Finite Group Databasebeta

← #472090

Subgroups — #472090

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2083C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1934C211yesmaximal, characteristic
5#15143C211yesmaximal, characteristic, complemented
6#15136C211yesmaximal, characteristic
7#11379C212yesmaximal, complemented
8#11373C212yesmaximal, complemented
9#11123C212yesmaximal
10#11049C212yesmaximal
11#10797C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#102C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#397—22no—
19C2×#395—22no—
20C2×#390C2211yescharacteristic
21C2×#390C2212yes—
22C2×#389C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#384C2212yes—
25#2083—24no—
26#2023—24no—
27#1990—24no—
28#1934—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1444—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×D4C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—24no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#107—24no—
68C2×#107—24no—
69C2×#107—42no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—42no—
77C2×#102—42no—
78C2×C4×C8—42no—
79C2×C4×C8D414yes—
80#397—48no—
81#395—48no—
82#390—24no—
83#390—22no—
84#390—22no—
85#390—42no—
86#389—22no—
87#389—22no—
88#384—24no—
89#384—22no—
90#384—22no—
91#384—42no—
92C2×#95—24no—
93C2×#95—24no—
94C2×#95—21no—
95C2×#95D412yes—
96C2×#95D412yes—
97#340—24no—
98#340—22no—
99#340—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—42no—
104#336—42no—
105#336D414yes—
106C25—21no—
107C22×Q8—44no—
108C22×Q8—42no—
109C22×Q8—44no—
110C22×Q8—82no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—21no—
115C22×D4—22no—
116C22×D4—21no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4#4112yes—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4C2×D414yes—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8C2×D414yes—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C4⋊Q8—28no—
148C4⋊Q8—48no—
149C4⋊Q8—44no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—84no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4#4114yes—
186C2×C4⋊C4#4112yes—
187C2×C22⋊C4—48no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—84no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204#107—48no—
205#102—28no—
206#102—48no—
207#102—44no—
208C4×C8—44no—
209#95—24no—
210#95—42no—
211#95—48no—
212#95—48no—
213#95D818yes—
214C24—22no—
215C24—41no—
216C24—42no—
217C24—42no—
218C24—41no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24—42no—
223C24C22×D411yescharacteristic
224C2×Q8—48no—
225C2×Q8—816no—
226C2×Q8—84no—
227C2×Q8—84no—
228C2×D4—22no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4#12712yes—
280C22×C4#12712yes—
281C22×C4C2×D818yes—
282C22×C4C2×D814yes—
283C22×C4#13714yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C2×#4111yescharacteristic
287C22×C4C2×#4112yes—
288C22×C4C2×#4111yescharacteristic
289C22×C4H(5,𝔽2)12yes—
290C22×C4H(5,𝔽2)11yescharacteristic
291C22×C42−1+412yes—
292C22×C42−1+412yes—
293C22×C42−1+411yescharacteristic
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—816no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C4⋊C4—84no—
334C22⋊C4—816no—
335C22⋊C4—84no—
336C22⋊C4—84no—
337C42—22no—
338C42—22no—
339C42—24no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—42no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—42no—
348C23—82no—
349C23—82no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—84no—
356C23—82no—
357C23—84no—
358C23—84no—
359C23—81no—
360C23—84no—
361C23C2×#12711yescharacteristic
362C23#53612yes—
363C23#54611yescharacteristic
364C23#54612yes—
365C23#54811yescharacteristic
366C23C22×D812yes—
367C23C2×#13712yes—
368C23#57712yes—
369C23#57712yes—
370Q8—816no—
371D4—42no—
372D4—42no—
373D4—44no—
374D4—81no—
375D4—82no—
376D4—81no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—28no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—28no—
385C2×C4—28no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—81no—
405C2×C4—82no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—81no—
412C2×C4—82no—
413C2×C4#49314yes—
414C2×C4#49714yes—
415C2×C4#52511yescharacteristic
416C2×C4#52511yescharacteristic
417C2×C4#53411yescharacteristic
418C2×C4#53611yescharacteristic
419C2×C4#53611yescharacteristic
420C2×C4#53612yes—
421C2×C4#53711yescharacteristic
422C2×C4#53912yes—
423C2×C4#54112yes—
424C2×C4#54212yes—
425C2×C4#54212yes—
426C2×C4#54612yes—
427C2×C4#54611yescharacteristic
428C2×C4#54811yescharacteristic
429C2×C4#55212yes—
430C2×C4#55612yes—
431C2×C4#56212yes—
432C2×C4#56312yes—
433C8—42no—
434C8—44no—
435C8—42no—
436C22—82no—
437C22—88no—
438C22—82no—
439C22—82no—
440C22—161no—
441C22—161no—
442C22—161no—
443C22—161no—
444C22—164no—
445C22#253911yescharacteristic
446C22#254011yescharacteristic
447C22#265811yescharacteristic
448C22#268712yes—
449C22#271612yes—
450C22C2×#49311yescharacteristic
451C22C2×#49711yescharacteristic
452C22#315411yescharacteristic
453C22#315411yescharacteristic
454C22#322614yes—
455C22#324414yes—
456C22#329412yes—
457C22#329414yes—
458C22#332014yes—
459C22#333112yes—
460C22#335812yes—
461C22#336912yes—
462C4—24no—
463C4—24no—
464C4—48no—
465C4—84no—
466C4—82no—
467C4—161no—
468C4—161no—
469C4—162no—
470C4#259911yescharacteristic
471C4#263711yescharacteristic
472C4#263811yescharacteristic
473C4#269012yes—
474C4#270012yes—
475C4#281411yescharacteristic
476C2—162no—
477C2#1681211yescharacteristic
478C2#2832911yescharacteristic
479C2#2834811yescharacteristic
480C2#2839711yescharacteristic
481C2#2841911yescharacteristic
482C2#3027611yescharacteristic
483C2#3029011yescharacteristic
484C2#3135014yes—
485C2#3222614yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2083C2semidirect
3. C2×#1990C2semidirect
5. #15143C2semidirect
7. #11379C2semidirect
8. #11373C2semidirect
11. #10797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20832
C2×#20232
C2×#19902
C2×#19342
#151432
#151362
#113792
#113732
#111232
#110492
#107972