
479C22#290111yescharacteristic
480C22#290111yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—42no—
487C4—42no—
488C4—41no—
489C4—41no—
490C4—42no—
491C4—82no—
492C4—81no—
493C4—81no—
494C4—81no—
495C4—81no—
496C4—82no—
497C4#146814yes—
498C2—42no—
499C2—44no—
500C2C2×#146811yescharacteristic
501C2#1746911yescharacteristic
502C2#1748411yescharacteristic
503C2#1760811yescharacteristic
504C2#1760811yescharacteristic
505C2#1769111yescharacteristic
506C2#1777811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7721C2semidirect
5. #7695C2semidirect
6. #7681C2semidirect
7. #7655C2semidirect
8. #7598C2semidirect
9. #7575C2semidirect
10. #7575C2semidirect
11. #7564C2semidirect

Maximal subgroups

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

subgroupindex
#353072
#224732
#224692
#77212
#76952
#76812
#76552
#75982
#75752
#75642