Finite Group Databasebeta

← #236056

Subgroups — #236056

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2C2×#3078C211yesmaximal, characteristic
3C2×#2568C211yesmaximal, characteristic
4#21385C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic
7#10077C214yesmaximal, complemented
8#10061C214yesmaximal, complemented
9#7901C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#3230—24no—
14#3230—24no—
15C2×#500C2212yes—
16C2×#487C2212yes—
17C2×#484—22no—
18#3078—24no—
19#3078—24no—
20C2×#479—22no—
21C2×#477—22no—
22C2×#477C2212yes—
23C2×#475C2212yes—
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#428—21no—
27C2×#423C2211yescharacteristic
28C2×#419—21no—
29#2568—22no—
30#2568—22no—
31#2568—22no—
32#2568C2214yes—
33C2×#400—22no—
34C2×#391—21no—
35C2×#391—22no—
36C2×#382—21no—
37C2×C4×C4⋊C4C2211yescharacteristic
38#1987C2212yes—
39#1981C2212yes—
40#1872—22no—
41C4×#103—22no—
42#1549—24no—
43C2×#332C2214yes—
44#1479—24no—
45C2×Q8⋊C8C2214yes—
46#1279C2214yes—
47C23×Q8—21no—
48C23×Q8D411yescharacteristic
49C22×M16—21no—
50#533—22no—
51#533—22no—
52#533—22no—
53#533D414yes—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8D412yes—
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×#125—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8—21no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—21no—
74C2×C4×Q8—22no—
75C2×C4×Q8D412yes—
76C2×C4×Q8C2312yes—
77C2×#117—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D412yes—
81C22×C4⋊C4D411yescharacteristic
82C22×C4⋊C4D412yes—
83C22×C42—21no—
84C22×C42—21no—
85C22×C42C2311yescharacteristic
86#500—24no—
87#500—24no—
88#487—24no—
89#487—24no—
90#484—216no—
91#479—216no—
92#477—216no—
93#477—24no—
94#477—24no—
95#475—24no—
96#475—24no—
97#431—22no—
98#431—22no—
99#431—22no—
100#431D414yes—
101#428—28no—
102C2×#107—24no—
103C2×#107—22no—
104C2×#106—22no—
105#423—22no—
106#423D414yes—
107C2×C4⋊C8—21no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8C2312yes—
110C2×C4⋊C8C2312yes—
111#419—28no—
112C2×#103—22no—
113C2×#103—22no—
114C2×#103—24no—
115C2×#103—24no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2312yes—
120#400—28no—
121#400—28no—
122#391—28no—
123#391—24no—
124#391—24no—
125#391—28no—
126#382—24no—
127#382—24no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D412yes—
131C4×C4⋊C4C2312yes—
132C2×#95—24no—
133C2×#95—24no—
134C2×#95—21no—
135#340—24no—
136#332—28no—
137Q8⋊C8—28no—
138C22×Q8—21no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×Q8—48no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8Q1614yes—
152C22×Q8C2×D412yes—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×M16—24no—
161C2×M16—42no—
162C2×M16—42no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—22no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8C2×D412yes—
183C4⋊Q8C2×D412yes—
184#125—24no—
185#125—42no—
186C22⋊Q8—24no—
187C22⋊Q8—24no—
188C22⋊Q8—42no—
189C22⋊Q8—42no—
190C4×Q8—24no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204#117—24no—
205#117—42no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—44no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4C2×D412yes—
229C2×C4⋊C4C2×D414yes—
230C2×C4⋊C4C2×D414yes—
231C2×C22⋊C4—41no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—41no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D412yes—
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D411yescharacteristic
247C2×C42#4112yes—
248C2×C42C2411yescharacteristic
249#107—28no—
250#107—216no—
251#106—28no—
252C4⋊C8—24no—
