Finite Group Databasebeta

← #165014

Subgroups — #165014

Order 512.

943 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36276C211yesmaximal, characteristic
2#22835C211yesmaximal, characteristic
3#22700C211yesmaximal, characteristic
4#7727C212yesmaximal, complemented
5#7687C212yesmaximal, complemented
6#7662C211yesmaximal, characteristic, complemented
7#7662C211yesmaximal, characteristic, complemented
8#7653C211yesmaximal, characteristic, complemented
9#7609C212yesmaximal, complemented
10#7575C212yesmaximal, complemented
11#7571C211yesmaximal, characteristic, complemented
12C2×#554C411yescharacteristic
13C2×D4×Q8C411yescharacteristic
14C2×#436—21no—
15#2862C2211yescharacteristic
16#2857C2212yes—
17#2567C411yescharacteristic
18#2550—21no—
19#2550—21no—
20#2549—22no—
21#2549—22no—
22#2546C411yescharacteristic
23#2502C412yes—
24#2434C412yes—
25#2349—21no—
26#2349—21no—
27C4×C4⋊Q8C2211yescharacteristic
28C2×#391C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1923—22no—
33#1851—22no—
34#1831C2211yescharacteristic
35#1831C2211yescharacteristic
36#1817C2212yes—
37#1797C2212yes—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42#1728C2212yes—
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2212yes—
46#1272C2212yes—
47#1272C2211yescharacteristic
48#1272C2212yes—
49#1272C2212yes—
50#1272C2211yescharacteristic
51#1272C2211yescharacteristic
52#1272C2212yes—
53C23×Q8C2×C412yes—
54C22×M16—22no—
55#554—44no—
56D4×Q8—44no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8C2×C411yescharacteristic
59C2×C4⋊Q8C2×C411yescharacteristic
60C2×C4⋊Q8C2×C412yes—
61C2×C4⋊Q8C2×C411yescharacteristic
62C2×#125—21no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8C2×C412yes—
66C2×C22⋊Q8C2×C412yes—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C22×C4⋊C4C2×C412yes—
75C22×C42C2312yes—
76#436—44no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
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×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8—22no—
88#398—22no—
89#398—22no—
90#398—22no—
91#395—22no—
92#395—22no—
93#395—22no—
94#391—21no—
95#391—21no—
96#391—22no—
97#391—22no—
98#391C2×C412yes—
99#391C2×C412yes—
100#391C2×C412yes—
101#391C2311yescharacteristic
102#391C2311yescharacteristic
103#391C2312yes—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—22no—
111#389—22no—
112#385—22no—
113#385—22no—
114#385—22no—
115#385—22no—
116#384C2×C411yescharacteristic
117#384C2×C412yes—
118#382—21no—
119#382—21no—
120#380—22no—
121#380—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C22⋊C4C2×C412yes—
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C2×#95—22no—
131C2×#95—22no—
132C43C2311yescharacteristic
133#336—22no—
134#336—21no—
135#336—21no—
136#336—22no—
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140#336C2312yes—
141#336C2311yescharacteristic
142#336C2311yescharacteristic
143C22×Q8—22no—
144C22×Q8—21no—
145C22×Q8—24no—
146C22×Q8—42no—
147C22×Q8—44no—
148C22×Q8—82no—
149C22×Q8C22×C411yescharacteristic
150C22×Q8C22×C412yes—
151C22×Q8C22×C411yescharacteristic
152C22×D4C22⋊C411yescharacteristic
153C23×C4—22no—
154C23×C4—22no—
155C23×C4C22×C412yes—
156C23×C4C2×D412yes—
157C23×C4C2×D412yes—
158C2×M16—44no—
159C2×M16—44no—
160C2×M16—82no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—21no—
166C22×C8—21no—
167C4⋊Q8—22no—
168C4⋊Q8—41no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—44no—
172C4⋊Q8—82no—
173#125—44no—
174C22⋊Q8—48no—
175C22⋊Q8—48no—
176C22⋊Q8—44no—
177C22⋊Q8—44no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×D4—24no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—44no—
184C4×D4—44no—
185C4×D4SD1614yes—
