Finite Group Databasebeta

← #197457

Subgroups — #197457

Order 512.

1263 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60123C211yesmaximal, characteristic
2#35316C211yesmaximal, characteristic
3#17742C212yesmaximal, complemented
4#17361C211yesmaximal, characteristic
5#9196C214yesmaximal, complemented
6#9195C214yesmaximal, complemented
7#7961C212yesmaximal, complemented
8C2×#536—21no—
9C22×C4⋊Q8C2211yescharacteristic
10C2×C4×#41C2211yescharacteristic
11#2923—24no—
12#2922—24no—
13#2921—21no—
14#2921C2212yes—
15#2920—21no—
16#2920C2212yes—
17#2920C2212yes—
18#2905C2212yes—
19#2879—21no—
20C2×C4×M16—21no—
21#2857—21no—
22#2857—21no—
23#2621C2212yes—
24#2621C2212yes—
25#2547—21no—
26#2547C412yes—
27#2546—21no—
28#2546C412yes—
29#2486—21no—
30#2486—21no—
31#2486C412yes—
32#2486C412yes—
33#2310—21no—
34#2285C2211yescharacteristic
35#1556—24no—
36#1549—24no—
37#1546C2218yes—
38#1467—21no—
39#1466—21no—
40#1465—21no—
41#1465—21no—
42#1458C2212yes—
43#1458C2212yes—
44#1286C2218yes—
45C22×#41D411yescharacteristic
46C23×Q8—21no—
47C22×M16—22no—
48#536—48no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8C2×C412yes—
57C2×C4⋊Q8C2312yes—
58C2×#124—22no—
59C2×#123—22no—
60C2×#123—22no—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C4×#41—22no—
64C4×#41—22no—
65C4×#41—22no—
66C4×#41—22no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×#117—21no—
76C2×#117D411yescharacteristic
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4C2×C412yes—
81C22×C42D411yescharacteristic
82C22×C42D411yescharacteristic
83C22×C42C2311yescharacteristic
84#433—28no—
85#433—28no—
86#432—24no—
87#432—22no—
88#432—22no—
89#432—22no—
90#432D414yes—
91#431—24no—
92#431—22no—
93#431—22no—
94#431D414yes—
95#431C2314yes—
96#424—28no—
97#423—22no—
98#423—22no—
99#423C2314yes—
100#408—28no—
101#407—22no—
102#407—22no—
103#407—22no—
104C4×M16—22no—
105C4×M16—22no—
106C4×M16—22no—
107C4×M16—22no—
108C4×M16—22no—
109C2×#97—22no—
110C2×C4×C8—22no—
111#398—22no—
112#398—22no—
113#398C2×C414yes—
114#398C2×C414yes—
115#397—22no—
116#397—22no—
117#397C2×C414yes—
118#397C2×C414yes—
119#391—22no—
120#391—22no—
121#391—22no—
122#391—22no—
123#389—22no—
124#389—22no—
125#389—22no—
126#389—22no—
127#384—21no—
128#382C2312yes—
129#381—22no—
130#381—22no—
131#380—22no—
132#380—22no—
133#380—22no—
134#380—22no—
135#380—21no—
136#380C2×C412yes—
137C4×C4⋊C4—21no—
138C4×C4⋊C4C2×C412yes—
139#332—216no—
140#324—22no—
141#324—22no—
142#323—22no—
143#323—22no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—24no—
147C2×#41—42no—
148C22×Q8—22no—
149C22×Q8—22no—
150C22×Q8—44no—
151C22×Q8—44no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×Q8C22⋊C411yescharacteristic
155C22×D4C22⋊C411yescharacteristic
156C22×D4C22⋊C411yescharacteristic
157C22×D4C22⋊C411yescharacteristic
158C23×C4—22no—
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C2×M16—24no—
169C2×M16—24no—
170C2×M16—44no—
171C2×M16—44no—
172C2×M16—42no—
173C22×C8—24no—
174C4⋊Q8—24no—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178C4⋊Q8—48no—
179C4⋊Q8—41no—
180C4⋊Q8—41no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—41no—
184C4⋊Q8—41no—
185#124—48no—
186#123—48no—
187#123—48no—
188C22⋊Q8—48no—
189C22⋊Q8—48no—
190C4×Q8—22no—
191C4×Q8—22no—
192C4×Q8—22no—
193C4×Q8—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—22no—
204C4×D4—22no—
205C4×D4—22no—
206#117—22no—
207#117—22no—
208#117—22no—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—44no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22×C414yes—
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—22no—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4C22⋊C411yescharacteristic
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22×C411yescharacteristic
238C2×C22⋊C4C22×C411yescharacteristic
239C2×C22⋊C4C2×D411yescharacteristic
240C2×C22⋊C4C2×D411yescharacteristic
