Finite Group Databasebeta

← #355474

Subgroups — #355474

Order 512.

2172 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34198C211yesmaximal, characteristic
2C2×#1889C212yesmaximal
3#20392C211yesmaximal, characteristic, complemented
4#20354C211yesmaximal, characteristic, complemented
5#10935C212yesmaximal, complemented
6#10915C212yesmaximal, complemented
7#10829C212yesmaximal, complemented
8#10754C212yesmaximal, complemented
9#10725C212yesmaximal, complemented
10C22×#120C2211yescharacteristic
11C2×#518C2212yes—
12C2×#518C2211yescharacteristic
13C23×C22⋊C4C2211yescharacteristic
14C22×#99C2212yes—
15#2358—24no—
16#2342—22no—
17D4×C22⋊C4—22no—
18#2321C414yescomplemented
19#2315C2211yescharacteristic
20#2308C2211yescharacteristic
21C4×#120C414yescomplemented
22C2×#379C2211yescharacteristic
23C2×#379C2214yes—
24#2273C2211yescharacteristic
25#2273C2212yes—
26#1889C4116yescomplemented
27C4×#99—24no—
28#1740—22no—
29C2×#342—22no—
30C2×#342C2212yes—
31C2×#342C2212yes—
32C2×#342C2212yes—
33C2×#342C2214yes—
34C2×#342C2214yes—
35C2×#342C2214yes—
36C26C2311yescharacteristic
37C23×D4—21no—
38C23×D4C2311yescharacteristic
39C24×C4C2×C412yescomplemented
40C2×#120—22no—
41C2×#120—24no—
42C2×#120C2×C412yes—
43C2×#120C2×C418yes—
44C2×#120C2×C412yes—
45#518—24no—
46#518—44no—
47#518—44no—
48#518C2×C418yescomplemented
49C2×C4×D4—24no—
50C2×C4×D4—44no—
51C2×C4×D4C2×C414yes—
52C2×C4×D4D414yes—
53C2×#117—22no—
54C2×#117D412yes—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—44no—
61C22×C22⋊C4—44no—
62C22×C22⋊C4C2×C412yescomplemented
63C22×C22⋊C4D412yes—
64C22×C22⋊C4C2311yescharacteristic
65C22×C22⋊C4C2311yescharacteristic
66C22×C22⋊C4C2314yes—
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2311yescharacteristic
69C22×C22⋊C4C2311yescharacteristic
70C22×C22⋊C4C2311yescharacteristic
71C2×#99—24no—
72C2×#99—24no—
73C2×#99C2×C418yescomplemented
74#390—24no—
75#388—24no—
76#388—24no—
77#386—22no—
78#386—24no—
79#386—24no—
80#386C2×C414yes—
81#385—22no—
82#385—24no—
83#385C2×C414yescomplemented
84#381—21no—
85#380—22no—
86#379—21no—
87#379—22no—
88#379—44no—
89#379C2×C412yescomplemented
90#379C2×C412yescomplemented
91#379C2×C4116yescomplemented
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—44no—
97C4×C22⋊C4C2×C414yescomplemented
98#342—24no—
99#342—24no—
100#342—24no—
101#342—44no—
102#342—44no—
103#342C2×C4116yescomplemented
104C25—21no—
105C25—21no—
106C25—24no—
107C25—22no—
108C25—22no—
109C25—22no—
110C25—84no—
111C25C22×C411yescharacteristic
112C25C22×C411yescharacteristic
113C25C22×C414yes—
114C25C2×D411yescharacteristic
115C25C2411yescharacteristic
116C22×D4—24no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—28no—
122C22×D4—84no—
123C22×D4C22×C412yes—
124C22×D4C22×C414yes—
125C22×D4C2×D412yes—
126C22×D4#4114yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—84no—
135C23×C4C22⋊C414yescomplemented
136C23×C4C22×C412yes—
137C23×C4C22×C412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140#120—24no—
141#120—24no—
142#120—216no—
143#120C4218yescomplemented
144#120C4218yescomplemented
145C4×D4—416no—
146C4×D4—48no—
147C4×D4—48no—
