Finite Group Databasebeta

← #381392

Subgroups — #381392

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1798C211yesmaximal, characteristic
2#14343C211yesmaximal, characteristic
3#13777C211yesmaximal, characteristic
4#11232C212yesmaximal, complemented
5#11231C212yesmaximal, complemented
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10C22×#117C2211yescharacteristic
11C22×#107C2212yes—
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C2×#380C2212yes—
17C2×#380C2211yescharacteristic
18C2×#380C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21#1798C418yescomplemented
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29#1445—21no—
30#1445—21no—
31#1440—21no—
32#1440—21no—
33#1437—21no—
34#1437—21no—
35#1436—21no—
36#1436—21no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#117—41no—
44C2×#117—41no—
45C2×#117—41no—
46C2×#117—41no—
47C2×#117C2×C414yes—
48C2×#117Q814yes—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42—21no—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#107—24no—
65C2×#107—42no—
66C2×#107—42no—
67C2×#106—24no—
68C2×#106—42no—
69C2×#106—42no—
70C2×C4⋊C8—41no—
71C2×C4⋊C8—41no—
72C2×C4⋊C8—41no—
73C2×C4⋊C8—41no—
74C2×C4⋊C8C2×C414yes—
75C2×C4⋊C8C2×C414yes—
76C2×C22⋊C8—42no—
77C2×C22⋊C8—42no—
78C2×C22⋊C8C2×C418yes—
79#380—24no—
80#380—24no—
81#380—41no—
82#380—41no—
83#380—41no—
84#380—41no—
85C4×C4⋊C4—41no—
86C4×C4⋊C4—41no—
87C4×C4⋊C4—41no—
88C4×C4⋊C4—41no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—21no—
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—42no—
103#340—42no—
104#340—42no—
105#340—42no—
106#336C2×C418yes—
107C25C2×D411yescharacteristic
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C22×C412yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×Q812yes—
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8C22×C414yes—
141C22×C8C22×C414yes—
142#117—44no—
143#117—44no—
144#117—44no—
145#117—44no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
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—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4C4⋊C414yes—
173C2×C4⋊C4C4⋊C414yes—
174C2×C4⋊C4#4114yes—
175C2×C4⋊C4#4114yes—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4—82no—
187C2×C22⋊C4C4⋊C414yes—
188C2×C22⋊C4C4⋊C414yes—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—82no—
200C2×C42—82no—
201C2×C42C22×C412yes—
202C2×C42C22×C412yes—
203C2×C42C2×Q812yes—
204C2×C42C2×Q812yes—
205#107—48no—
206#106—48no—
207C4⋊C8—44no—
208C4⋊C8—44no—
209C22⋊C8—48no—
210#95—28no—
211#95—42no—
212#95—42no—
213#95—42no—
214#95—42no—
215#95—82no—
216#95—82no—
217#95—82no—
218#95—82no—
219#95—82no—
220#95—82no—
221C24—22no—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24C2×C4⋊C412yes—
230C24C22×D411yescharacteristic
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
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—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
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—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×C4C2×C4⋊C412yes—
298C22×C4C2×C4⋊C412yes—
299C22×C4C2×C4⋊C412yes—
300C22×C4C2×C4⋊C412yes—
301C22×C4C2×C4⋊C412yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×Q811yescharacteristic
304C22×C4C2×#4112yes—
305C22×C4C2×#4112yes—
306C22×C4H(5,𝔽2)12yes—
307C22×C4H(5,𝔽2)11yescharacteristic
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C4H(5,𝔽2)11yescharacteristic
310C22×C4H(5,𝔽2)11yescharacteristic
311C22×C42−1+412yes—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—88no—
347C22⋊C4—84no—
348C22⋊C4—84no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—84no—
358C42—84no—
359C23—22no—
360C23—22no—
361C23—24no—
362C23—24no—
363C23—22no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23C22×C4⋊C411yescharacteristic
381C23#52512yes—
382C23#53411yescharacteristic
383C23#53411yescharacteristic
384C23#53712yes—
385C23#57612yes—
386C23#57712yes—
387C23#57712yes—
388C23#57812yes—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—42no—
401C2×C4—42no—
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—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
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—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×C4C4×#4111yescharacteristic
449C2×C4C4×#4111yescharacteristic
450C2×C4#51811yescharacteristic
451C2×C4#51811yescharacteristic
452C2×C4#51811yescharacteristic
453C2×C4#51911yescharacteristic
454C2×C4#53312yes—
455C2×C4#53812yes—
456C2×C4#53812yes—
457C2×C4#53912yes—
458C2×C4#53912yes—
459C2×C4#54012yes—
460C2×C4#54012yes—
461C2×C4#54212yes—
462C2×C4#54212yes—
463C2×C4#54311yescharacteristic
464C2×C4#54311yescharacteristic
465C2×C4#54412yes—
466C8—44no—
467C8—44no—
468C22—44no—
469C22—42no—
470C22—42no—
471C22—44no—
472C22—42no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22#229611yescharacteristic
478C22#231511yescharacteristic
479C22#231611yescharacteristic
480C22#263012yes—
481C22#263012yes—
482C22#291212yes—
483C22#291312yes—
484C22#311812yes—
485C22#311912yes—
486C22#312012yes—
487C22#312112yes—
488C22#319911yescharacteristic
489C22#320011yescharacteristic
490C22#320011yescharacteristic
491C22#320211yescharacteristic
492C22#320312yes—
493C22#320412yes—
494C22#325712yes—
495C22#325712yes—
496C22#325812yes—
497C22#325812yes—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—84no—
504C4—84no—
505C4—162no—
506C4—162no—
507C4#233211yescharacteristic
508C4#233211yescharacteristic
509C4#233311yescharacteristic
510C4#233311yescharacteristic
511C4#266512yes—
512C4#266712yes—
513C2—82no—
514C2#1429711yescharacteristic
515C2#2136712yes—
516C2#2194611yescharacteristic
517C2#2194711yescharacteristic
518C2#2196612yes—
519C2#3002212yes—
520C2#3002412yes—
521C2#3002612yes—
522C2#3002812yes—
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. #11232C2semidirect
5. #11231C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
21. #1798C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17982
#143432
#137772
#112322
#112312
#110962
#110952
#109082
#109072