Finite Group Databasebeta

← #198346

Subgroups — #198346

Order 512.

1210 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
1C22×#462C211yesmaximal, characteristic
2#60248C211yesmaximal, characteristic
3#21829C211yesmaximal, characteristic
4#17746C211yesmaximal, characteristic
5#17449C211yesmaximal, characteristic
6#9248C218yesmaximal, complemented
7#8002C211yesmaximal, characteristic
8#7935C211yesmaximal, characteristic
9C23×Q16C412yes—
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#462—21no—
13C2×#462—21no—
14C2×#462—22no—
15C2×#462—21no—
16C2×#462—21no—
17C2×#462—22no—
18C2×#462C414yes—
19C2×#462C2214yes—
20#2984—22no—
21#2984—22no—
22#2983—22no—
23#2983—21no—
24#2982—21no—
25Q8×M16—22no—
26Q8×M16C2214yes—
27#2920—22no—
28#2920C2211yescharacteristic
29#2920C2211yescharacteristic
30#2920C2211yescharacteristic
31#2920C2211yescharacteristic
32C22×C4⋊C8C2211yescharacteristic
33C22×#103C412yes—
34#1997—22no—
35#1997C2214yes—
36#1821—22no—
37#1558—22no—
38#1548—22no—
39#1548C2214yes—
40#1491—22no—
41#1487C2214yes—
42#1479—21no—
43#1479—21no—
44#1292C2214yes—
45#1283C2214yes—
46C23×Q8C2×C411yescharacteristic
47C23×Q8C2×C412yes—
48C22×Q16—22no—
49C22×Q16—24no—
50C22×Q16—22no—
51C22×Q16—48no—
52C22×Q16C2×C418yes—
53C23×C8C2×C412yes—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8C2×C412yes—
65C2×C4⋊Q8C2312yes—
66C2×C4×Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2×C412yes—
73C2×C4×Q8D411yescharacteristic
74C2×C4×Q8D411yescharacteristic
75C2×C4×Q8D412yes—
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×Q8D412yes—
79C2×C4×Q8C2312yes—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4C2×C411yescharacteristic
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#462—24no—
87#462—24no—
88#462—24no—
89#462—24no—
90#462—24no—
91#462—24no—
92#462—24no—
93#462—24no—
94#446—28no—
95#446—24no—
96#446—24no—
97#446—28no—
98C8×Q8—24no—
99C8×Q8—24no—
100#432—24no—
101#432—24no—
102#432—21no—
103#432—21no—
104#432—21no—
105#432—21no—
106#432—21no—
107#432—21no—
108#432—21no—
109#432D412yes—
110#431—24no—
111#431—24no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431—21no—
118#431D412yes—
119#431C2312yes—
120#423—24no—
121#423—21no—
122#423D412yes—
123#423C2312yes—
124#423C2312yes—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8—21no—
129C2×C4⋊C8C2×C412yes—
130C2×C4⋊C8C2312yes—
131C2×#103—24no—
132C2×#103—22no—
133C2×#103—22no—
134C2×#103C2×C418yes—
135#407—21no—
136#407—21no—
137#407—21no—
138#407D412yes—
139C4×M16—24no—
140C4×M16—21no—
141C4×M16—21no—
142C4×M16—21no—
143C4×M16C2312yes—
144#343—28no—
145#333—28no—
146Q8⋊C8—24no—
147Q8⋊C8—24no—
148C22×Q8—21no—
149C22×Q8—21no—
150C22×Q8—21no—
151C22×Q8—22no—
152C22×Q8—24no—
153C22×Q8—22no—
154C22×Q8—22no—
155C22×Q8—22no—
156C22×Q8—22no—
157C22×Q8—42no—
158C22×Q8—44no—
159C22×Q8—44no—
160C22×Q8C22⋊C411yescharacteristic
161C22×Q8C22⋊C412yes—
162C22×Q8C22⋊C411yescharacteristic
163C22×Q8C22×C412yes—
164C22×Q8C22×C414yes—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—22no—
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4#4111yescharacteristic
171C2×Q16—28no—
172C2×Q16—28no—
173C2×Q16—416no—
174C2×Q16—48no—
175C2×Q16—48no—
176C2×Q16—416no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—41no—
180C2×M16C2×D414yes—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—44no—
184C22×C8C22×C414yes—
185C4⋊Q8—22no—
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191C4⋊Q8—22no—
192C4⋊Q8—22no—
193C4⋊Q8—41no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C4⋊Q8—41no—
197C4⋊Q8—41no—
198C4⋊Q8—41no—
199C4⋊Q8—41no—
200C4⋊Q8—41no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—21no—
204C4×Q8—22no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208C4×Q8—22no—
209C4×Q8—22no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8—21no—
213C4×Q8—21no—
214C4×Q8—22no—
215C4×Q8—22no—
216C4×Q8—22no—
217C4×Q8C22⋊C412yes—
218C4×Q8C22⋊C412yes—
219C4×Q8C22⋊C412yes—
220C4×Q8C22⋊C412yes—
221C4×Q8C2×D412yes—
222C4×Q8C2×D412yes—
223C4×Q8C2×D412yes—
224C4×Q8C2×D412yes—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4—41no—
245C2×C4⋊C4—41no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4C22⋊C412yes—
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22×C412yes—
252C2×C42—22no—
253C2×C42—21no—
254C2×C42—22no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—22no—
258C2×C42—21no—
259C2×C42—22no—
260C2×C42C22×C411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42#4111yescharacteristic
269C2×C42#4111yescharacteristic
