Finite Group Databasebeta

← #240346

Subgroups — #240346

Order 512.

1192 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3039C211yesmaximal, characteristic
2#21373C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#20648C211yesmaximal, characteristic
5C2×#1567C211yesmaximal, characteristic, complemented
6C2×#1546C211yesmaximal, characteristic, complemented
7#10478C214yesmaximal, complemented
8#10476C214yesmaximal, complemented
9#8043C211yesmaximal, characteristic, complemented
10C22×#137—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13#3039—24no—
14#3039—24no—
15C2×#462—21no—
16C2×#462C2212yes—
17C2×#461—21no—
18C2×#461C2212yes—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#417—21no—
23C2×#419—21no—
24C22×C4×C8C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#1990—21no—
27#1990—21no—
28#1990C2214yes—
29#1984C2212yes—
30#1954C2212yes—
31#1869—22no—
32#1803—21no—
33#1803C2212yes—
34#1802—21no—
35#1802C2212yes—
36C4×C22⋊C8—21no—
37C4×C22⋊C8—21no—
38#1736—21no—
39C2×#340—22no—
40#1567—24no—
41C2×#335C2214yes—
42#1546—24no—
43C2×#332C2214yes—
44#1296C2214yes—
45C22×#41D411yescharacteristic
46C23×Q8—21no—
47C2×#137—28no—
48C2×#137—44no—
49C2×#137—44no—
50C22×Q16—22no—
51C22×SD16—22no—
52C22×M16—21no—
53C22×M16—21no—
54C23×C8D412yes—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8D414yes—
62C2×C4⋊Q8C2312yes—
63C4×#41—22no—
64C4×#41—22no—
65C4×#41—22no—
66C4×#41—22no—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4D411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117D411yescharacteristic
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C42—21no—
81C22×C42C2311yescharacteristic
82#462—28no—
83#462—24no—
84#462—24no—
85#461—28no—
86#461—24no—
87#461—24no—
88#431—22no—
89#431—22no—
90#423—22no—
91#423—22no—
92#423—22no—
93#423D414yes—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99#419—28no—
100#417—44no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—22no—
104C2×#102—22no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—22no—
108C2×C4×C8—22no—
109C2×C4×C8—22no—
110C2×C4×C8—21no—
111C2×C4×C8D411yescharacteristic
112C2×C4×C8D411yescharacteristic
113C2×C4×C8C2312yes—
114#384—21no—
115#384—21no—
116#384—21no—
117#384C2312yes—
118#340—24no—
119#340—24no—
120#340—22no—
121#340—22no—
122#336—22no—
123#336—22no—
124#335—28no—
125#332—28no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41—22no—
130C2×#41—22no—
131C2×#41D814yes—
132C22×Q8—24no—
133C22×Q8—22no—
134C22×Q8—41no—
135C22×Q8—42no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8—42no—
140C22×Q8C2×D411yescharacteristic
141C22×D4—21no—
142C22×D4C2×D411yescharacteristic
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4#4111yescharacteristic
151#137—432no—
152C2×Q16—28no—
153C2×Q16—44no—
154C2×Q16—44no—
155C2×SD16—28no—
156C2×SD16—44no—
157C2×SD16—44no—
158C2×M16—24no—
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—42no—
164C22×C8—24no—
165C22×C8—22no—
166C22×C8—24no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—42no—
170C22×C8C2×D412yes—
171C22×C8C2×D412yes—
172C4⋊Q8—24no—
173C4⋊Q8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—22no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8C2×D414yes—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×D4—22no—
189C4×D4—22no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—21no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197C4×D4—42no—
198#117—22no—
199#117—22no—
200#117—42no—
201#117—42no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42#4112yes—
237C2×C42#4112yes—
238C2×C42#4111yescharacteristic
239C2×C42C2411yescharacteristic
240C4⋊C8—24no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—42no—
247C4⋊C8C2×D414yes—
248#103—28no—
249#103—28no—
250#103—44no—
251#102—44no—
252C22⋊C8—44no—
253C22⋊C8—42no—
254C22⋊C8—44no—
255C4×C8—22no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—42no—
260C24—21no—
