Finite Group Databasebeta

← #188401

Subgroups — #188401

Order 512.

1060 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#34125C211yesmaximal, characteristic
2#21349C211yesmaximal, characteristic, complemented
3#20815C211yesmaximal, characteristic, complemented
4C2×#1488C211yesmaximal, characteristic
5C2×#1476C211yesmaximal, characteristic
6#8569C212yesmaximal, complemented
7#8567C212yesmaximal, complemented
8#8566C212yesmaximal, complemented
9#8564C212yesmaximal, complemented
10#7898C211yesmaximal, characteristic, complemented
11#7894C211yesmaximal, characteristic, complemented
12C2×C4×#41C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2357—22no—
18#2357C412yes—
19#2351C412yes—
20#2347—22no—
21#2347C412yes—
22#2341C412yes—
23#2293—21no—
24#2293—21no—
25#2293C2211yescharacteristic
26#2293C2211yescharacteristic
27#2283—21no—
28#2283C2211yescharacteristic
29C42×Q8C2211yescharacteristic
30C42×Q8C2211yescharacteristic
31C4×#117—21no—
32C4×#117C2211yescharacteristic
33#1939—22no—
34#1850—22no—
35#1842C2212yes—
36#1831C2212yes—
37#1798—21no—
38#1798—21no—
39#1768—21no—
40#1768—21no—
41C4×C4⋊C8C2212yes—
42C4×C4⋊C8C2212yes—
43#1488—24no—
44#1476—24no—
45C2×Q8⋊C8C2212yes—
46C2×Q8⋊C8C2212yes—
47C2×D4⋊C8C2212yes—
48C2×D4⋊C8C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53C22×#41C2×C411yescharacteristic
54C23×C8—21no—
55C4×#41—22no—
56C4×#41—22no—
57C4×#41D414yes—
58C4×#41Q814yes—
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2×C412yes—
61C2×C4×Q8C2×C411yescharacteristic
62C2×C4×Q8C2311yescharacteristic
63C2×C4×D4—21no—
64C2×C4×D4—21no—
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2311yescharacteristic
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117C2×C411yescharacteristic
71C2×#117D411yescharacteristic
72C2×#117D411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#432—22no—
76#432—22no—
77#431—22no—
78#431—22no—
79#431—22no—
80#431—22no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×C22⋊C8—21no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×#97—21no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8C2312yes—
97C2×C4×C8C2312yes—
98#390—22no—
99#390—22no—
100#390C2×C412yes—
101#390C2×C412yes—
102#389—22no—
103#389—22no—
104#389C2×C412yes—
105#389C2×C412yes—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#383—21no—
113#383—22no—
114#382—22no—
115#382C2×C412yes—
116#381—21no—
117#381—22no—
118#380—22no—
119#380C2×C412yes—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4C2×C412yes—
127C4×C4⋊C4C2×C412yes—
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4C2×C412yes—
138C4×C22⋊C4D412yes—
139#376—21no—
140#376D411yescharacteristic
141#376D411yescharacteristic
142C43—21no—
143C43—21no—
144C43C2311yescharacteristic
145C43C2311yescharacteristic
146#336—22no—
147#336—22no—
148#336—22no—
149#336—22no—
150Q8⋊C8—24no—
151Q8⋊C8—24no—
152D4⋊C8—24no—
153D4⋊C8—24no—
154C2×#41—22no—
155C2×#41—24no—
156C2×#41—42no—
157C2×#41C4⋊C414yes—
158C22×Q8C22×C411yescharacteristic
159C22×D4—21no—
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—44no—
172C22×C8—42no—
173C4×Q8—22no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8C4⋊C414yes—
181C4×Q8C2×D412yes—
182C4×Q8C2×Q812yes—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—22no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4C4⋊C414yes—
191C4×D4C2×D412yes—
192C4×D4C2×Q812yes—
193#117—22no—
194#117—24no—
195#117—24no—
196#117—24no—
197#117—42no—
198#117—44no—
199#117—44no—
200#117C4⋊C414yes—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4C22⋊C412yes—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—22no—
221C2×C22⋊C4C22⋊C412yes—
222C2×C22⋊C4C22⋊C412yes—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—24no—
227C2×C42—24no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42C22×C412yes—