148C4×D4—416no—
149C4×D4—88no—
150#117—28no—
151#117—44no—
152#117—44no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C22⋊C414yescomplemented
156C2×C22⋊C4—21no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—21no—
176C2×C22⋊C4—21no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—48no—
186C2×C22⋊C4—48no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—416no—
191C2×C22⋊C4—84no—
192C2×C22⋊C4—816no—
193C2×C22⋊C4C22⋊C414yescomplemented
194C2×C22⋊C4C22×C418yes—
195C2×C22⋊C4C22×C418yes—
196C2×C22⋊C4C22×C412yes—
197C2×C22⋊C4C22×C412yes—
198C2×C22⋊C4C22×C414yes—
199C2×C22⋊C4C22×C412yes—
200C2×C22⋊C4C2×D412yes—
201C2×C22⋊C4C2×D412yes—
202C2×C22⋊C4C2×D412yes—
203C2×C42—24no—
204C2×C42—42no—
205C2×C42—44no—
206C2×C42C22⋊C414yescomplemented
207#99—216no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—44no—
212C24—21no—
213C24—22no—
214C24—22no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—22no—
220C24—22no—
221C24—24no—
222C24—24no—
223C24—28no—
224C24—28no—
225C24—21no—
226C24—22no—
227C24—44no—
228C24—44no—
229C24—416no—
230C24—44no—
231C24—42no—
232C24—42no—
233C24—41no—
234C24—41no—
235C24—42no—
236C24—44no—
237C24—44no—
238C24—82no—
239C24—88no—
240C24—82no—
241C24—82no—
242C24—88no—
243C24—82no—
244C24—88no—
245C24—88no—
246C24—88no—
247C24—82no—
248C24C2×C4212yes—
249C24C2×C4212yes—
250C24C2×C22⋊C412yes—
251C24C4×D414yes—
252C24C23×C411yescharacteristic
253C24C23×C412yes—
254C24C22×D411yescharacteristic
255C24C22×D411yescharacteristic
256C24C2×#4112yes—
257C24H(5,𝔽2)14yes—
258C24H(5,𝔽2)12yes—
259C242−1+412yes—
260C2×D4—24no—
261C2×D4—28no—
262C2×D4—24no—
263C2×D4—28no—
264C2×D4—24no—
265C2×D4—24no—
266C2×D4—28no—
267C2×D4—28no—
268C2×D4—28no—
269C2×D4—28no—
270C2×D4—48no—
271C2×D4—84no—
272C2×D4—816no—
273C2×D4C2×C4214yes—
274C2×D4#11714yes—
275C2×D4C4×D418yes—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—28no—
287C22×C4—28no—
288C22×C4—44no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—44no—
292C22×C4—44no—
293C22×C4—48no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—44no—
297C22×C4—84no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—84no—
304C22×C4—88no—
305C22×C4—88no—
306C22×C4—88no—
307C22×C4—88no—
308C22×C4—164no—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C414yes—
311C22×C4C2×C22⋊C414yes—
312C22×C4C4×D414yes—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—84no—
316C22⋊C4—22no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—28no—
323C22⋊C4—28no—
324C22⋊C4—28no—
325C22⋊C4—28no—
326C22⋊C4—28no—
327C22⋊C4—24no—
328C22⋊C4—28no—
329C22⋊C4—44no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—41no—
336C22⋊C4—44no—
337C22⋊C4—41no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—84no—
342C22⋊C4—84no—
343C22⋊C4—832no—
344C22⋊C4C2×C4214yes—
345C42—44no—
346C42—48no—
347C42—84no—
348C42—164no—
349C23—24no—
350C23—28no—
351C23—22no—
352C23—22no—
353C23—24no—
354C23—22no—
355C23—24no—
356C23—24no—
357C23—22no—
358C23—24no—
359C23—42no—
360C23—42no—
361C23—48no—
362C23—42no—
363C23—44no—
364C23—42no—
365C23—44no—
366C23—44no—
367C23—48no—
368C23—44no—
369C23—42no—
370C23—416no—
