Finite Group Databasebeta

← #362806

Subgroups — #362806

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2C2×#1750C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#11208C212yesmaximal, complemented
5#11207C212yesmaximal, complemented
6#10881C212yesmaximal, complemented
7#10880C212yesmaximal, complemented
8#10731C212yesmaximal
9#10730C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×C22⋊C8C2212yes—
14C22×#97C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2316—24no—
17#2316C418yescomplemented
18C2×#389—22no—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380—22no—
24C2×#380C2212yes—
25#1750—24no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1441—22no—
36#1439—22no—
37C4⋊#95C2212yes—
38C24×C4—21no—
39C24×C4C2311yescharacteristic
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×#117—22no—
43C2×#117—41no—
44C2×#117—41no—
45C2×#117—41no—
46C2×#117—41no—
47C2×#117—41no—
48C2×#117—41no—
49C2×#117—41no—
50C2×#117—41no—
51C2×#117C2×C414yes—
52C2×#117C2×C414yes—
53C2×#117D414yes—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2×C414yes—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D412yes—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C22⋊C8—24no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8Q818yes—
76C2×#97—22no—
77C2×#97—41no—
78C2×#97—41no—
79C2×C4×C8—22no—
80C2×C4×C8—41no—
81C2×C4×C8—41no—
82#389—216no—
83#389—24no—
84#389—24no—
85#389C2×C418yes—
86#389C2×C418yes—
87#384—41no—
88#384—41no—
89#382—41no—
90#382—41no—
91#380—216no—
92#380—24no—
93#380—24no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340—42no—
107#340—42no—
108#336—24no—
109C25C2×D411yescharacteristic
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—82no—
124C23×C4—82no—
125C23×C4C22×C412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×Q814yes—
131C23×C4#4114yes—
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8C2×Q814yes—
146C22×C8C2×Q814yes—
147#117—44no—
148#117—44no—
149#117—44no—
150#117—44no—
151#117—44no—
152#117—44no—
153#117—44no—
154#117—44no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—216no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—216no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4—82no—
189C2×C4⋊C4—82no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4C22⋊C418yes—
192C2×C4⋊C4C22×C414yes—
193C2×C4⋊C4C22×C414yes—
194C2×C4⋊C4#4114yes—
195C2×C4⋊C4#4114yes—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4C22⋊C418yes—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—24no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42C22×C412yes—
217C2×C42C22×C412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C22⋊C8—216no—
221#97—44no—
222C4×C8—44no—
223#95—28no—
224#95—28no—
225#95—44no—
226#95—44no—
227#95—82no—
228#95—82no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24C2×C22⋊C412yes—
238C24#12412yes—
239C24C4⋊Q812yes—
240C24C22×D411yescharacteristic
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—216no—
275C22×C4—24no—
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—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—88no—
301C22×C4—88no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4C2×C22⋊C414yes—
305C22×C4C2×C22⋊C414yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C4×Q818yes—
308C22×C4C22⋊Q818yes—
309C22×C4#12412yes—
310C22×C4C4⋊Q812yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×Q812yes—
314C22×C4C2×#4112yes—
315C22×C4C2×#4112yes—
316C22×C4C2×#4112yes—
317C22×C4H(5,𝔽2)11yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C4H(5,𝔽2)11yescharacteristic
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C4⋊C4—216no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—48no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—88no—
353C4⋊C4—88no—
354C4⋊C4—88no—
355C4⋊C4#11718yes—
356C22⋊C4—48no—
357C22⋊C4—48no—
358C22⋊C4—88no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—48no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—24no—
374C23—24no—
375C23—24no—
376C23—24no—
377C23—24no—
378C23—24no—
379C23—22no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—42no—
390C23#39114yes—
391C23C22×C22⋊C411yescharacteristic
392C23#52511yescharacteristic
393C23#52511yescharacteristic
394C23C2×#12411yescharacteristic
395C23C2×C4⋊Q811yescharacteristic
396C23#53411yescharacteristic
397C23#53411yescharacteristic
398C23#57612yes—
399C23#57712yes—
400C23#57712yes—
401C23#57812yes—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—28no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—28no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—816no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4#39114yes—
456C2×C4C2×#11712yes—
457C2×C4C2×C4×Q812yes—
458C2×C4#51811yescharacteristic
459C2×C4#51811yescharacteristic
460C2×C4C2×C22⋊Q812yes—
461C2×C4#52512yes—
462C2×C4#53411yescharacteristic
463C2×C4#53411yescharacteristic
464C2×C4#55312yes—
465C2×C4#55312yes—
466C2×C4#55312yes—
467C2×C4#55312yes—
468C2×C4#55412yes—
469C2×C4#55412yes—
470C2×C4#55412yes—
471C2×C4#55412yes—
472C8—44no—
473C8—44no—
474C22—28no—
475C22—24no—
476C22—42no—
477C22—42no—
478C22—44no—
479C22—42no—
480C22—42no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—44no—
485C22#228011yescharacteristic
486C22C2×#39111yescharacteristic
487C22#230811yescharacteristic
488C22#258311yescharacteristic
489C22#258311yescharacteristic
490C22#259111yescharacteristic
491C22#259111yescharacteristic
492C22#289712yes—
493C22#289812yes—
494C22#311812yes—
495C22#312112yes—
496C22#312211yescharacteristic
497C22#312311yescharacteristic
498C22#312611yescharacteristic
499C22#312711yescharacteristic
500C22#313612yes—
501C22#313712yes—
502C22#316312yes—
503C22#316412yes—
504C22#319812yes—
505C22#320112yes—
506C22#320312yes—
507C22#320412yes—
508C4—24no—
509C4—48no—
510C4—44no—
511C4—44no—
512C4—84no—
513C4—162no—
514C4—162no—
515C4#234512yes—
516C4#234512yes—
517C4#257912yes—
518C4#257912yes—
519C2—44no—
520C2#1392311yescharacteristic
521C2#2074411yescharacteristic
522C2#2074511yescharacteristic
523C2#2132612yes—
524C2#2177212yes—
525C2#2930611yescharacteristic
526C2#2930711yescharacteristic
527C2#2930812yes—
528C2#2934011yescharacteristic
529C2#2934111yescharacteristic
530C2#2934212yes—
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. #11208C2semidirect
5. #11207C2semidirect
6. #10881C2semidirect
7. #10880C2semidirect
17. #2316C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
C2×#17502
#143402
#112082
#112072
#108812
#108802
#107312
#107302