Finite Group Databasebeta

← #239503

Subgroups — #239503

Order 512.

1254 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35178C211yesmaximal, characteristic
2#21386C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5C2×#1461C211yesmaximal, characteristic, complemented
6#13942C211yesmaximal, characteristic, complemented
7#10397C218yesmaximal, complemented
8#8026C211yesmaximal, characteristic, complemented
9C2×#536—21no—
10C2×#532C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2854—22no—
15#2618C2214yes—
16#2548—21no—
17#2546—21no—
18#2545—22no—
19#2538—22no—
20#2536—22no—
21#2311—21no—
22C2×#391—21no—
23#2285C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1988—21no—
27#1982—21no—
28#1947—22no—
29#1874C2214yes—
30#1852C2214yes—
31C4×#107—22no—
32C4×#102—22no—
33#1517—24no—
34C2×#327C2214yes—
35#1461—24no—
36C2×#324C2214yes—
37#1441C2214yes—
38C4⋊#95—21no—
39#1434—22no—
40#1294C2214yes—
41C22×#41—21no—
42C23×Q8D411yescharacteristic
43#536—44no—
44#536—44no—
45#532—22no—
46#532—22no—
47#532—22no—
48#532—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8D411yescharacteristic
51C2×#127D411yescharacteristic
52C2×#124—22no—
53C2×#124—21no—
54C2×#123—24no—
55C2×C22⋊Q8—24no—
56C2×#121—21no—
57C2×#121C2312yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8—22no—
66C2×C4×D4—21no—
67C2×C4×D4D412yes—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2312yes—
73C22×C42—21no—
74C22×C42D412yes—
75C22×C42C2311yescharacteristic
76#431—22no—
77#431—22no—
78#431—22no—
79#431D414yes—
80C2×#107—24no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×#102—24no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88#398—24no—
89#396—28no—
90#394—24no—
91#391—22no—
92#391—44no—
93#390—22no—
94#390—24no—
95#389—24no—
96#388—28no—
97#386—24no—
98#384D412yes—
99#382—21no—
100#382—22no—
101#382—22no—
102#380—24no—
103#380—21no—
104#380C2312yes—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D412yes—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—24no—
114C2×#95—24no—
115C2×#95C2312yes—
116#340—24no—
117#340—24no—
118#327—28no—
119#324—28no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—42no—
123C2×#41—42no—
124C2×#41—42no—
125C2×#41—42no—
126C22×Q8—21no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8D814yes—
135C22×D4—22no—
136C22×D4—21no—
137C22×D4—21no—
138C22×D4C2×D412yes—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—48no—
152#127—22no—
153#127—22no—
154#124—48no—
155#124—42no—
156#124—42no—
157#123—48no—
158#123—48no—
159C22⋊Q8—416no—
160#121—22no—
161#121—22no—
162#121—22no—
163#121—22no—
164#121—42no—
165#121—42no—
166C4×Q8—42no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×D4—24no—
175C4×D4—24no—
176C4×D4—42no—
177C4×D4—42no—
178#117—42no—
179#117—42no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D414yes—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—28no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—24no—
211C2×C42—24no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224#107—48no—
225C4⋊C8—24no—
226C4⋊C8—42no—
227#102—48no—
228C22⋊C8—22no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8C2×D414yes—
232C4×C8—24no—
233C4×C8—42no—
234#95—28no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—48no—
240#95—41no—
241#95—41no—
242#95—42no—
243#95—48no—
244C24—22no—
245C24C22×D411yescharacteristic
246#41—84no—
247#41—84no—
248#41—84no—
249#41—84no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—28no—
253C2×Q8—28no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—44no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4#12011yescharacteristic
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12112yes—
311C22×C4#12112yes—
312C22×C4#12312yes—
313C22×C4#12312yes—
314C22×C4C2×D814yes—
315C22×C4C2×D812yes—
316C22×C4#13512yes—
317C22×C4AGL(1,ℤ/8)12yes—
318C22×C4#13712yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C2×C8—24no—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—42no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—416no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—44no—
346C4⋊C4—84no—
347C22⋊C4—416no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—44no—
353C22⋊C4—88no—
354C42—44no—
355C42—44no—
356C42—44no—
357C42—82no—
358C42—82no—
359C42#12112yes—
360C42#12112yes—
361C42#12112yes—
362C42#12112yes—
363C42#12312yes—
364C42#12312yes—
365C42#12412yes—
366C42#12412yes—
367C23—22no—
368C23—22no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23—44no—
373C23—44no—
374C23C2×#12011yescharacteristic
375C23C2×#12111yescharacteristic
376C23C2×#12311yescharacteristic
377C23C22×D811yescharacteristic
378C23C2×#13511yescharacteristic
379C23C2×AGL(1,ℤ/8)11yescharacteristic
380C23C2×#13711yescharacteristic
381Q8—48no—
382Q8—48no—
383Q8—84no—
384D4—44no—
385D4—82no—
386D4—84no—
387D4—82no—
388D4—82no—
389D4—82no—
390D4—82no—
391D4—84no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—88no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4#39412yes—
433C2×C4#39412yes—
434C2×C4#39412yes—
435C2×C4#39412yes—
436C2×C4#44712yes—
437C2×C4#44712yes—
438C2×C4#45212yes—
439C2×C4#45212yes—
440C2×C4#46614yes—
441C2×C4#46814yes—
442C2×C4#48014yes—
443C2×C4#48214yes—
444C2×C4C2×#12111yescharacteristic
445C2×C4C2×#12111yescharacteristic
446C2×C4C2×#12311yescharacteristic
447C2×C4C2×#12411yescharacteristic
448C8—44no—
449C8—82no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—84no—
459C22—84no—
460C22—82no—
461C22—82no—
462C22—82no—
463C22C2×#39411yescharacteristic
464C22C2×#44711yescharacteristic
465C22C2×#45211yescharacteristic
466C22#302012yes—
467C22#302512yes—
468C22C2×#46611yescharacteristic
469C22C2×#46811yescharacteristic
470C22C2×#48011yescharacteristic
471C22C2×#48211yescharacteristic
472C4—22no—
473C4—22no—
474C4—21no—
475C4—21no—
476C4—21no—
477C4—21no—
478C4—44no—
479C4—84no—
480C4—84no—
481C4—82no—
482C4—82no—
483C4—162no—
484C4#203314yes—
485C4#203314yes—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—162no—
491C2#1821512yes—
492C2#1821812yes—
493C2C2×#203311yescharacteristic
494C2#2349411yescharacteristic
495C2#2351411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21358C2semidirect
5. C2×#1461C2semidirect
6. #13942C2semidirect
7. #10397C2semidirect
8. #8026C2semidirect

Maximal subgroups

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

subgroupindex
#351782
#213862
#213582
C2×#15172
C2×#14612
#139422
#103972
#80262