Finite Group Databasebeta

← #364360

Subgroups — #364360

Order 512.

1302 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1750C211yesmaximal, characteristic
2#14340C211yesmaximal, characteristic
3#13795C211yesmaximal, characteristic
4#11268C212yesmaximal, complemented
5#11267C212yesmaximal, complemented
6#10821C212yesmaximal, complemented
7#10820C212yesmaximal, complemented
8#10731C212yesmaximal
9#10730C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#384C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#380C2212yes—
19#1750C418yescomplemented
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29#1441—22no—
30#1439—22no—
31C4⋊#95—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C22⋊#95—22no—
36C24×C4—21no—
37C24×C4C2311yescharacteristic
38C23×C8C2312yes—
39C23×C8C2312yes—
40C2×#117—24no—
41C2×#117—24no—
42C2×#117—41no—
43C2×#117—41no—
44C2×#117—41no—
45C2×#117—41no—
46C2×#117—41no—
47C2×#117—41no—
48C2×#117C2×C414yes—
49C2×#117Q814yes—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4—41no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4D412yes—
64C22×C42—21no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8C2×C418yes—
70C2×#97—41no—
71C2×#97—41no—
72C2×#97C2×C414yescomplemented
73C2×C4×C8—41no—
74C2×C4×C8—41no—
75C2×C4×C8C2×C414yescomplemented
76#384—41no—
77#384—41no—
78#382—41no—
79#382—41no—
80#380—24no—
81#380—24no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90#340—24no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#336C2×C418yescomplemented
99C25C2×D411yescharacteristic
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—24no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—82no—
114C23×C4—82no—
115C23×C4C22×C412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×Q812yes—
120C23×C4#4114yes—
121C23×C4#4114yes—
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8C22×C414yes—
136C22×C8C22×C414yes—
137#117—44no—
138#117—44no—
139#117—44no—
140#117—44no—
141#117—48no—
142#117—48no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4C4⋊C418yes—
176C2×C4⋊C4#4114yes—
177C2×C4⋊C4#4114yes—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—42no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4C4⋊C418yes—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—24no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42C22×C412yes—
199C2×C42C22×C412yes—
200C2×C42C2×Q812yes—
201C2×C42C2×Q812yes—
202C22⋊C8—216no—
203#97—44no—
204C4×C8—44no—
205#95—44no—
206#95—44no—
207#95—82no—
208#95—82no—
209#95—82no—
210#95—82no—
211#95—82no—
212#95—82no—
213C24—21no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—22no—
221C24C2×C4⋊C412yes—
222C24#12412yes—
223C24#12412yes—
224C24C22×D411yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—48no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
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—42no—
283C22×C4—42no—
284C22×C4—84no—
285C22×C4—84no—
286C22×C4—88no—
287C22×C4—88no—
288C22×C4C2×C4⋊C414yes—
289C22×C4C2×C4⋊C414yes—
290C22×C4C2×C4⋊C412yes—
291C22×C4#11718yes—
292C22×C4#12412yes—
293C22×C4#12412yes—
294C22×C4#12518yes—
295C22×C4C23×C411yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C2×#4112yes—
298C22×C4C2×#4112yes—
299C22×C4C2×#4112yes—
300C22×C4C2×#4112yes—
301C22×C4H(5,𝔽2)11yescharacteristic
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C4H(5,𝔽2)11yescharacteristic
304C22×C4H(5,𝔽2)11yescharacteristic
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—48no—
324C4⋊C4—48no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—88no—
338C22⋊C4—48no—
339C22⋊C4—48no—
340C22⋊C4—88no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—48no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23#38214yes—
373C23C22×C4⋊C411yescharacteristic
374C23#52511yescharacteristic
375C23#52511yescharacteristic
376C23C2×#12411yescharacteristic
377C23C2×#12411yescharacteristic
378C23#53411yescharacteristic
379C23#53411yescharacteristic
380C23#57612yes—
381C23#57712yes—
382C23#57712yes—
383C23#57812yes—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—816no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4#38214yes—
438C2×C4C2×#11712yes—
439C2×C4C4×#4112yes—
440C2×C4#51811yescharacteristic
441C2×C4#51811yescharacteristic
442C2×C4C2×#12512yes—
443C2×C4#53312yes—
444C2×C4#54311yescharacteristic
445C2×C4#54311yescharacteristic
446C2×C4#55112yes—
447C2×C4#55112yes—
448C2×C4#55112yes—
449C2×C4#55112yes—
450C2×C4#55312yes—
451C2×C4#55312yes—
452C2×C4#55312yes—
453C2×C4#55312yes—
454C8—44no—
455C8—44no—
456C22—28no—
457C22—24no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—42no—
462C22—44no—
463C22—44no—
464C22—42no—
465C22—42no—
466C22—44no—
467C22C2×#38211yescharacteristic
468C22#229611yescharacteristic
469C22#231511yescharacteristic
470C22#256111yescharacteristic
471C22#256111yescharacteristic
472C22#259111yescharacteristic
473C22#259111yescharacteristic
474C22#291212yes—
475C22#291312yes—
476C22#312211yescharacteristic
477C22#312311yescharacteristic
478C22#312412yes—
479C22#312512yes—
480C22#312611yescharacteristic
481C22#312711yescharacteristic
482C22#313612yes—
483C22#313612yes—
484C22#313712yes—
485C22#313712yes—
486C22#319812yes—
487C22#319812yes—
488C22#320112yes—
489C22#320112yes—
490C4—24no—
491C4—44no—
492C4—44no—
493C4—84no—
494C4—84no—
495C4—162no—
496C4—162no—
497C4#235312yes—
498C4#235312yes—
499C4#256912yes—
500C4#256912yes—
501C2—44no—
502C2#1396611yescharacteristic
503C2#2076312yes—
504C2#2136811yescharacteristic
505C2#2136911yescharacteristic
506C2#2194512yes—
507C2#2911511yescharacteristic
508C2#2911711yescharacteristic
509C2#2912012yes—
510C2#2934011yescharacteristic
511C2#2934111yescharacteristic
512C2#2934512yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11268C2semidirect
5. #11267C2semidirect
6. #10821C2semidirect
7. #10820C2semidirect
19. #1750C4semidirect
72. C2×#97C2×C4semidirect
75. C2×C4×C8C2×C4semidirect
98. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17502
#143402
#137952
#112682
#112672
#108212
#108202
#107312
#107302