Finite Group Databasebeta

← #128132

Subgroups — #128132

Order 512.

843 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21190C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#7322C214yesmaximal, complemented
5C22×C4×D4C2211yescharacteristic
6C2×#423C411yescharacteristic, complemented
7C22×#99C412yes—
8C2×#407—21no—
9C2×#407—21no—
10C2×#407—21no—
11C2×C4×M16C411yescharacteristic, complemented
12C2×#380—21no—
13#1955C412yescomplemented
14#1955C2212yes—
15#1954—21no—
16#1954—21no—
17#1799—21no—
18#1799—21no—
19#1799C412yes—
20#1799C2212yes—
21C2×#342—21no—
22#1465—21no—
23#1465C2212yes—
24#1463C412yescomplemented
25#1462—21no—
26#1459—21no—
27#1459C412yescomplemented
28C23×D4C2×C411yescharacteristic
29C24×C4D411yescharacteristic
30C24×C4D411yescharacteristic
31C22×M16—21no—
32C22×M16—21no—
33C22×M16—41no—
34C2×C4×D4—41no—
35C2×C4×D4—41no—
36C2×C4×D4—41no—
37C2×C4×D4—41no—
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4C2×C411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4D411yescharacteristic
44C2×C4×D4Q811yescharacteristic
45C2×C4×D4C2311yescharacteristic
46C22×C4⋊C4—41no—
47C22×C4⋊C4C2×C411yescharacteristic
48C22×C22⋊C4—22no—
49C22×C22⋊C4D411yescharacteristic
50C22×C22⋊C4Q811yescharacteristic
51C22×C42C2×C411yescharacteristic
52#423—41no—
53#423—41no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8C2×C412yescomplemented
56C2×#99—22no—
57C2×#99—44no—
58C2×#99C2×C414yes—
59#407—22no—
60#407—22no—
61#407—44no—
62#407—44no—
63#407—41no—
64#407—41no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—41no—
71C4×M16—41no—
72C4×M16—41no—
73C2×#97C2×C412yescomplemented
74C2×C4×C8—21no—
75#380—22no—
76#380—22no—
77#380—41no—
78#380—41no—
79C2×#95—41no—
80#342—22no—
81#342—22no—
82#324—22no—
83#324C2×C414yescomplemented
84#323—22no—
85#323—22no—
86C25C22⋊C411yescharacteristic
87C25C4⋊C411yescharacteristic
88C22×D4—22no—
89C22×D4—22no—
90C22×D4—41no—
91C22×D4—41no—
92C22×D4—42no—
93C22×D4C4211yescharacteristic
94C22×D4C22⋊C411yescharacteristic
95C22×D4C4⋊C411yescharacteristic
96C22×D4C22×C411yescharacteristic
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—42no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—42no—
109C23×C4—42no—
110C23×C4C4211yescharacteristic
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22⋊C411yescharacteristic
114C23×C4C22⋊C411yescharacteristic
115C23×C4C22⋊C411yescharacteristic
116C23×C4C22⋊C411yescharacteristic
117C23×C4C4⋊C411yescharacteristic
118C23×C4C4⋊C411yescharacteristic
119C23×C4C4⋊C411yescharacteristic
120C23×C4C4⋊C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C2×M16—24no—
124C2×M16—42no—
125C2×M16—41no—
126C2×M16—41no—
127C2×M16—42no—
128C2×M16—42no—
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—41no—
132C2×M16—41no—
133C2×M16—82no—
134C2×M16—82no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—42no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144C4×D4—41no—
145C4×D4—41no—
146C4×D4—41no—
147C4×D4—41no—
148C4×D4—41no—
149C4×D4—41no—
150C4×D4—41no—
151C4×D4—41no—
152C4×D4—41no—
153C4×D4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4C4211yescharacteristic
161C2×C4⋊C4C22⋊C411yescharacteristic
162C2×C4⋊C4C4⋊C411yescharacteristic
163C2×C4⋊C4C22×C411yescharacteristic
164C2×C22⋊C4—24no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—41no—
168C2×C22⋊C4—41no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4C22⋊C411yescharacteristic
173C2×C22⋊C4C22⋊C411yescharacteristic
174C2×C22⋊C4C4⋊C411yescharacteristic
175C2×C22⋊C4C4⋊C411yescharacteristic
176C2×C22⋊C4C4⋊C411yescharacteristic
177C2×C22⋊C4C4⋊C411yescharacteristic
178C2×C22⋊C4C2×D411yescharacteristic
179C2×C22⋊C4C2×Q811yescharacteristic