186C4×D4Q1614yes—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C22⋊C4C22⋊C412yes—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—82no—
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2411yescharacteristic
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231C4⋊C8—42no—
232C4⋊C8—82no—
233C22⋊C8—44no—
234C22⋊C8—44no—
235C22⋊C8—82no—
236C4×C8—42no—
237C4×C8—42no—
238C4×C8—82no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95—22no—
248#95—22no—
249#95—21no—
250#95—21no—
251#95C22⋊C412yes—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254C24C2×C22⋊C412yes—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—22no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—44no—
261C2×Q8—42no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—41no—
267C2×Q8—42no—
268C2×Q8—44no—
269C2×Q8—82no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×Q8—88no—
273C2×Q8—88no—
274C2×Q8#13112yes—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4#10314yes—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—21no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—82no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C412yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×SD1614yes—
308C22×C4C2×Q1614yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312M16—88no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—42no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—41no—
335C4⋊C4—42no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4#10014yes—
352C4⋊C4#10314yes—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4#10318yes—
358C42—21no—
359C42—21no—
360C42—22no—
361C42—42no—
362C42—41no—
363C42—41no—
364C42—42no—
365C42—41no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42C2×SD1612yes—
370C42C2×Q1612yes—
371C42#13712yes—
372C42#13712yes—
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23C2×#10314yes—
380C23C22×C22⋊C411yescharacteristic
381Q8—42no—
382Q8—42no—
383Q8—88no—
384Q8—81no—
385Q8—81no—
386Q8—82no—
387Q8—82no—
388Q8—162no—
389D4—28no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—88no—
435C2×C4—164no—
436C2×C4C2×#10014yes—
437C2×C4C2×#10012yes—
438C2×C4C2×#10312yes—
439C2×C4C2×#10314yes—
440C2×C4C2×#10312yes—
441C2×C4#41912yes—
442C2×C4#41912yes—
443C2×C4C2×#13111yescharacteristic
444C2×C4#56811yescharacteristic
445C2×C4C22×SD1611yescharacteristic
446C2×C4C22×Q1611yescharacteristic
447C2×C4C2×#13711yescharacteristic
448C2×C4C2×#13711yescharacteristic
449C8—42no—
450C8—42no—
451C8—81no—
452C8—81no—
453C8—82no—
454C8—82no—
455C22—24no—
456C22—24no—
457C22—24no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22#288011yescharacteristic
463C22#288311yescharacteristic
464C22C22×#10011yescharacteristic
465C22C22×#10311yescharacteristic
466C22C2×#41911yescharacteristic
467C22#290111yescharacteristic
468C22#290111yescharacteristic
469C4—21no—
470C4—21no—
471C4—24no—
472C4—22no—
473C4—22no—
474C4—21no—
475C4—21no—
476C4—24no—
477C4—21no—
478C4—21no—
479C4—42no—
480C4—41no—
481C4—42no—
482C4—82no—
483C4—81no—
484C4—81no—
485C4—81no—
486C4—82no—
487C4#154814yes—
488C2—44no—
489C2#1742111yescharacteristic
490C2#1750911yescharacteristic
491C2#1751411yescharacteristic
492C2#1761411yescharacteristic
493C2#1762711yescharacteristic
494C2#1768811yescharacteristic
495C2C2×#154811yescharacteristic
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. #7727C2semidirect
5. #7687C2semidirect
6. #7662C2semidirect
7. #7662C2semidirect
8. #7653C2semidirect
9. #7609C2semidirect
10. #7575C2semidirect
11. #7571C2semidirect

Maximal subgroups

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

subgroupindex
#362762
#228352
#227002
#77272
#76872
#76622
#76532
#76092
#75752
#75712