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42C22⋊C411yescharacteristic
251C2×C42C22⋊C411yescharacteristic
252C2×C42C22⋊C411yescharacteristic
253C2×C42C22⋊C411yescharacteristic
254C2×C42C22×C411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D412yes—
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264C4⋊C8—28no—
265C4⋊C8—24no—
266C4⋊C8—24no—
267C4⋊C8—42no—
268C4⋊C8C2×D418yes—
269C22⋊C8—28no—
270C22⋊C8—24no—
271C22⋊C8—24no—
272C22⋊C8—24no—
273C22⋊C8—42no—
274#97—24no—
275#97—24no—
276#97—42no—
277C4×C8—24no—
278C4×C8—24no—
279C4×C8—24no—
280#95—24no—
281#95—24no—
282#95—24no—
283#95—24no—
284#95—22no—
285#95C22×C414yes—
286C24C2×C22⋊C411yescharacteristic
287C24C2×C22⋊C411yescharacteristic
288C24C2×C22⋊C411yescharacteristic
289#41—48no—
290#41—48no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—44no—
295C2×Q8—44no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—44no—
299C2×Q8—44no—
300C2×Q8—44no—
301C2×Q8—88no—
302C2×Q8—88no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—41no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—44no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C411yescharacteristic
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C411yescharacteristic
344C22×C4C2×C22⋊C411yescharacteristic
345C22×C4C2×C22⋊C411yescharacteristic
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C4×D412yes—
350C22×C4C4×D412yes—
351C22×C4#12012yes—
352C22×C4#12112yes—
353C22×C4#12112yes—
354C22×C4#12312yes—
355C22×C4AGL(1,ℤ/8)14yes—
356C22×C4C23×C411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359M16—48no—
360M16—44no—
361M16—44no—
362M16—84no—
363C2×C8—28no—
364C2×C8—48no—
365C2×C8—44no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—28no—
373C4⋊C4—44no—
374C4⋊C4—44no—
375C4⋊C4—48no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—48no—
383C22⋊C4—22no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—22no—
389C22⋊C4—44no—
390C22⋊C4—48no—
391C22⋊C4—48no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—21no—
399C42—21no—
400C42—21no—
401C42—21no—
402C42—21no—
403C42—21no—
404C42—44no—
405C42C4×D412yes—
406C42C4×D412yes—
407C42#12011yescharacteristic
408C42#12011yescharacteristic
409C42#12112yes—
410C42#12112yes—
411C42#12311yescharacteristic
412C42#12311yescharacteristic
413C42C22×D412yes—
414C42C2×#4112yes—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—41no—
423C23—41no—
424C23—41no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23#38612yes—
429C23#41812yes—
430C23C22×C22⋊C411yescharacteristic
431Q8—44no—
432Q8—88no—
433Q8—84no—
434Q8—84no—
435D4—44no—
436D4—44no—
437D4—44no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—22no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—88no—
482C2×C4#38611yescharacteristic
483C2×C4#38611yescharacteristic
484C2×C4#38612yes—
485C2×C4#38611yescharacteristic
486C2×C4#38611yescharacteristic
487C2×C4#41812yes—
488C2×C4C2×C4×D412yes—
489C2×C4C2×#12011yescharacteristic
490C2×C4C2×#12112yes—
491C2×C4C2×#12311yescharacteristic
492C2×C4C2×AGL(1,ℤ/8)12yes—
493C2×C4#57512yes—
494C8—48no—
495C8—84no—
496C22—21no—
497C22—21no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—81no—
512C22—81no—
513C22C2×#38611yescharacteristic
514C22C2×#41811yescharacteristic
515C22#289611yescharacteristic
516C4—22no—
517C4—21no—
518C4—21no—
519C4—22no—
520C4—22no—
521C4—22no—
522C4—41no—
523C4—41no—
524C4—41no—
525C4—41no—
526C4—41no—
527C4—84no—
528C4—84no—
529C4#190912yes—
530C4#191512yes—
531C4#300612yes—
532C4#309412yes—
533C2—21no—
534C2—21no—
535C2—42no—
536C2—42no—
537C2—81no—
538C2#2091211yescharacteristic
539C2C2×#190911yescharacteristic
540C2C2×#191511yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17742C2semidirect
5. #9196C2semidirect
6. #9195C2semidirect
7. #7961C2semidirect

Maximal subgroups

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

subgroupindex
#601232
#353162
#177422
#173612
#91962
#91952
#79612