Finite Group Databasebeta

← #106572

Subgroups — #106572

Order 512.

989 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60124C211yesmaximal, characteristic
2#17357C211yesmaximal, characteristic
3#13777C211yesmaximal, characteristic
4#8098C212yesmaximal, complemented
5#7950C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7#2924—24no—
8#2924—24no—
9#2921—21no—
10#2921C412yescomplemented
11#2920—21no—
12#2920C2212yes—
13#2920C2212yes—
14#2905C412yescomplemented
15C2×#407—21no—
16#2873—21no—
17C2×#380—21no—
18C2×C4×C4⋊C4C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1464C412yescomplemented
21#1464C412yescomplemented
22#1463—21no—
23#1462—21no—
24#1458—21no—
25#1458—21no—
26#1437—21no—
27#1437C412yes—
28#1436—21no—
29#1436C412yes—
30C4⋊#95—21no—
31#1303—24no—
32#1285—24no—
33C24×C4D411yescharacteristic
34C22×M16—22no—
35C2×#117—22no—
36C2×#117—22no—
37C2×#117—22no—
38C2×#117—22no—
39C2×#117C2×C411yescharacteristic
40C2×#117C2×C411yescharacteristic
41C2×#117C2×C411yescharacteristic
42C2×#117C2×C411yescharacteristic
43C2×#117D411yescharacteristic
44C2×#117D411yescharacteristic
45C2×#117Q811yescharacteristic
46C2×#117Q811yescharacteristic
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4C2×C411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C22⋊C4—22no—
52C22×C22⋊C4C2×C411yescharacteristic
53C22×C22⋊C4D411yescharacteristic
54C22×C42—21no—
55C22×C42C2×C412yes—
56C22×C42D411yescharacteristic
57C22×C42C2311yescharacteristic
58#433—28no—
59#433—28no—
60#432—22no—
61#432—22no—
62#432—22no—
63#432—22no—
64#432C2×C414yes—
65#432D414yes—
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70#431—22no—
71#431D414yes—
72#424—28no—
73#423—22no—
74#423—22no—
75#423C2×C414yes—
76#407—22no—
77#407—22no—
78#407—22no—
79#407—22no—
80#407—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83#405—28no—
84C4×M16—22no—
85C4×M16—22no—
86C4×M16—22no—
87#380—28no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95C2×C412yes—
97#324—22no—
98#324—22no—
99#323—22no—
100#323—22no—
101C25C22⋊C411yescharacteristic
102C23×C4—24no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—42no—
110C23×C4C22⋊C411yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C4⋊C411yescharacteristic
116C23×C4C4⋊C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—44no—
123C2×M16—44no—
124C2×M16—42no—
125C22×C8—24no—
126#117—22no—
127#117—22no—
128#117—22no—
129#117—22no—
130#117—22no—
131#117—22no—
132#117—22no—
133#117—22no—
134#117—22no—
135#117—22no—
136#117—22no—
137#117—22no—
138#117—22no—
139#117—22no—
140#117—22no—
141#117—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4C4211yescharacteristic
155C2×C4⋊C4C4211yescharacteristic
156C2×C4⋊C4C22⋊C411yescharacteristic
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C4⋊C411yescharacteristic
159C2×C4⋊C4C4⋊C411yescharacteristic
160C2×C4⋊C4C4⋊C411yescharacteristic
161C2×C4⋊C4C4⋊C411yescharacteristic
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—82no—
168C2×C22⋊C4—82no—
169C2×C22⋊C4C4211yescharacteristic
170C2×C22⋊C4C4211yescharacteristic
171C2×C22⋊C4C22⋊C411yescharacteristic
172C2×C22⋊C4C22⋊C411yescharacteristic
173C2×C22⋊C4C4⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C4⋊C411yescharacteristic
176C2×C22⋊C4C4⋊C411yescharacteristic
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—24no—
182C2×C42—42no—
183C2×C42—82no—
184C2×C42—82no—
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C4⋊C411yescharacteristic
