Finite Group Databasebeta

← #236134

Subgroups — #236134

Order 512.

1356 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic
3C2×#2568C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6C2×#1487C211yesmaximal, characteristic
7#10069C214yesmaximal, complemented
8#10061C214yesmaximal, complemented
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13#3229—24no—
14#3229—24no—
15C2×#498C2212yes—
16C2×#487C2212yes—
17#3104—24no—
18#3104—24no—
19C2×#484—21no—
20C2×#483—22no—
21C2×#477C2212yes—
22C2×#475—22no—
23C2×#462C2212yes—
24C2×#451—21no—
25C2×C4×Q16—22no—
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#423C2211yescharacteristic
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#1989—22no—
39#1988C2212yes—
40#1987C2212yes—
41#1736—22no—
42#1546—24no—
43C2×#332C2214yes—
44#1487—24no—
45C2×Q8⋊C8C2214yes—
46#1279C2214yes—
47C23×Q8—21no—
48C23×Q8D411yescharacteristic
49C22×Q16—22no—
50C22×M16—21no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533D414yes—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8D412yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2312yes—
65C2×#125—21no—
66C2×C22⋊Q8—21no—
67C2×C22⋊Q8—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×C4×Q8—21no—
75C2×C4×Q8—22no—
76C2×C4×Q8D412yes—
77C2×C4×Q8C2312yes—
78C2×#117—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4D412yes—
82C22×C4⋊C4D411yescharacteristic
83C22×C4⋊C4D412yes—
84C22×C42—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#498—24no—
88#498—24no—
89#487—24no—
90#487—24no—
91#484—28no—
92#483—216no—
93#477—24no—
94#477—24no—
95#475—216no—
96#462—24no—
97#462—24no—
98#451—28no—
99C4×Q16—216no—
100#431—22no—
101#431—22no—
102#431D414yes—
103#431D414yes—
104C2×#107—22no—
105C2×#106—24no—
106#423—22no—
107#423—22no—
108C2×C4⋊C8—21no—
109C2×C4⋊C8—21no—
110C2×C4⋊C8C2312yes—
111C2×C4⋊C8C2312yes—
112C2×#103—22no—
113C2×#103—24no—
114C2×#103—22no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C4×C8—21no—
118C2×C4×C8C2312yes—
119#400—28no—
120#400—28no—
121#391—28no—
122#391—24no—
123#391—24no—
124#391—28no—
125#382—24no—
126#382—24no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4C2312yes—
131C2×#95—24no—
132C2×#95—24no—
133C2×#95—21no—
134#340—24no—
135#340—24no—
136#332—28no—
137Q8⋊C8—28no—
138C22×Q8—24no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—21no—
145C22×Q8—21no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—44no—
150C22×Q8—44no—
151C22×Q8C2×D412yes—
152C23×C4—21no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×Q16—28no—
160C2×Q16—44no—
161C2×Q16—44no—
162C2×M16—42no—
163C2×M16—44no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—22no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8C2×D412yes—
184C4⋊Q8C2×D412yes—
185#125—24no—
186#125—42no—
187C22⋊Q8—24no—
188C22⋊Q8—24no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191C4×Q8—24no—
192C4×Q8—24no—
193C4×Q8—24no—
194C4×Q8—24no—
195C4×Q8—22no—
196C4×Q8—22no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—42no—
200C4×Q8—42no—
201C4×Q8—42no—
202C4×Q8—42no—
203C4×Q8—42no—
204C4×Q8—42no—
205#117—24no—
206#117—42no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D414yes—
232C2×C4⋊C4C2×D414yes—
233C2×C22⋊C4—41no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—44no—
239C2×C42—41no—
240C2×C42—42no—
241C2×C42—42no—
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×D411yescharacteristic
248C2×C42#4112yes—
249C2×C42C2411yescharacteristic
250#107—28no—
251#106—216no—
252C4⋊C8—24no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—22no—
256C4⋊C8—42no—
257C4⋊C8C2×D414yes—
258#103—28no—
259#103—28no—
260#103—216no—
261C22⋊C8—24no—
262C22⋊C8—24no—
263C4×C8—24no—
264C4×C8—22no—
265C4×C8C2×D414yes—
266#95—216no—
267#95—216no—
268#95—22no—
269#95—22no—
270C24#12011yescharacteristic
271C2×Q8—24no—
272C2×Q8—22no—
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—48no—
280C2×Q8—48no—
281C2×Q8—42no—
282C2×Q8—44no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—42no—
290C2×Q8—44no—
291C2×Q8—48no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—88no—
295C2×Q8—84no—
296C2×Q8—84no—
297C2×Q8#13714yes—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—42no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—48no—
317C22×C4#12012yes—
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12012yes—
321C22×C4#12011yescharacteristic
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4#12112yes—
325C22×C4#12712yes—
326C22×C4#13514yes—
327C22×C4#13714yes—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331Q16—416no—
332M16—84no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—44no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—44no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—24no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C22⋊C4—44no—
366C42—44no—
367C42—42no—
368C42—42no—
369C42—44no—
370C42—44no—
371C42—44no—
372C42#12012yes—
373C42#12012yes—
374C42#12112yes—
375C42#12112yes—
376C42#12112yes—
377C42#12712yes—
378C42C22×D411yescharacteristic
379C42C22×D411yescharacteristic
380C42C22×D411yescharacteristic
381C42C2×#4111yescharacteristic
382C23—21no—
383C23—21no—
384C23—22no—
385C23#39212yes—
386C23#45214yes—
387C23C2×#12011yescharacteristic
388Q8—24no—
389Q8—24no—
390Q8—48no—
391Q8—44no—
392Q8—42no—
393Q8—42no—
394Q8—44no—
395Q8—82no—
396Q8—82no—
397Q8—82no—
398Q8—84no—
399Q8—88no—
400Q8—82no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—48no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—48no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—88no—
423C2×C4—84no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—82no—
427C2×C4—88no—
428C2×C4#39212yes—
429C2×C4#39212yes—
430C2×C4#39212yes—
431C2×C4#45214yes—
432C2×C4#46418yes—
433C2×C4#49514yes—
434C2×C4#49714yes—
435C2×C4C2×#12011yescharacteristic
436C2×C4C2×#12011yescharacteristic
437C2×C4C2×#12111yescharacteristic
438C2×C4C2×#12111yescharacteristic
439C2×C4C2×#12111yescharacteristic
440C2×C4C2×#12711yescharacteristic
441C2×C4C2×#13512yes—
442C2×C4C2×#13712yes—
443C2×C4C2×#13712yes—
444C2×C4#57512yes—
445C8—44no—
446C8—44no—
447C8—82no—
448C22—24no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22#201514yes—
453C22C2×#39211yescharacteristic
454C22C2×#45212yes—
455C22#300411yescharacteristic
456C22#300511yescharacteristic
457C22#302012yes—
458C4—22no—
459C4—21no—
460C4—21no—
461C4—22no—
462C4—44no—
463C4—84no—
464C4—82no—
465C4—82no—
466C4—82no—
467C4—84no—
468C4#201514yes—
469C4#300712yes—
470C4C2×#46412yes—
471C4#305612yes—
472C4C2×#49511yescharacteristic
473C4C2×#49711yescharacteristic
474C2—24no—
475C2#1876112yes—
476C2#1878112yes—
477C2#2339211yescharacteristic
478C2#2340311yescharacteristic
479C2C2×#201511yescharacteristic
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. #10069C2semidirect
8. #10061C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
C2×#31042
C2×#25682
#213122
C2×#15462
C2×#14872
#100692
#100612
#78922