Finite Group Databasebeta

← #410705

Subgroups — #410705

Order 512.

988 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×#1834C211yesmaximal, characteristic
2C2×C8×C4⋊C4C211yesmaximal, characteristic
3#15152C211yesmaximal, characteristic
4#15151C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic
6#11357C211yesmaximal, characteristic
7#11315C212yesmaximal
8#11227C212yesmaximal
9#10880C212yesmaximal
10#10820C212yesmaximal
11#10820C211yesmaximal, characteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×#400—21no—
17C2×#398—21no—
18C2×#398—21no—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1834—24no—
24C8×C4⋊C4—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1445—21no—
36#1445—21no—
37#1445—21no—
38#1442—21no—
39#1441—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—22no—
66C2×C4×C8—22no—
67C2×C4×C8—22no—
68C2×C4×C8—24no—
69C2×C4×C8—22no—
70C2×C4×C8—22no—
71C2×C4×C8—24no—
72C2×C4×C8D414yes—
73#400—44no—
74#398—44no—
75#398—44no—
76#389—24no—
77#389—24no—
78#384—22no—
79#384—22no—
80#384—22no—
81#384—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95D411yescharacteristic
96C2×#95D411yescharacteristic
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#336—22no—
106#336—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2411yescharacteristic
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8#4112yes—
136C22×C8#4112yes—
137C22×C8#4114yes—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
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—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—44no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4D814yes—
180C2×C4⋊C4SD1614yes—
181C2×C4⋊C4#4114yes—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C2×D412yes—
193C2×C42#4112yes—
194C2×C42#4112yes—
195C2×C42#4112yes—
196C4⋊C8—28no—
197C4×C8—28no—
198C4×C8—28no—
199C4×C8—44no—
200#95—22no—
201#95—22no—
202#95—44no—
203#95—42no—
204#95—44no—
205#95—42no—
206#95—44no—
207#95—44no—
208#95—42no—
209#95—44no—
210#95—42no—
211#95—82no—
212#95—82no—
213C24C22×D411yescharacteristic
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
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#12112yes—
263C22×C4#12112yes—
264C22×C4#12412yes—
265C22×C4#12412yes—
266C22×C4C2×D812yes—
267C22×C4C2×D812yes—
268C22×C4C2×D812yes—
269C22×C4C2×SD1612yes—
270C22×C4C2×SD1612yes—
271C22×C4C2×SD1612yes—
272C22×C4#13512yes—
273C22×C4#13512yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4C2×#4111yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4112yes—
281C22×C4C2×#4111yescharacteristic
282C22×C42−1+412yes—
283C22×C42−1+412yes—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—24no—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—42no—
303C2×C8#12314yes—
304C4⋊C4—28no—
305C4⋊C4—28no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C4⋊C4—84no—
325C4⋊C4—84no—
326C4⋊C4—88no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—24no—
334C42—41no—
335C42—41no—
336C42—42no—
337C23C2×#12111yescharacteristic
338C23C2×#12411yescharacteristic
339C23#53211yescharacteristic
340C23#54712yes—
341C23#54812yes—
342C23C22×D811yescharacteristic
343C23C22×SD1611yescharacteristic
344C23C2×#13511yescharacteristic
345C23C2×#13511yescharacteristic
346C23#57712yes—
347C23#57812yes—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4#46514yes—
406C2×C4#46614yes—
407C2×C4#48614yes—
408C2×C4C2×#12311yescharacteristic
409C2×C4#52511yescharacteristic
410C2×C4#52511yescharacteristic
411C2×C4#52912yes—
412C2×C4#52912yes—
413C2×C4#52911yescharacteristic
414C2×C4#52911yescharacteristic
415C2×C4#53211yescharacteristic
416C2×C4#53611yescharacteristic
417C2×C4#53611yescharacteristic
418C2×C4#54112yes—
419C2×C4#54112yes—
420C2×C4#55212yes—
421C2×C4#55212yes—
422C2×C4#55212yes—
423C2×C4#55212yes—
424C2×C4#55212yes—
425C2×C4#55612yes—
426C8—24no—
427C8—42no—
428C8—42no—
429C8—42no—
430C22#248611yescharacteristic
431C22#248912yes—
432C22#251912yes—
433C22#254511yescharacteristic
434C22#285011yescharacteristic
435C22C2×#46511yescharacteristic
436C22C2×#46611yescharacteristic
437C22#305711yescharacteristic
438C22#305911yescharacteristic
439C22C2×#48611yescharacteristic
440C22#313111yescharacteristic
441C22#313712yes—
442C22#316711yescharacteristic
443C22#316911yescharacteristic
444C22#317812yes—
445C22#333412yes—
446C22#333512yes—
447C22#333612yes—
448C22#333712yes—
449C22#335812yes—
450C22#335912yes—
451C22#336412yes—
452C22#336512yes—
453C4—24no—
454C4—24no—
455C4—24no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—84no—
460C4—162no—
461C4—162no—
462C4—162no—
463C4#239511yescharacteristic
464C4#248411yescharacteristic
465C4#252412yes—
466C4#252812yes—
467C4#261111yescharacteristic
468C4#263411yescharacteristic
469C2#1508511yescharacteristic
470C2#2765811yescharacteristic
471C2#2766111yescharacteristic
472C2#2772712yes—
473C2#2773212yes—
474C2#2803012yes—
475C2#2803212yes—
476C2#2863711yescharacteristic
477C2#2863911yescharacteristic
478C2#3304211yescharacteristic
479C2#3304611yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#18342
C2×C8×C4⋊C42
#151522
#151512
#113732
#113572
#113152
#112272
#108802
#108202