Finite Group Databasebeta

← #275379

Subgroups — #275379

Order 512.

552 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33153C211yesmaximal, characteristic
2#27949C211yesmaximal, characteristic
3#22632C211yesmaximal, characteristic
4#21364C211yesmaximal, characteristic
5#10653C211yesmaximal, characteristic, complemented
6#10649C211yesmaximal, characteristic, complemented
7#10593C211yesmaximal, characteristic, complemented
8#10589C211yesmaximal, characteristic, complemented
9#9615C211yesmaximal, characteristic, complemented
10#9612C211yesmaximal, characteristic, complemented
11#9567C211yesmaximal, characteristic, complemented
12#9567C211yesmaximal, characteristic, complemented
13#9136C211yesmaximal, characteristic
14#8834C211yesmaximal, characteristic
15#8726C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17#2857C2211yescharacteristic
18#2285C2211yescharacteristic
19C4×#117C2211yescharacteristic
20#2085—21no—
21#2085—21no—
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1987C2211yescharacteristic
25#1987C2211yescharacteristic
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1939C2211yescharacteristic
33#1939C2211yescharacteristic
34#1923—21no—
35#1923—21no—
36#1919—21no—
37#1918—21no—
38#1854—21no—
39#1854—21no—
40#1854—21no—
41#1854—21no—
42#1854—21no—
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1831—21no—
46#1825—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1776—21no—
50#1776—21no—
51C8×C4⋊C4C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1744—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C22⋊Q8—21no—
69C2×#117—21no—
70C2×#117D411yescharacteristic
71C2×#117D411yescharacteristic
72C22×C42C2311yescharacteristic
73#432—22no—
74#432—22no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8D411yescharacteristic
97C2×C22⋊C8D411yescharacteristic
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#391—21no—
104#389—21no—
105#389—21no—
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#382—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122#376—21no—
123C43—21no—
124#340—21no—
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#335—22no—
147#334—22no—
148#333—22no—
149#333—22no—
150C22×Q8—21no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4#4111yescharacteristic
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C22⋊Q8—82no—
169#117—22no—
170#117—22no—
171#117—22no—
172#117—41no—
173#117—81no—
174#117—81no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4C2×D411yescharacteristic
188C2×C4⋊C4C2×D411yescharacteristic
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—41no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42#4111yescharacteristic
209C2×C42C2411yescharacteristic
210#107—42no—
211#107—42no—
212#107—42no—
213#107—81no—
214#107—81no—
215#106—42no—
216#106—42no—
217#106—42no—
218#106—81no—
219#106—81no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—81no—
231C4⋊C8—81no—
232C4⋊C8#4112yes—
233C4⋊C8#4112yes—
234C22⋊C8—24no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—42no—
238C22⋊C8—81no—
239#97—24no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—42no—
245C4×C8—42no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254C24#12111yescharacteristic
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—81no—
258C2×Q8—81no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8#13712yes—
318C2×C8#13712yes—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4—82no—
344C4⋊C4#13512yes—
345C4⋊C4#13512yes—
346C4⋊C4#13712yes—
347C22⋊C4—24no—
348C22⋊C4—42no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C42—24no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—42no—
359C42—42no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—81no—
365C42—82no—
366C42C2×#4111yescharacteristic
367C42C2×#4111yescharacteristic
368C422−1+411yescharacteristic
369C422−1+411yescharacteristic
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23C2×#12111yescharacteristic
378Q8—161no—
379Q8—161no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4#52511yescharacteristic
426C2×C4#53611yescharacteristic
427C2×C4#54711yescharacteristic
428C2×C4#54711yescharacteristic
429C2×C4#55211yescharacteristic
430C2×C4#55211yescharacteristic
431C2×C4C2×#13511yescharacteristic
432C2×C4C2×#13511yescharacteristic
433C2×C4C2×#13711yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4C2×#13711yescharacteristic
436C2×C4#57511yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57711yescharacteristic
439C8—22no—
440C8—22no—
441C8—41no—
442C8—41no—
443C8—42no—
444C8—42no—
445C8—81no—
446C8—81no—
447C8—81no—
448C8—81no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22#241311yescharacteristic
457C22#303911yescharacteristic
458C22#304511yescharacteristic
459C22#305011yescharacteristic
460C22#305611yescharacteristic
461C22#306711yescharacteristic
462C22#306811yescharacteristic
463C4—21no—
464C4—21no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—42no—
469C4—42no—
470C4—42no—
471C4—81no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4#310211yescharacteristic
477C4#310711yescharacteristic
478C4#323111yescharacteristic
479C4#323311yescharacteristic
480C4#334411yescharacteristic
481C4#334411yescharacteristic
482C4#335711yescharacteristic
483C4#335711yescharacteristic
484C2—81no—
485C2#1927311yescharacteristic
486C2#1931511yescharacteristic
487C2#1941211yescharacteristic
488C2#1949311yescharacteristic
489C2#2459811yescharacteristic
490C2#2460411yescharacteristic
491C2#2462511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10653C2semidirect
6. #10649C2semidirect
7. #10593C2semidirect
8. #10589C2semidirect
9. #9615C2semidirect
10. #9612C2semidirect
11. #9567C2semidirect
12. #9567C2semidirect

Maximal subgroups

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

subgroupindex
#331532
#279492
#226322
#213642
#106532
#106492
#105932
#105892
#96152
#96122
#95672
#91362
#88342
#87262