Finite Group Databasebeta

← #415132

Subgroups — #415132

Order 512.

1138 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1993C211yesmaximal, characteristic
2C2×#1988C211yesmaximal, characteristic
3C2×#1940C211yesmaximal, characteristic
4C2×#1776C211yesmaximal, characteristic
5#14161C211yesmaximal, characteristic
6#13954C211yesmaximal, characteristic
7#11228C212yesmaximal
8#11227C212yesmaximal
9#11220C212yesmaximal
10#11219C212yesmaximal
11#10850C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#97C2211yescharacteristic
18C2×#389C2212yes—
19C2×#389C2212yes—
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24#1993—24no—
25#1988—24no—
26#1940—24no—
27#1776—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37#1446—21no—
38#1446—21no—
39#1444—22no—
40C4⋊#95—21no—
41C4⋊#95—21no—
42C4⋊#95—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×Q8—21no—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8Q814yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×#107—22no—
70C2×#107—22no—
71C2×#107Q814yescomplemented
72C2×#106—22no—
73C2×#106—22no—
74C2×#106Q814yescomplemented
75C2×C4⋊C8—24no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8—42no—
81C2×#97—22no—
82C2×#97—22no—
83C2×#97D414yes—
84#389—24no—
85#389—24no—
86#389—42no—
87#389—42no—
88#382—22no—
89#382—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D414yes—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—22no—
113#340—22no—
114#340—42no—
115#340—42no—
116#336—24no—
117C22×Q8—24no—
118C22×Q8—42no—
119C22×Q8—42no—
120C22×Q8—82no—
121C22×Q8—82no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8C2×D412yes—
151C22×C8C2×D412yes—
152C22×C8C2×Q812yes—
153C22×C8C2×Q812yes—
154C4⋊Q8—28no—
155C4⋊Q8—28no—
156C4⋊Q8—44no—
157C4⋊Q8—44no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×Q814yes—
200C2×C4⋊C4C2×Q812yes—
201C2×C4⋊C4C2×Q812yes—
202C2×C4⋊C4C2×Q812yes—
203C2×C4⋊C4C2×Q812yes—
204C2×C4⋊C4#4112yes—
205C2×C4⋊C4#4112yes—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—44no—
214C2×C42—42no—
215C2×C42—82no—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222#107—28no—
223#106—28no—
224C4⋊C8—48no—
225C4⋊C8—44no—
226#97—28no—
227#95—24no—
228#95—22no—
229#95—22no—
230#95—44no—
231#95—44no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—82no—
236#95—82no—
237#95—82no—
238#95—82no—
239C24C22×D411yescharacteristic
240C2×Q8—48no—
241C2×Q8—88no—
242C2×Q8—84no—
243C2×Q8—84no—
244C2×Q8—88no—
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—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
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—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
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#12112yes—
297C22×C4#12112yes—
298C22×C4#12112yes—
299C22×C4#12112yes—
300C22×C4C22⋊Q812yes—
301C22×C4C22⋊Q812yes—
302C22×C4#13714yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×Q811yescharacteristic
306C22×C4C22×Q811yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C42−1+411yescharacteristic
316C22×C42−1+411yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8C4⋊Q814yes—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—48no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—48no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—88no—
363C4⋊C4—88no—
364C4⋊C4C22⋊Q814yes—
365C4⋊C4C22⋊Q814yes—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—84no—
371C42—84no—
372C42#12114yes—
373C42C22⋊Q814yes—
374C42C22⋊Q814yes—
375C42#12614yes—
376C23C2×#12111yescharacteristic
377C23C2×#12111yescharacteristic
378C23C2×C22⋊Q811yescharacteristic
379C23#54611yescharacteristic
380C23#54611yescharacteristic
381C23D4×Q811yescharacteristic
382C23D4×Q811yescharacteristic
383C23C2×#13712yes—
384C23#57512yes—
385C23#57612yes—
386C23#57712yes—
387Q8—816no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—82no—
431C2×C4—81no—
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#39614yes—
445C2×C4#39614yes—
446C2×C4C2×#12111yescharacteristic
447C2×C4C2×C22⋊Q811yescharacteristic
448C2×C4C2×C22⋊Q811yescharacteristic
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×C22⋊Q811yescharacteristic
451C2×C4C2×#12611yescharacteristic
452C2×C4C2×C4⋊Q811yescharacteristic
453C2×C4#53211yescharacteristic
454C2×C4#53311yescharacteristic
455C2×C4#53311yescharacteristic
456C2×C4#53511yescharacteristic
457C2×C4#53611yescharacteristic
458C2×C4#53711yescharacteristic
459C2×C4#53711yescharacteristic
460C2×C4#54211yescharacteristic
461C2×C4#54211yescharacteristic
462C2×C4#55211yescharacteristic
463C2×C4#55211yescharacteristic
464C2×C4#55411yescharacteristic
465C2×C4#55411yescharacteristic
466C2×C4#55711yescharacteristic
467C2×C4#55711yescharacteristic
468C2×C4#55711yescharacteristic
469C2×C4#55711yescharacteristic
470C2×C4Q8211yescharacteristic
471C2×C4Q8211yescharacteristic
472C2×C4#55911yescharacteristic
473C2×C4#55911yescharacteristic
474C8—24no—
475C8—42no—
476C8—82no—
477C22C2×#39611yescharacteristic
478C22#241711yescharacteristic
479C22#241711yescharacteristic
480C22#245111yescharacteristic
481C22#245111yescharacteristic
482C22#254411yescharacteristic
483C22#254611yescharacteristic
484C22#303911yescharacteristic
485C22#304011yescharacteristic
486C22#304111yescharacteristic
487C22#304311yescharacteristic
488C22#305612yes—
489C22#306212yes—
490C22#307812yes—
491C22#308112yes—
492C22#331412yes—
493C22#331612yes—
494C22#331912yes—
495C22#332112yes—
496C22#338512yes—
497C22#338712yes—
498C22#338812yes—
499C22#338912yes—
500C4—24no—
501C4—24no—
502C4—44no—
503C4—44no—
504C4—44no—
505C4—84no—
506C4—82no—
507C4—82no—
508C4—82no—
509C4—162no—
510C4#251111yescharacteristic
511C4#251111yescharacteristic
512C4#256511yescharacteristic
513C4#256711yescharacteristic
514C4#258211yescharacteristic
515C4#258211yescharacteristic
516C4#262011yescharacteristic
517C4#285711yescharacteristic
518C2#1518411yescharacteristic
519C2#2361411yescharacteristic
520C2#2361911yescharacteristic
521C2#2482811yescharacteristic
522C2#2483011yescharacteristic
523C2#2483111yescharacteristic
524C2#2483311yescharacteristic
525C2#2645312yes—
526C2#2645412yes—
527C2#2861212yes—
528C2#2868612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
71. C2×#107Q8semidirect
74. C2×#106Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19932
C2×#19882
C2×#19402
C2×#17762
#141612
#139542
#112282
#112272
#112202
#112192
#108502