Finite Group Databasebeta

← #474274

Subgroups — #474274

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 505 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#15180C211yesmaximal, characteristic
2#15152C211yesmaximal, characteristic
3#11316C212yesmaximal
4#11190C212yesmaximal
5#11077C212yesmaximal
6#11050C212yesmaximal
7#11010C212yesmaximal
8#10926C212yesmaximal
9#10834C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×#103C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×C4×C8C2211yescharacteristic
17C2×#400—21no—
18C2×#391C2212yes—
19C2×#391C2212yes—
20C2×#384C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2212yes—
27C2×#336C2212yes—
28#1445—21no—
29#1442—21no—
30#1442—21no—
31#1442—21no—
32#1441—21no—
33#1441—21no—
34#1441—21no—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C23×Q8C2312yes—
40C23×C8C2312yes—
41C23×C8C2312yes—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—42no—
44C2×C4⋊Q8—41no—
45C2×C4⋊Q8—41no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×C4⋊C8—22no—
59C2×C4⋊C8—41no—
60C2×C4⋊C8—41no—
61C2×#103—24no—
62C2×#103—24no—
63C2×#103—24no—
64C2×#103—42no—
65C2×#103—42no—
66C2×#103—42no—
67C2×#103—42no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103—42no—
72C2×#103—42no—
73C2×C4×C8—22no—
74C2×C4×C8—41no—
75C2×C4×C8—41no—
76#400—44no—
77#391—24no—
78#391—24no—
79#391—24no—
80#391—42no—
81#384—41no—
82#384—41no—
83C4×C4⋊C4—41no—
84C4×C4⋊C4—41no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95D411yescharacteristic
98C2×#95D411yescharacteristic
99#340—24no—
100#340—42no—
101#340—42no—
102#340—42no—
103#336—42no—
104#336—42no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8#4114yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2411yescharacteristic
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
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—22no—
143C22×C8#4114yes—
144C4⋊Q8—44no—
145C4⋊Q8—48no—
146C4⋊Q8—44no—
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—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4SD1614yes—
179C2×C4⋊C4Q1614yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42#4112yes—
193C2×C42#4112yes—
194C4⋊C8—44no—
195#103—48no—
196#103—48no—
197#103—48no—
198#103—48no—
199C4×C8—44no—
200#95—41no—
201#95—41no—
202#95—44no—
203#95—44no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—44no—
208#95—44no—
209#95—42no—
210#95—44no—
211#95—44no—
212#95—82no—
213#95—82no—
214C24C22×D411yescharacteristic
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4#12412yes—
296C22×C4#12412yes—
297C22×C4C2×SD1612yes—
298C22×C4C2×SD1612yes—
299C22×C4C2×SD1612yes—
300C22×C4C2×Q1612yes—
301C22×C4C2×Q1612yes—
302C22×C4C2×Q1612yes—
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4112yes—
305C22×C4C2×#4112yes—
306C22×C4C2×#4111yescharacteristic
307C22×C4H(5,𝔽2)11yescharacteristic
308C22×C4H(5,𝔽2)11yescharacteristic
309C22×C4H(5,𝔽2)12yes—
310C22×C4H(5,𝔽2)11yescharacteristic
311C22×C4H(5,𝔽2)12yes—
312C22×C42−1+411yescharacteristic
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—84no—
344C4⋊C4—84no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—42no—
350C42—42no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C23C2×#12411yescharacteristic
356C23#53411yescharacteristic
357C23#53711yescharacteristic
358C23#54612yes—
359C23#55012yes—
360C23C22×SD1611yescharacteristic
361C23C22×Q1611yescharacteristic
362C23#57511yescharacteristic
363C23#57511yescharacteristic
364C23#57712yes—
365C23#57812yes—
366Q8—44no—
367Q8—44no—
368Q8—44no—
369Q8—44no—
370Q8—82no—
371Q8—82no—
372Q8—82no—
373Q8—82no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#48714yes—
434C2×C4#52912yes—
435C2×C4#53811yescharacteristic
436C2×C4#53812yes—
437C2×C4#53812yes—
438C2×C4#53811yescharacteristic
439C2×C4#53911yescharacteristic
440C2×C4#53912yes—
441C2×C4#53911yescharacteristic
442C2×C4#54012yes—
443C2×C4#54212yes—
444C2×C4#55112yes—
445C2×C4#55111yescharacteristic
446C2×C4#55312yes—
447C2×C4#55312yes—
448C2×C4#55511yescharacteristic
449C2×C4#56012yes—
450C2×C4#56011yescharacteristic
451C2×C4H(3,𝔽4)11yescharacteristic
452C8—44no—
453C8—82no—
454C22#249012yes—
455C22#263911yescharacteristic
456C22#269711yescharacteristic
457C22#270512yes—
458C22#273311yescharacteristic
459C22C2×#48711yescharacteristic
460C22#313511yescharacteristic
461C22#313712yes—
462C22#318711yescharacteristic
463C22#318911yescharacteristic
464C22#320011yescharacteristic
465C22#320211yescharacteristic
466C22#324611yescharacteristic
467C22#324911yescharacteristic
468C22#325812yes—
469C22#329812yes—
470C22#329912yes—
471C22#332612yes—
472C22#332712yes—
473C22#339212yes—
474C22#339512yes—
475C22#340112yes—
476C22#340212yes—
477C4—24no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—84no—
484C4—162no—
485C4—162no—
486C4—162no—
487C4#257211yescharacteristic
488C4#264912yes—
489C4#277812yes—
490C4#278211yescharacteristic
491C4#279511yescharacteristic
492C4#282811yescharacteristic
493C2#1693211yescharacteristic
494C2#2774912yes—
495C2#2775412yes—
496C2#3018711yescharacteristic
497C2#3019311yescharacteristic
498C2#3166811yescharacteristic
499C2#3166911yescharacteristic
500C2#3190012yes—
501C2#3190412yes—
502C2#3266911yescharacteristic
503C2#3267011yescharacteristic
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
#151802
#151522
#113162
#111902
#110772
#110502
#110102
#109262
#108342