188C2×C42C4⋊C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2×D411yescharacteristic
194C2×C42C2×Q811yescharacteristic
195C2×C42#4112yes—
196C2×C42#4112yes—
197C4⋊C8—24no—
198C4⋊C8—24no—
199C4⋊C8—24no—
200C4⋊C8—24no—
201C4⋊C8—42no—
202C4⋊C8C22⋊C418yescomplemented
203C22⋊C8—24no—
204C22⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C22⋊C8—42no—
209#97—24no—
210#97—24no—
211#97—24no—
212C4×C8—24no—
213C4×C8—24no—
214C4×C8—42no—
215#95—28no—
216#95—24no—
217#95—42no—
218#95—82no—
219#95—82no—
220C24—22no—
221C24—21no—
222C24—21no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—41no—
228C24#9511yescharacteristic
229C24#9511yescharacteristic
230C24C2×C22⋊C411yescharacteristic
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4#9511yescharacteristic
283C22×C4#9511yescharacteristic
284C22×C4#9511yescharacteristic
285C22×C4#9511yescharacteristic
286C22×C4#9511yescharacteristic
287C22×C4#9511yescharacteristic
288C22×C4#10414yes—
289C22×C4C2×C4211yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C4⋊C411yescharacteristic
293C22×C4C2×C4⋊C411yescharacteristic
294C22×C4C2×C4⋊C411yescharacteristic
295C22×C4C4×D412yes—
296C22×C4C4×D412yes—
297C22×C4#12011yescharacteristic
298C22×C4#12011yescharacteristic
299C22×C4#12112yes—
300C22×C4#12112yes—
301C22×C4#12311yescharacteristic
302C22×C4#12311yescharacteristic
303M16—48no—
304M16—44no—
305M16—44no—
306M16—82no—
307M16—82no—
308C2×C8—28no—
309C2×C8—48no—
310C2×C8—44no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C22⋊C4—22no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—22no—
331C22⋊C4—88no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—84no—
341C42—84no—
342C42C2×C22⋊C412yes—
343C42#11712yes—
344C42C4×D412yes—
345C42C4×D412yes—
346C42#12012yes—
347C42C22⋊Q812yes—
348C42C22⋊Q812yes—
349C42#12312yes—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23—81no—
380C23—81no—
381C23#33912yes—
382C23C2×#9511yescharacteristic
383C23#38611yescharacteristic
384C23#38611yescharacteristic
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4#33912yes—
445C2×C4C4×C22⋊C412yes—
446C2×C4#37911yescharacteristic
447C2×C4#38012yes—
448C2×C4#38111yescharacteristic
449C2×C4#38511yescharacteristic
450C2×C4#38511yescharacteristic
451C2×C4#38512yes—
452C2×C4#38612yes—
453C2×C4C2×#10412yes—
454C2×C4#42112yes—
455C8—48no—
456C8—84no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—42no—
473C22—42no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—81no—
480C22—82no—
481C22—82no—
482C22—82no—
483C22—82no—
484C22—81no—
485C22C22⋊#9511yescharacteristic
486C22C2×#33911yescharacteristic
487C22#173311yescharacteristic
488C22#190911yescharacteristic
489C22#191011yescharacteristic
490C22#191411yescharacteristic
491C22#191511yescharacteristic
492C4—22no—
493C4—22no—
494C4—22no—
495C4—21no—
496C4—21no—
497C4—22no—
498C4—22no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—84no—
505C4—84no—
506C4#178912yes—
507C4#182612yes—
508C2—21no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—21no—
514C2—42no—
515C2—81no—
516C2#1073311yescharacteristic
517C2#1074211yescharacteristic
518C2#1075511yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8098C2semidirect
5. #7950C2semidirect
10. #2921C4semidirect
14. #2905C4semidirect
20. #1464C4semidirect
21. #1464C4semidirect
202. C4⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#601242
#173572
#137772
#80982
#79502