371C23—42no—
372C23—48no—
373C23—48no—
374C23—48no—
375C23—416no—
376C23—41no—
377C23—41no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—42no—
382C23—44no—
383C23—44no—
384C23—48no—
385C23—84no—
386C23—88no—
387C23—88no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—84no—
394C23—88no—
395C23—84no—
396C23—82no—
397C23—84no—
398C23—88no—
399C23—84no—
400C23—88no—
401C23—816no—
402C23—816no—
403C23—84no—
404C23—84no—
405C23—161no—
406C23—161no—
407C23C4×C22⋊C414yes—
408C23C22×C4211yescharacteristic
409C23C22×C22⋊C411yescharacteristic
410C23C2×#11711yescharacteristic
411C23C2×C4×D412yes—
412C23C2×C4×D412yes—
413C23#51812yes—
414C23#51814yes—
415C23#52012yes—
416C23#52014yes—
417C23#53412yes—
418C23#53412yes—
419C23#53511yescharacteristic
420C23#53611yescharacteristic
421C23#53712yes—
422C23#53814yes—
423C23#53914yes—
424D4—28no—
425D4—216no—
426D4—28no—
427D4—416no—
428D4—88no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—48no—
444C2×C4—48no—
445C2×C4—84no—
446C2×C4—84no—
447C2×C4—84no—
448C2×C4—816no—
449C2×C4—88no—
450C2×C4—88no—
451C2×C4—84no—
452C2×C4—88no—
453C2×C4—88no—
454C2×C4—164no—
455C2×C4—162no—
456C2×C4—164no—
457C2×C4—164no—
458C2×C4—168no—
459C2×C4C4×C22⋊C414yes—
460C22—22no—
461C22—24no—
462C22—24no—
463C22—24no—
464C22—24no—
465C22—28no—
466C22—24no—
467C22—44no—
468C22—42no—
469C22—42no—
470C22—44no—
471C22—44no—
472C22—41no—
473C22—42no—
474C22—42no—
475C22—44no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—44no—
480C22—42no—
481C22—44no—
482C22—44no—
483C22—48no—
484C22—42no—
485C22—42no—
486C22—48no—
487C22—44no—
488C22—44no—
489C22—42no—
490C22—42no—
491C22—84no—
492C22—82no—
493C22—84no—
494C22—88no—
495C22—816no—
496C22—88no—
497C22—82no—
498C22—84no—
499C22—82no—
500C22C2×C4×C22⋊C411yescharacteristic
501C22#227512yes—
502C22#230811yescharacteristic
503C22#231111yescharacteristic
504C22#231812yes—
505C22#232112yes—
506C22#232212yes—
507C22#232514yes—
508C4—24no—
509C4—44no—
510C4—44no—
511C4—44no—
512C4—82no—
513C4—88no—
514C4—88no—
515C4—82no—
516C2—21no—
517C2—22no—
518C2—21no—
519C2—24no—
520C2—22no—
521C2—22no—
522C2—22no—
523C2—44no—
524C2—44no—
525C2—82no—
526C2#1378411yescharacteristic
527C2#2177612yes—
subgroupG/H
center, socleC22#2308
commutatorC23C22×C42
Frattini, cosocleC25C24
prime radicalC22×#120C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20392C2semidirect
4. #20354C2semidirect
5. #10935C2semidirect
6. #10915C2semidirect
7. #10829C2semidirect
8. #10754C2semidirect
9. #10725C2semidirect
18. #2321C4semidirect
21. C4×#120C4semidirect
26. #1889C4semidirect
39. C24×C4C2×C4semidirect
48. #518C2×C4semidirect
62. C22×C22⋊C4C2×C4semidirect
73. C2×#99C2×C4semidirect
83. #385C2×C4semidirect
89. #379C2×C4semidirect
90. #379C2×C4semidirect
91. #379C2×C4semidirect
97. C4×C22⋊C4C2×C4semidirect
103. #342C2×C4semidirect
135. C23×C4C22⋊C4semidirect
143. #120C42semidirect
144. #120C42semidirect
155. C2×C4⋊C4C22⋊C4semidirect
193. C2×C22⋊C4C22⋊C4semidirect
206. C2×C42C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
#341982
C2×#18892
#203922
#203542
#109352
#109152
#108292
#107542
#107252