Finite Group Databasebeta

← #472095

Subgroups — #472095

Order 512.

1174 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×#2084C211yesmaximal, characteristic, complemented
2C2×#2023C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4C2×#1934C211yesmaximal, characteristic
5#15230C211yesmaximal, characteristic, complemented
6#15136C211yesmaximal, characteristic
7#11378C212yesmaximal, complemented
8#11372C212yesmaximal, complemented
9#11121C212yesmaximal
10#11051C212yesmaximal
11#10797C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×#102C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#397—22no—
19C2×#390C2211yescharacteristic
20C2×#390C2212yes—
21C2×#389C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2212yes—
24#2084—24no—
25#2023—24no—
26#1990—24no—
27#1934—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1446—22no—
38#1441—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—21no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—24no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66C2×#106—24no—
67C2×#106—24no—
68C2×#106—42no—
69C2×#102—22no—
70C2×#102—22no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—42no—
76C2×#102—42no—
77C2×C4×C8—42no—
78C2×C4×C8D414yes—
79#397—48no—
80#390—24no—
81#390—22no—
82#390—22no—
83#390—42no—
84#389—22no—
85#389—22no—
86#384—22no—
87#384—22no—
88#382—24no—
89#382—42no—
90C2×#95—24no—
91C2×#95—24no—
92C2×#95—21no—
93C2×#95—24no—
94C2×#95D412yes—
95C2×#95D412yes—
96#340—24no—
97#340—22no—
98#340—22no—
99#340—24no—
100#340—24no—
101#340—24no—
102#336—42no—
103#336—42no—
104#336D414yes—
105C25—21no—
106C22×Q8—44no—
107C22×Q8—42no—
108C22×Q8—44no—
109C22×Q8—82no—
110C22×D4—21no—
111C22×D4—21no—
112C22×D4—22no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4#4112yes—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4C2×D414yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8C2×D414yes—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C4⋊Q8—28no—
147C4⋊Q8—48no—
148C4⋊Q8—44no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
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—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4#4114yes—
182C2×C4⋊C4#4112yes—
183C2×C22⋊C4—48no—
184C2×C22⋊C4—48no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—84no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200#106—48no—
201#102—28no—
202#102—48no—
203#102—44no—
204C4×C8—44no—
205#95—24no—
206#95—24no—
207#95—48no—
208#95—48no—
209#95—42no—
210#95—84no—
211C24—22no—
212C24—41no—
213C24—42no—
214C24—42no—
215C24—41no—
216C24—42no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24C22×D411yescharacteristic
221C2×Q8—48no—
222C2×Q8—816no—
223C2×Q8—84no—
224C2×Q8—84no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—24no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—42no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—41no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4#12712yes—
278C22×C4#12712yes—
279C22×C4#13514yes—
280C22×C4AGL(1,ℤ/8)14yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4112yes—
285C22×C4C2×#4111yescharacteristic
286C22×C4H(5,𝔽2)12yes—
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C42−1+412yes—
289C22×C42−1+412yes—
290C22×C42−1+411yescharacteristic
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—24no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—48no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—84no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C22⋊C4—816no—
331C22⋊C4—84no—
332C22⋊C4—84no—
333C42—22no—
334C42—22no—
335C42—24no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C23—44no—
341C23—44no—
342C23—42no—
343C23—44no—
344C23—82no—
345C23—82no—
346C23—81no—
347C23—81no—
348C23—81no—
349C23—81no—
350C23—81no—
351C23—82no—
352C23—84no—
353C23—84no—
354C23—84no—
355C23—84no—
356C23—81no—
357C23C2×#12711yescharacteristic
358C23#53612yes—
359C23#54611yescharacteristic
360C23#54612yes—
361C23#54811yescharacteristic
362C23C2×#13512yes—
363C23C2×AGL(1,ℤ/8)12yes—
364C23#57712yes—
365C23#57712yes—
366Q8—816no—
367D4—42no—
368D4—42no—
369D4—44no—
370D4—81no—
371D4—82no—
372D4—81no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—28no—
376C2×C4—28no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—81no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—84no—
405C2×C4—84no—
406C2×C4—81no—
407C2×C4—82no—
408C2×C4—84no—
409C2×C4#49514yes—
410C2×C4#49614yes—
411C2×C4#52511yescharacteristic
412C2×C4#52511yescharacteristic
413C2×C4#53411yescharacteristic
414C2×C4#53611yescharacteristic
415C2×C4#53611yescharacteristic
416C2×C4#53612yes—
417C2×C4#53711yescharacteristic
418C2×C4#53912yes—
419C2×C4#54112yes—
420C2×C4#54212yes—
421C2×C4#54212yes—
422C2×C4#54612yes—
423C2×C4#54611yescharacteristic
424C2×C4#54811yescharacteristic
425C2×C4#55212yes—
426C2×C4#55612yes—
427C2×C4#56212yes—
428C2×C4#56312yes—
429C8—42no—
430C8—44no—
431C8—42no—
432C22—82no—
433C22—88no—
434C22—82no—
435C22—82no—
436C22—161no—
437C22—161no—
438C22—161no—
439C22—161no—
440C22—164no—
441C22#253911yescharacteristic
442C22#254011yescharacteristic
443C22#265811yescharacteristic
444C22#268712yes—
445C22#271612yes—
446C22C2×#49511yescharacteristic
447C22C2×#49611yescharacteristic
448C22#315411yescharacteristic
449C22#315411yescharacteristic
450C22#322814yes—
451C22#324414yes—
452C22#330114yes—
453C22#330312yes—
454C22#330614yes—
455C22#332412yes—
456C22#336012yes—
457C22#336812yes—
458C4—24no—
459C4—24no—
460C4—84no—
461C4—84no—
462C4—82no—
463C4—162no—
464C4—161no—
465C4—161no—
466C4#259911yescharacteristic
467C4#263711yescharacteristic
468C4#263811yescharacteristic
469C4#269012yes—
470C4#270012yes—
471C4#281411yescharacteristic
472C2—162no—
473C2#1681211yescharacteristic
474C2#2833411yescharacteristic
475C2#2834311yescharacteristic
476C2#2839811yescharacteristic
477C2#2842011yescharacteristic
478C2#3027811yescharacteristic
479C2#3029011yescharacteristic
480C2#3135514yes—
481C2#3223114yes—
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×#2084C2semidirect
3. C2×#1990C2semidirect
5. #15230C2semidirect
7. #11378C2semidirect
8. #11372C2semidirect
11. #10797C2semidirect

Maximal subgroups

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

subgroupindex
C2×#20842
C2×#20232
C2×#19902
C2×#19342
#152302
#151362
#113782
#113722
#111212
#110512
#107972