Finite Group Databasebeta

← #164388

Subgroups — #164388

Order 512.

971 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36406C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic, complemented
3#20558C211yesmaximal, characteristic, complemented
4#7707C212yesmaximal, complemented
5#7682C211yesmaximal, characteristic, complemented
6#7682C211yesmaximal, characteristic, complemented
7#7667C212yesmaximal, complemented
8#7635C211yesmaximal, characteristic, complemented
9#7612C212yesmaximal, complemented
10#7578C212yesmaximal, complemented
11#7567C211yesmaximal, characteristic, complemented
12C2×#556C411yescharacteristic
13C2×#547C411yescharacteristic
14C2×C8×D4—21no—
15#2862C2211yescharacteristic
16#2857C2212yes—
17#2553—22no—
18#2553—22no—
19#2546C411yescharacteristic
20#2542C411yescharacteristic
21#2499C412yes—
22#2432C412yes—
23#2427—21no—
24#2427—21no—
25#2360—21no—
26#2360—21no—
27C4×#124C2211yescharacteristic
28#2297C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1817C2212yes—
34#1797C2212yes—
35#1770—21no—
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1752—22no—
39#1749C2212yes—
40#1744C2211yescharacteristic
41#1744C2211yescharacteristic
42C2×#336C2212yes—
43#1272C2212yes—
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2211yescharacteristic
48#1272C2212yes—
49#1272C2211yescharacteristic
50#1270C2211yescharacteristic
51#1270C2211yescharacteristic
52#1270C2212yes—
53C22×#41C2×C412yes—
54C23×C8—22no—
55#556—44no—
56#547—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×#126—22no—
61C2×#124—21no—
62C2×#124C2×C411yescharacteristic
63C2×#123—22no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8C2×C412yes—
67C2×C22⋊Q8C2×C412yes—
68C2×#121—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×#117C2×C412yes—
76C22×C4⋊C4—21no—
77C22×C42C2312yes—
78C8×D4—28no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×#97—21no—
86C2×#97—22no—
87C2×#97—21no—
88C2×C4×C8—21no—
89C2×C4×C8—21no—
90#399—22no—
91#398—22no—
92#397—22no—
93#397—22no—
94#394—22no—
95#394—22no—
96#391—21no—
97#391—21no—
98#391C2×C412yes—
99#391C2×C412yes—
100#391C2311yescharacteristic
101#391C2312yes—
102#390C2×C412yes—
103#390C2311yescharacteristic
104#389—21no—
105#389—21no—
106#389—22no—
107#389—22no—
108#388—21no—
109#388—21no—
110#387—22no—
111#387—22no—
112#386—22no—
113#386—22no—
114#385—22no—
115#385—22no—
116#384C2×C412yes—
117#382—21no—
118#382—21no—
119#382C2×C411yescharacteristic
120#380—22no—
121#380—22no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4C2×C412yes—
130C4×C22⋊C4D412yes—
131C4×C22⋊C4D412yes—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—22no—
136#336—21no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2311yescharacteristic
141#336C2312yes—
142#336C2311yescharacteristic
143C2×#41—24no—
144C2×#41—42no—
145C2×#41—44no—
146C2×#41—82no—
147C22×Q8—21no—
148C22×Q8C22×C411yescharacteristic
149C22×Q8C22×C412yes—
150C22×D4—22no—
151C22×D4C22⋊C411yescharacteristic
152C22×D4C22×C411yescharacteristic
153C23×C4—21no—
154C23×C4—22no—
155C23×C4C22×C412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—28no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—82no—
166C22×C8—82no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—82no—
170#126—48no—
171#124—22no—
172#124—41no—
173#124—44no—
174#123—48no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177C22⋊Q8—44no—
178C22⋊Q8—44no—
179#121—44no—
180C4×Q8—42no—
181C4×Q8—44no—
182C4×D4—24no—
183C4×D4—24no—
184C4×D4—42no—
185C4×D4—42no—
186#117—44no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4C22⋊C412yes—
215C2×C22⋊C4C22⋊C412yes—
216C2×C22⋊C4C22⋊C412yes—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—82no—
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2411yescharacteristic
230C4⋊C8—24no—
231C4⋊C8—42no—
232C4⋊C8—82no—
233C22⋊C8—28no—
234C22⋊C8—82no—
235C22⋊C8—82no—
236#97—81no—
237#97—81no—
238#97—82no—
239C4×C8—24no—
240C4×C8—42no—
241#95—21no—
242#95—21no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—22no—
250#95—22no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C24—21no—
255C24C2×C22⋊C412yes—
256#41—88no—
257#41—88no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—42no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×D4—21no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—82no—
277C2×D4#10014yes—
278C2×D4#13112yes—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—82no—
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—24no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—88no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—41no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4#10414yes—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C22⋊C4—41no—
360C22⋊C4—41no—
361C22⋊C4—44no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—42no—
372C42—42no—
373C42—41no—
374C42—42no—
375C42—44no—
376C42—42no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23C2×#10014yes—
391C23C22×C22⋊C411yescharacteristic
392Q8—81no—
393Q8—81no—
394Q8—82no—
395Q8—82no—
396Q8—84no—
397D4—28no—
398D4—42no—
399D4—42no—
400D4—88no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—41no—
429C2×C4—42no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—88no—
446C2×C4—84no—
447C2×C4C2×#10012yes—
448C2×C4C2×#10414yes—
449C2×C4C2×#10412yes—
450C2×C4#42112yes—
451C2×C4C2×#13111yescharacteristic
452C2×C4#56811yescharacteristic
453C2×C4#57711yescharacteristic
454C2×C4#57711yescharacteristic
455C2×C4#57811yescharacteristic
456C2×C4#57811yescharacteristic
457C8—42no—
458C8—44no—
459C8—42no—
460C8—81no—
461C8—82no—
462C8—81no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—81no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22#288011yescharacteristic
478C22#288311yescharacteristic
479C22C22×#10011yescharacteristic
480C22#289811yescharacteristic
481C22#289811yescharacteristic
482C22C22×#10411yescharacteristic
483C22C2×#42111yescharacteristic
484C4—21no—
485C4—21no—
486C4—21no—
487C4—21no—
488C4—24no—
489C4—21no—
490C4—21no—
491C4—42no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—42no—
496C4—82no—
497C4—81no—
498C4—81no—
499C4—81no—
500C4—82no—
501C4#153014yes—
502C2—44no—
503C2—82no—
504C2#1742111yescharacteristic
505C2#1751811yescharacteristic
506C2#1752311yescharacteristic
507C2#1754811yescharacteristic
508C2#1760011yescharacteristic
509C2C2×#153011yescharacteristic
510C2#1777911yescharacteristic
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. #22012C2semidirect
3. #20558C2semidirect
4. #7707C2semidirect
5. #7682C2semidirect
6. #7682C2semidirect
7. #7667C2semidirect
8. #7635C2semidirect
9. #7612C2semidirect
10. #7578C2semidirect
11. #7567C2semidirect

Maximal subgroups

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

subgroupindex
#364062
#220122
#205582
#77072
#76822
#76672
#76352
#76122
#75782
#75672