261C24#12111yescharacteristic
262#41—28no—
263#41—28no—
264#41—44no—
265#41—44no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—24no—
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—42no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—48no—
279C2×Q8—48no—
280C2×Q8—82no—
281C2×Q8—82no—
282C2×Q8—84no—
283C2×Q8—84no—
284C2×Q8C2×D812yes—
285C2×D4—24no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4C2×D812yes—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—44no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4#12012yes—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12111yescharacteristic
325C22×C4#12312yes—
326C22×C4#12312yes—
327C22×C4#12412yes—
328C22×C4C2×D812yes—
329C22×C4#13512yes—
330C22×C4AGL(1,ℤ/8)12yes—
331C22×C4#13712yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335Q16—416no—
336SD16—416no—
337M16—48no—
338M16—84no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—48no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8#13714yescomplemented
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—44no—
366C4⋊C4—44no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—44no—
371C4⋊C4—82no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—42no—
379C42—41no—
380C42—41no—
381C42—42no—
382C42—42no—
383C42#12012yes—
384C42#12112yes—
385C42#12112yes—
386C42#12312yes—
387C42#12312yes—
388C42#12412yes—
389C42C22×D411yescharacteristic
390C42C22×D411yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—22no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23#39412yes—
405C23#46612yes—
406C23#46812yes—
407C23C2×#12111yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—41no—
411Q8—41no—
412Q8—44no—
413Q8—44no—
414Q8—82no—
415Q8—82no—
416Q8—88no—
417D4—22no—
418D4—22no—
419D4—48no—
420D4—41no—
421D4—41no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—48no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—44no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—88no—
444C2×C4—84no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—164no—
451C2×C4#39412yes—
452C2×C4#39412yes—
453C2×C4#39412yes—
454C2×C4#44714yes—
455C2×C4#45214yes—
456C2×C4#46612yes—
457C2×C4#46812yes—
458C2×C4#47112yes—
459C2×C4#47312yes—
460C2×C4#48014yes—
461C2×C4#48214yes—
462C2×C4C2×#12011yescharacteristic
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×#12111yescharacteristic
465C2×C4C2×#12311yescharacteristic
466C2×C4C2×#12311yescharacteristic
467C2×C4C2×#12411yescharacteristic
468C2×C4C22×D811yescharacteristic
469C2×C4C2×#13511yescharacteristic
470C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
471C2×C4C2×#13711yescharacteristic
472C2×C4C2×#13712yes—
473C2×C4#57512yes—
474C8—24no—
475C8—44no—
476C8—42no—
477C8—82no—
478C8—82no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22#203314yes—
493C22C2×#39411yescharacteristic
494C22#303911yescharacteristic
495C22#304011yescharacteristic
496C22C2×#46611yescharacteristic
497C22C2×#46811yescharacteristic
498C22C2×#47111yescharacteristic
499C22C2×#47311yescharacteristic
500C4—21no—
501C4—21no—
502C4—22no—
503C4—42no—
504C4—84no—
505C4—81no—
506C4—81no—
507C4—162no—
508C4#203314yes—
509C4#205214yes—
510C4C2×#44711yescharacteristic
511C4C2×#45211yescharacteristic
512C4#303712yes—
513C4#305612yes—
514C4C2×#48011yescharacteristic
515C4C2×#48211yescharacteristic
516C2—24no—
517C2—82no—
518C2#1927111yescharacteristic
519C2#1927611yescharacteristic
520C2#1929511yescharacteristic
521C2#1930011yescharacteristic
522C2C2×#203311yescharacteristic
523C2C2×#205211yescharacteristic
524C2#2354711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1990C2semidirect
5. C2×#1567C2semidirect
6. C2×#1546C2semidirect
7. #10478C2semidirect
8. #10476C2semidirect
9. #8043C2semidirect
354. C2×C8#137semidirect

Maximal subgroups

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

subgroupindex
C2×#30392
#213732
C2×#19902
#206482
C2×#15672
C2×#15462
#104782
#104762
#80432