253C4⋊C8—24no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—22no—
257C4⋊C8C2×D414yes—
258#103—28no—
259#103—28no—
260#103—216no—
261#103—48no—
262C22⋊C8—24no—
263C22⋊C8—42no—
264C4×C8—22no—
265C4×C8—42no—
266C4×C8C2×D414yes—
267#95—216no—
268#95—216no—
269#95—22no—
270#95—22no—
271C24#12011yescharacteristic
272C2×Q8—22no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—28no—
281C2×Q8—28no—
282C2×Q8—48no—
283C2×Q8—44no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—41no—
287C2×Q8—42no—
288C2×Q8—48no—
289C2×Q8—42no—
290C2×Q8—44no—
291C2×Q8—42no—
292C2×Q8—44no—
293C2×Q8—44no—
294C2×Q8—44no—
295C2×Q8—88no—
296C2×Q8—88no—
297C2×Q8C2×Q1614yes—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—24no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—48no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12012yes—
319C22×C4#12012yes—
320C22×C4#12011yescharacteristic
321C22×C4#12112yes—
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4#12712yes—
325C22×C4C2×Q1612yes—
326C22×C4#13512yes—
327C22×C4#13712yes—
328C22×C4#13712yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332M16—48no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—44no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—44no—
340C4⋊C4—22no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—44no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C22⋊C4—44no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—44no—
371C42—44no—
372C42—44no—
373C42#12012yes—
374C42#12012yes—
375C42#12112yes—
376C42#12112yes—
377C42#12112yes—
378C42#12712yes—
379C42C22×D411yescharacteristic
380C42C22×D411yescharacteristic
381C42C22×D411yescharacteristic
382C42C2×#4111yescharacteristic
383C23—22no—
384C23—22no—
385C23#39212yes—
386C23#45112yes—
387C23#45212yes—
388C23C2×#12011yescharacteristic
389Q8—24no—
390Q8—24no—
391Q8—44no—
392Q8—42no—
393Q8—48no—
394Q8—42no—
395Q8—44no—
396Q8—82no—
397Q8—88no—
398Q8—82no—
399Q8—84no—
400Q8—84no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—48no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—88no—
422C2×C4—84no—
423C2×C4—88no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4#39212yes—
427C2×C4#39212yes—
428C2×C4#39212yes—
429C2×C4#45114yes—
430C2×C4#45112yes—
431C2×C4#45214yes—
432C2×C4#45212yes—
433C2×C4#46714yes—
434C2×C4#47014yes—
435C2×C4C2×#12011yescharacteristic
436C2×C4C2×#12011yescharacteristic
437C2×C4C2×#12111yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4C2×#12111yescharacteristic
440C2×C4C2×#12711yescharacteristic
441C2×C4C22×Q1611yescharacteristic
442C2×C4C2×#13511yescharacteristic
443C2×C4C2×#13711yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C2×C4#57514yes—
446C8—44no—
447C8—44no—
448C8—82no—
449C22—24no—
450C22—24no—
451C22—22no—
452C22—22no—
453C22#200114yes—
454C22C2×#39211yescharacteristic
455C22C2×#45111yescharacteristic
456C22C2×#45211yescharacteristic
457C22#300512yes—
458C22#302012yes—
459C4—21no—
460C4—21no—
461C4—22no—
462C4—22no—
463C4—44no—
464C4—84no—
465C4—82no—
466C4—82no—
467C4—84no—
468C4—82no—
469C4#200114yes—
470C4C2×#45111yescharacteristic
471C4C2×#45211yescharacteristic
472C4#304012yes—
473C4C2×#46711yescharacteristic
474C4C2×#47011yescharacteristic
475C4#315312yes—
476C2—24no—
477C2#1876312yes—
478C2#1877912yes—
479C2C2×#200111yescharacteristic
480C2#2340311yescharacteristic
481C2#2343111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10077C2semidirect
8. #10061C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C2×#30782
C2×#25682
#213852
C2×#15492
C2×#14792
#100772
#100612
#79012