Finite Group Databasebeta

← #106709

Subgroups — #106709

Order 512.

976 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2974C211yesmaximal, characteristic
2#17362C211yesmaximal, characteristic
3C2×#1459C211yesmaximal, characteristic
4C2×#1298C212yesmaximal, complemented
5#8008C212yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7#2974—24no—
8#2974—24no—
9C2×#436—21no—
10C2×#435—21no—
11C2×#435—21no—
12C2×C8×D4—21no—
13C2×#432C411yescharacteristic
14C2×#431C411yescharacteristic
15C2×#407C411yescharacteristic
16C2×#407C2212yes—
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C411yescharacteristic
19C22×C22⋊C8C2212yes—
20C22×C22⋊C8C2211yescharacteristic
21#1465C412yes—
22#1465C412yes—
23#1463—21no—
24#1462—21no—
25#1459—21no—
26#1459—21no—
27#1459—21no—
28#1459—21no—
29#1459C412yes—
30#1459C412yes—
31C2×#324—21no—
32C2×#323—21no—
33#1298—24no—
34#1293—24no—
35C23×D4Q811yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2311yescharacteristic
38C22×M16—21no—
39C22×M16—21no—
40C22×M16—22no—
41C23×C8—21no—
42C23×C8—21no—
43C23×C8—22no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4C2×C411yescharacteristic
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C22⋊C4D411yescharacteristic
58C22×C22⋊C4Q811yescharacteristic
59C22×C42C2×C411yescharacteristic
60#436—28no—
61#435—28no—
62#435—28no—
63C8×D4—28no—
64#432—22no—
65#432—22no—
66#431—22no—
67#431—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70#407—22no—
71#407—22no—
72#407—22no—
73#407—22no—
74#407—22no—
75#407—22no—
76#407C2×C414yes—
77#407D414yes—
78C2×C22⋊C8—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—21no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8C2×C414yes—
93C2×C22⋊C8C2×C411yescharacteristic
94C2×C22⋊C8C2×C411yescharacteristic
95C2×C22⋊C8C2×C411yescharacteristic
96C2×C22⋊C8C2×C411yescharacteristic
97C2×C22⋊C8D414yes—
98C2×#97—21no—
99C2×C4×C8—21no—
100#324—22no—
101#324—22no—
102#324—22no—
103#324—22no—
104#323—22no—
105#323—22no—
106#323—22no—
107#323—22no—
108C25C4⋊C411yescharacteristic
109C25C2×Q811yescharacteristic
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—24no—
113C22×D4—42no—
114C22×D4C4⋊C411yescharacteristic
115C22×D4C4⋊C411yescharacteristic
116C22×D4C4⋊C411yescharacteristic
117C22×D4C4⋊C411yescharacteristic
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—82no—
123C23×C4—82no—
124C23×C4C4211yescharacteristic
125C23×C4C4211yescharacteristic
126C23×C4C22⋊C411yescharacteristic
127C23×C4C22⋊C411yescharacteristic
128C23×C4C4⋊C411yescharacteristic
129C23×C4C4⋊C411yescharacteristic
130C23×C4C4⋊C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4#4112yes—
140C23×C4#4112yes—
141C2×M16—24no—
142C2×M16—24no—
143C2×M16—24no—
144C2×M16—42no—
145C2×M16—82no—
146C2×M16—82no—
147C2×M16—84no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—24no—
153C22×C8—24no—
154C22×C8—24no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8—82no—
159C22×C8—82no—
160C22×C8—84no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4C22⋊C411yescharacteristic
188C2×C22⋊C4C22⋊C411yescharacteristic
189C2×C22⋊C4C22⋊C411yescharacteristic
190C2×C22⋊C4C22⋊C411yescharacteristic
191C2×C22⋊C4C4⋊C411yescharacteristic
192C2×C22⋊C4C4⋊C411yescharacteristic
193C2×C22⋊C4C4⋊C411yescharacteristic
194C2×C22⋊C4C4⋊C411yescharacteristic
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42C4211yescharacteristic
200C2×C42C4211yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C4⋊C8—24no—
204C4⋊C8—24no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C22⋊C8—21no—
208C22⋊C8—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—24no—
217C22⋊C8—42no—
218C22⋊C8—42no—