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42C4211yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C4⋊C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C4⋊C8—42no—
189C4⋊C8—42no—
190#99—44no—
191#99—44no—
192C22⋊C8—24no—
193C22⋊C8—44no—
194C22⋊C8—44no—
195C22⋊C8—44no—
196C22⋊C8—44no—
197C22⋊C8—42no—
198C22⋊C8—42no—
199C22⋊C8—42no—
200#97—42no—
201C4×C8—42no—
202#95—42no—
203#95—42no—
204C24—21no—
205C24—21no—
206C24—22no—
207C24—21no—
208C24—21no—
209C24—21no—
210C24—21no—
211C24—22no—
212C24—41no—
213C24—42no—
214C24—42no—
215C24—81no—
216C24—81no—
217C24#9511yescharacteristic
218C24#9511yescharacteristic
219C24#9511yescharacteristic
220C24C2×C22⋊C411yescharacteristic
221C24C2×C4⋊C411yescharacteristic
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—22no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—44no—
235C2×D4—44no—
236C2×D4—44no—
237C2×D4—44no—
238C2×D4#10012yes—
239C2×D4#10112yes—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—44no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—81no—
296C22×C4—84no—
297C22×C4—84no—
298C22×C4—84no—
299C22×C4#9511yescharacteristic
300C22×C4#9511yescharacteristic
301C22×C4#9511yescharacteristic
302C22×C4#9511yescharacteristic
303C22×C4#9511yescharacteristic
304C22×C4#9911yescharacteristic
305C22×C4#9911yescharacteristic
306C22×C4#9911yescharacteristic
307C22×C4#9911yescharacteristic
308C22×C4C2×C4211yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C4⋊C411yescharacteristic
312C22×C4C2×C4⋊C411yescharacteristic
313M16—44no—
314M16—44no—
315M16—82no—
316M16—84no—
317M16—82no—
318M16—82no—
319M16—82no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—44no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—84no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—84no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—84no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—84no—
390C23—82no—
391C23—82no—
392C23—84no—
393C23—84no—
394C23#34211yescharacteristic
395C23#34211yescharacteristic
396C23#34312yes—
397C23#34412yes—
398C23C2×#9511yescharacteristic
399C23C2×#10011yescharacteristic
400C23C2×#10111yescharacteristic
401D4—42no—
402D4—42no—
403D4—44no—
404D4—44no—
405D4—42no—
406D4—42no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—84no—
460C2×C4—84no—
461C2×C4—84no—
462C2×C4—168no—
463C2×C4#34211yescharacteristic
464C2×C4#34211yescharacteristic
465C2×C4#34312yes—
466C2×C4#35312yes—
467C2×C4#35412yes—
468C2×C4C2×#9911yescharacteristic
469C2×C4C2×#9911yescharacteristic
470C8—44no—
471C8—84no—
472C8—82no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—21no—
481C22—22no—
482C22—21no—
483C22—22no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—41no—
489C22—41no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—41no—
501C22—41no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—84no—
506C22—81no—
507C22—82no—
508C22—82no—
509C22—84no—
510C22#139312yes—
511C22C2×#34211yescharacteristic
512C22C2×#34311yescharacteristic
513C22C2×#34411yescharacteristic
514C4—21no—
515C4—21no—
516C4—22no—
517C4—41no—
518C4—41no—
519C4—41no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—41no—
524C4—164no—
525C4#139312yes—
526C4C2×#35311yescharacteristic
527C4C2×#35411yescharacteristic
528C2—22no—
529C2—21no—
530C2—21no—
531C2—41no—
532C2—41no—
533C2—42no—
534C2—41no—
535C2—42no—
536C2—81no—
537C2#810711yescharacteristic
538C2C2×#139311yescharacteristic
539C2#1258911yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7322C2semidirect
6. C2×#423C4semidirect
11. C2×C4×M16C4semidirect
13. #1955C4semidirect
24. #1463C4semidirect
27. #1459C4semidirect
55. C2×C4⋊C8C2×C4semidirect
73. C2×#97C2×C4semidirect
83. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
#211902
C2×#17992
#173622
#73222