Finite Group Databasebeta

← #413565

Subgroups — #413565

Order 512.

1008 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
1C2×#2085C212yesmaximal
2C2×#1948C211yesmaximal, characteristic
3C2×#1947C211yesmaximal, characteristic
4#14276C211yesmaximal, characteristic
5#13779C211yesmaximal, characteristic
6#11257C212yesmaximal
7#11256C212yesmaximal
8#11207C212yesmaximal
9#10850C211yesmaximal, characteristic
10#10850C211yesmaximal, characteristic
11#10833C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×#389C2212yes—
17C2×#384C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×#382C2211yescharacteristic
20C2×#382C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23C2×#376C2211yescharacteristic
24#2085—28no—
25#1948—24no—
26#1947—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1441—22no—
36#1441—22no—
37#1436—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C23×C8C2312yes—
43C23×C8C2312yes—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2311yescharacteristic
54C22×C42—21no—
55C22×C42—21no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×#107—24no—
60C2×#107—24no—
61C2×#107—42no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—42no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8Q814yes—
70C2×C4⋊C8Q814yes—
71#389—24no—
72#389—24no—
73#384—22no—
74#384—22no—
75#384—22no—
76#384D414yes—
77#382—22no—
78#382—22no—
79#382—22no—
80#382D414yes—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85#376—22no—
86#376—22no—
87C2×#95—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D412yes—
98C2×#95D412yes—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#336—24no—
106#336—42no—
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4C2×D412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C2×Q814yes—
131C22×C8C2×Q814yes—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4#4112yes—
177C2×C4⋊C4#4112yes—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42—82no—
191C2×C42—82no—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×Q812yes—
195C2×C42C2×Q812yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198C2×C42#4112yes—
199C2×C42#4112yes—
200#107—48no—
201#106—48no—
202C4⋊C8—28no—
203C4⋊C8—28no—
204#95—28no—
205#95—24no—
206#95—24no—
207#95—42no—
208#95—42no—
209#95—44no—
210#95—42no—
211#95—42no—
212#95—82no—
213#95—82no—
214#95—82no—
215#95—82no—
216C24C22×D411yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4#12114yes—
269C22×C4#12112yes—
270C22×C4#12112yes—
271C22×C4#12312yes—
272C22×C4#12312yes—
273C22×C4C4⋊Q812yes—
274C22×C4C4⋊Q812yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×Q811yescharacteristic
278C22×C4C22×Q811yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4H(5,𝔽2)11yescharacteristic
286C22×C42−1+411yescharacteristic
287C22×C42−1+411yescharacteristic
288C22×C42−1+411yescharacteristic
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—22no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
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—88no—
330C4⋊C4—88no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—84no—
340C42—84no—
341C42C22⋊Q814yes—
342C42C22⋊Q814yes—
343C42#12514yes—
344C42#12514yes—
345C23C2×#12111yescharacteristic
346C23C2×#12111yescharacteristic
347C23C2×#12311yescharacteristic
348C23#52511yescharacteristic
349C23C2×C4⋊Q811yescharacteristic
350C23#53611yescharacteristic
351C23#53711yescharacteristic
352C23#57612yes—
353C23#57712yes—
354C23#57712yes—
355C23#57812yes—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4#39814yes—
413C2×C4#39814yes—
414C2×C4C2×C22⋊Q811yescharacteristic
415C2×C4C2×C22⋊Q811yescharacteristic
416C2×C4C2×#12511yescharacteristic
417C2×C4C2×#12511yescharacteristic
418C2×C4#52911yescharacteristic
419C2×C4#52911yescharacteristic
420C2×C4#53211yescharacteristic
421C2×C4#53211yescharacteristic
422C2×C4#54111yescharacteristic
423C2×C4#54111yescharacteristic
424C2×C4#54211yescharacteristic
425C2×C4#54211yescharacteristic
426C2×C4D4211yescharacteristic
427C2×C4#54711yescharacteristic
428C2×C4#54711yescharacteristic
429C2×C4#54811yescharacteristic
430C2×C4#54811yescharacteristic
431C2×C4D4×Q811yescharacteristic
432C2×C4D4×Q811yescharacteristic
433C2×C4#55211yescharacteristic
434C2×C4#55211yescharacteristic
435C2×C4#55411yescharacteristic
436C2×C4#55411yescharacteristic
437C2×C4#55511yescharacteristic
438C2×C4#55711yescharacteristic
439C2×C4#55711yescharacteristic
440C2×C4#55711yescharacteristic
441C2×C4#55711yescharacteristic
442C8—44no—
443C8—44no—
444C22C2×#39811yescharacteristic
445C22#253611yescharacteristic
446C22#253711yescharacteristic
447C22#254611yescharacteristic
448C22#258411yescharacteristic
449C22#262511yescharacteristic
450C22#285411yescharacteristic
451C22#304111yescharacteristic
452C22#304211yescharacteristic
453C22#304311yescharacteristic
454C22#304411yescharacteristic
455C22#304512yes—
456C22#304612yes—
457C22#309812yes—
458C22#309912yes—
459C22#311812yes—
460C22#312112yes—
461C22#316312yes—
462C22#316412yes—
463C22#324112yes—
464C22#324212yes—
465C22#325712yes—
466C22#325812yes—
467C4—24no—
468C4—24no—
469C4—44no—
470C4—82no—
471C4—82no—
472C4—84no—
473C4—82no—
474C4—82no—
475C4—82no—
476C4—162no—
477C4#245211yescharacteristic
478C4#245211yescharacteristic
479C4#245311yescharacteristic
480C4#245311yescharacteristic
481C4#250611yescharacteristic
482C4#250611yescharacteristic
483C4#250811yescharacteristic
484C4#250811yescharacteristic
485C2#1514911yescharacteristic
486C2#2366011yescharacteristic
487C2#2366111yescharacteristic
488C2#2821011yescharacteristic
489C2#2821311yescharacteristic
490C2#2825711yescharacteristic
491C2#2825811yescharacteristic
492C2#2869812yes—
493C2#2932512yes—
494C2#2994612yes—
495C2#3311512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20852
C2×#19482
C2×#19472
#142762
#137792
#112572
#112562
#112072
#108502
#108332