219C22⋊C8—42no—
220C22⋊C8—42no—
221C22⋊C8C22⋊C412yes—
222C22⋊C8C22⋊C412yes—
223C22⋊C8C22⋊C412yes—
224C22⋊C8C22⋊C412yes—
225#97—24no—
226C4×C8—24no—
227C24—22no—
228C24—22no—
229C24—21no—
230C24—21no—
231C24—22no—
232C24—41no—
233C24—42no—
234C24—42no—
235C24—81no—
236C24—81no—
237C24#9511yescharacteristic
238C24#9511yescharacteristic
239C24C2×C4⋊C411yescharacteristic
240C24C2×C4⋊C411yescharacteristic
241C24C2×C4⋊C411yescharacteristic
242C24C22⋊Q812yes—
243C24C22⋊Q812yes—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—48no—
256C2×D4—48no—
257C2×D4#10814yes—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—81no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—81no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4#9511yescharacteristic
295C22×C4#9511yescharacteristic
296C22×C4#9511yescharacteristic
297C22×C4#9511yescharacteristic
298C22×C4#9511yescharacteristic
299C22×C4#9511yescharacteristic
300C22×C4C2×C4211yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4#11711yescharacteristic
307C22×C4#11711yescharacteristic
308C22×C4C4×D412yes—
309C22×C4C4×D412yes—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4#12012yes—
313C22×C4#12012yes—
314C22×C4#12112yes—
315C22×C4#12112yes—
316C22×C4#12312yes—
317C22×C4#12312yes—
318M16—84no—
319M16—84no—
320M16—84no—
321M16—84no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—84no—
333C2×C8—84no—
334C2×C8—84no—
335C2×C8—84no—
336C2×C8—84no—
337C2×C8—84no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—22no—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C23—22no—
359C23—24no—
360C23—24no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—24no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—41no—
372C23—41no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—41no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—41no—
382C23—41no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—42no—
387C23—81no—
388C23—81no—
389C23—84no—
390C23—84no—
391C23—84no—
392C23—84no—
393C23#34112yes—
394C23C2×#9511yescharacteristic
395C23#38011yescharacteristic
396C23#38011yescharacteristic
397C23#38512yes—
398C23#38512yes—
399C23C2×#10812yes—
400C23#43012yes—
401D4—28no—
402D4—44no—
403D4—44no—
404D4—44no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4#34112yes—
451C2×C4C4×C22⋊C411yescharacteristic
452C2×C4C4×C22⋊C411yescharacteristic
453C2×C4#37911yescharacteristic
454C2×C4#38111yescharacteristic
455C2×C4#38612yes—
456C2×C4#38612yes—
457C8—42no—
458C8—42no—
459C8—42no—
460C8—42no—
461C8—84no—
462C8—84no—
463C22—21no—
464C22—22no—
465C22—21no—
466C22—21no—
467C22—22no—
468C22—21no—
469C22—21no—
470C22—22no—
471C22—21no—
472C22—22no—
473C22—21no—
474C22—22no—
475C22—22no—
476C22—42no—
477C22—42no—
478C22—48no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—81no—
489C22—81no—
490C22—84no—
491C22—84no—
492C22—81no—
493C22—81no—
494C22—84no—
495C22C22⋊#9511yescharacteristic
496C22C2×#34111yescharacteristic
497C22#173811yescharacteristic
498C22#181312yes—
499C22#185514yes—
500C4—21no—
501C4—22no—
502C4—22no—
503C4—21no—
504C4—21no—
505C4—22no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—41no—
511C4#182812yes—
512C2—22no—
513C2—21no—
514C2—21no—
515C2—24no—
516C2—42no—
517C2—41no—
518C2—42no—
519C2—81no—
520C2#1073411yescharacteristic
521C2#1076011yescharacteristic
522C2#1076811yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C2×C42
Frattini, cosocleC24×C4C23
socleC22C22⋊#95
prime radicalC23×D4Q8
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1298C2semidirect
5. #8008C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29742
#173622
C2×#14592
C2×#12982
#80082