235C2×C42C22×C412yes—
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D412yes—
241C2×C42C2×D412yes—
242C2×C42C2×D412yes—
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C2×C42C2×Q812yes—
247C2×C42#4112yes—
248C2×C42#4112yes—
249C2×C42C2411yescharacteristic
250C4⋊C8—22no—
251C4⋊C8—22no—
252C4⋊C8—22no—
253C4⋊C8—22no—
254C4⋊C8—44no—
255C4⋊C8—44no—
256C4⋊C8—44no—
257C4⋊C8—42no—
258C4⋊C8—44no—
259C22⋊C8—42no—
260C22⋊C8—42no—
261C22⋊C8—42no—
262#97—42no—
263C4×C8—22no—
264C4×C8—22no—
265C4×C8—22no—
266C4×C8—22no—
267C4×C8—42no—
268C4×C8—42no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—21no—
277#95—21no—
278#95C22⋊C412yes—
279#95C22⋊C412yes—
280#95C22⋊C412yes—
281#95C22⋊C412yes—
282C24—21no—
283C24C2×C22⋊C411yescharacteristic
284#41—28no—
285#41—44no—
286#41—48no—
287C2×Q8—21no—
288C2×Q8—22no—
289C2×Q8C2×C4⋊C412yes—
290C2×Q8#13114yes—
291C2×D4—24no—
292C2×D4—21no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—44no—
296C2×D4C2×C4⋊C412yes—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C4⋊C412yes—
323C22×C4#11712yes—
324C22×C4#12112yes—
325C22×C4#12112yes—
326C22×C4C22⋊Q812yes—
327C22×C4C22⋊Q812yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C2×C8—24no—
332C2×C8—42no—
333C2×C8—48no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4C2×C4⋊C412yes—
360C4⋊C4C2×C4⋊C412yes—
361C22⋊C4—24no—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—42no—
367C22⋊C4—44no—
368C22⋊C4—44no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—82no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—24no—
382C42—24no—
383C42—42no—
384C42—42no—
385C42—44no—
386C42—44no—
387C42—42no—
388C42—42no—
389C42—82no—
390C42—82no—
391C42C2×C4⋊C412yes—
392C42#11712yes—
393C42#12112yes—
394C42#12112yes—
395C42C22⋊Q812yes—
396C42C22⋊Q812yes—
397C42#13514yes—
398C42C22×D411yescharacteristic
399C42C22×Q811yescharacteristic
400C42C2×#4111yescharacteristic
401C42C2×#4111yescharacteristic
402C23—22no—
403C23—21no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—42no—
408C23—41no—
409C23—41no—
410C23—42no—
411C23#38012yes—
412C23C22×C22⋊C411yescharacteristic
413Q8—22no—
414Q8—24no—
415Q8#42414yes—
416D4—22no—
417D4—22no—
418D4—48no—
419D4—42no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—48no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—84no—
460C2×C4#38012yes—
461C2×C4#38012yes—
462C2×C4#38012yes—
463C2×C4#41614yes—
464C2×C4C22×C4⋊C411yescharacteristic
465C2×C4C2×#11711yescharacteristic
466C2×C4C2×#12111yescharacteristic
467C2×C4C2×#12111yescharacteristic
468C2×C4C2×C22⋊Q811yescharacteristic
469C2×C4C2×C22⋊Q811yescharacteristic
470C2×C4C2×#13112yes—
471C2×C4#56812yes—
472C2×C4C2×#13512yes—
473C2×C4#57512yes—
474C8—44no—
475C8—82no—
476C8—82no—
477C8—82no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—82no—
487C22—82no—
488C22C2×#38011yescharacteristic
489C22#288012yes—
490C22C2×#41611yescharacteristic
491C22#289611yescharacteristic
492C22#290112yes—
493C4—24no—
494C4—22no—
495C4—24no—
496C4—22no—
497C4—21no—
498C4—22no—
499C4—21no—
500C4—42no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—82no—
506C4—82no—
507C4C2×#42411yescharacteristic
508C4#290511yescharacteristic
509C4#292412yes—
510C4#303512yes—
511C4#307612yes—
512C2—42no—
513C2—82no—
514C2#1751012yes—
515C2#1751612yes—
516C2#2062811yescharacteristic
517C2#2064511yescharacteristic
518C2#2065511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21349C2semidirect
3. #20815C2semidirect
6. #8569C2semidirect
7. #8567C2semidirect
8. #8566C2semidirect
9. #8564C2semidirect
10. #7898C2semidirect
11. #7894C2semidirect

Maximal subgroups

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

subgroupindex
#341252
#213492
#208152
C2×#14882
C2×#14762
#85692
#85672
#85662
#85642
#78982
#78942