270C2×C42#4111yescharacteristic
271C2×C42C2411yescharacteristic
272C4⋊C8—28no—
273C4⋊C8—28no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8—22no—
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—22no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—41no—
285C4⋊C8C2×D414yes—
286#103—28no—
287#103—28no—
288C22⋊C8—22no—
289C22⋊C8—22no—
290C22⋊C8—22no—
291C22⋊C8—22no—
292C22⋊C8—22no—
293C22⋊C8—41no—
294C22⋊C8—41no—
295C22⋊C8—41no—
296#97—28no—
297#97—22no—
298#97—22no—
299#97—41no—
300#97—41no—
301C4×C8—28no—
302C4×C8—22no—
303C4×C8—22no—
304C4×C8—22no—
305C4×C8—41no—
306C24C4×D411yescharacteristic
307C2×Q8—22no—
308C2×Q8—21no—
309C2×Q8—22no—
310C2×Q8—21no—
311C2×Q8—22no—
312C2×Q8—24no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—24no—
316C2×Q8—22no—
317C2×Q8—22no—
318C2×Q8—44no—
319C2×Q8—42no—
320C2×Q8—42no—
321C2×Q8—44no—
322C2×Q8—44no—
323C2×Q8—44no—
324C2×Q8—44no—
325C2×Q8—44no—
326C2×Q8—44no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8—44no—
330C2×Q8—44no—
331C2×Q8—84no—
332C2×Q8—84no—
333C2×Q8—84no—
334C2×Q8#10014yes—
335C2×Q8C2×C22⋊C412yes—
336C2×Q8C2×C22⋊C412yes—
337C22×C4—21no—
338C22×C4—24no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—24no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—82no—
354C22×C4—84no—
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C4×D411yescharacteristic
358C22×C4C4×D411yescharacteristic
359C22×C4C4×D411yescharacteristic
360C22×C4C4×D411yescharacteristic
361C22×C4C4×D411yescharacteristic
362C22×C4#12011yescharacteristic
363C22×C4#12011yescharacteristic
364C22×C4#12111yescharacteristic
365C22×C4#12111yescharacteristic
366C22×C4#12111yescharacteristic
367C22×C4#12111yescharacteristic
368C22×C4#12311yescharacteristic
369C22×C4#12311yescharacteristic
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4C2×#4111yescharacteristic
373Q16—416no—
374Q16—416no—
375M16—28no—
376M16—44no—
377M16—44no—
378M16—82no—
379C2×C8—24no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—44no—
383C2×C8—44no—
384C2×C8—44no—
385C2×C8—42no—
386C2×C8—42no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4C2×C22⋊C412yes—
408C4⋊C4C2×C22⋊C412yes—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—22no—
416C42—22no—
417C42C2×C22⋊C411yescharacteristic
418C42C2×C22⋊C411yescharacteristic
419C42C4×D411yescharacteristic
420C42C4×D411yescharacteristic
421C42#12011yescharacteristic
422C42#12011yescharacteristic
423C42#12111yescharacteristic
424C42#12111yescharacteristic
425C42#12111yescharacteristic
426C42#12111yescharacteristic
427C42#12311yescharacteristic
428C42#12311yescharacteristic
429C42C22×D411yescharacteristic
430C42C22×D411yescharacteristic
431C42C22×D411yescharacteristic
432C42C2×#4111yescharacteristic
433C23—22no—
434C23—22no—
435C23—22no—
436C23#38611yescharacteristic
437C23#38611yescharacteristic
438C23C2×C4×D411yescharacteristic
439Q8—22no—
440Q8—24no—
441Q8—22no—
442Q8—22no—
443Q8—44no—
444Q8—44no—
445Q8—42no—
446Q8—44no—
447Q8—44no—
448Q8—84no—
449Q8—84no—
450Q8—84no—
451Q8C2×#10014yes—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—82no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—84no—
484C2×C4—84no—
485C2×C4—84no—
486C2×C4#38611yescharacteristic
487C2×C4#38611yescharacteristic
488C2×C4#38611yescharacteristic
489C2×C4#38611yescharacteristic
490C2×C4#38611yescharacteristic
491C2×C4#38611yescharacteristic
492C2×C4C2×#10012yes—
493C2×C4#41312yes—
494C2×C4C22×C22⋊C411yescharacteristic
495C2×C4C2×C4×D411yescharacteristic
496C2×C4C2×#12011yescharacteristic
497C2×C4C2×#12111yescharacteristic
498C2×C4C2×#12111yescharacteristic
499C2×C4C2×#12311yescharacteristic
500C2×C4#57814yes—
501C8—44no—
502C8—44no—
503C8—44no—
504C8—82no—
505C8—82no—
506C22—28no—
507C22—21no—
508C22—21no—
509C22—21no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22#190012yes—
514C22#190212yes—
515C22C2×#38611yescharacteristic
516C22#295012yes—
517C4—21no—
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—21no—
525C4—24no—
526C4—42no—
527C4—42no—
528C4—42no—
529C4—84no—
530C4#190012yes—
531C4C22×#10011yescharacteristic
532C4C2×#41311yescharacteristic
533C4#301112yes—
534C4#304412yes—
535C2—22no—
536C2—22no—
537C2—22no—
538C2#2093511yescharacteristic
539C2C2×#190011yescharacteristic
540C2C2×#190211yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C4×D4
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
6. #9248C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
#602482
#218292
#177462
#174492
#92482
#80022
#79352