Finite Group Databasebeta

← #224527

Subgroups — #224527

Order 512.

972 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2453C211yesmaximal, characteristic
2#22510C212yesmaximal
3#9654C212yesmaximal
4#9654C212yesmaximal
5#9650C212yesmaximal
6#8562C212yesmaximal
7#8560C211yesmaximal, characteristic
8#8154C212yesmaximal
9#8153C211yesmaximal, characteristic
10C22×#125C411yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#446—22no—
13#2453—22no—
14#2453—22no—
15#2453—22no—
16#2453—22no—
17C2×#400C412yes—
18C2×#398C412yes—
19C2×#391C411yescharacteristic
20C2×#391C411yescharacteristic
21C2×#389C411yescharacteristic
22C2×#389C2212yes—
23C2×#389C2211yescharacteristic
24C2×#387C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1985—22no—
27#1985—22no—
28#1984—22no—
29#1797C2212yes—
30#1797C2212yes—
31#1797C2212yes—
32#1749C2212yes—
33#1749C2212yes—
34C4×C22⋊C8C2212yes—
35C2×#324C2212yes—
36C2×#324C2212yes—
37C2×#324C2212yes—
38C2×#324C2211yescharacteristic
39C2×#324C2211yescharacteristic
40C2×#324C2212yes—
41C2×#324C2212yes—
42#1280C2212yes—
43#1280C2211yescharacteristic
44#1280C2211yescharacteristic
45C23×Q8C2×C411yescharacteristic
46C2×#125—22no—
47C2×#125—21no—
48C2×#125—21no—
49C2×#125—44no—
50C2×C4×Q8—44no—
51C2×C4×Q8D412yes—
52C2×C4×Q8D412yes—
53C2×C4×Q8D412yes—
54C2×C4×Q8D412yes—
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#446—48no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8C2312yes—
74C2×C22⋊C8C2312yes—
75C2×C22⋊C8C2312yes—
76C2×C22⋊C8C2312yes—
77C2×#97—22no—
78C2×#97—22no—
79C2×C4×C8—22no—
80#400—22no—
81#400—22no—
82#400—22no—
83#400—22no—
84#398—22no—
85#398—22no—
86#398—22no—
87#398—22no—
88#391—22no—
89#391—21no—
90#391—21no—
91#391—22no—
92#391—21no—
93#391—21no—
94#389—21no—
95#389—22no—
96#389—21no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#387—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C2×#95C2×C412yes—
110C2×#95C2×C412yes—
111C2×#95C2×C412yes—
112C2×#95C2×C411yescharacteristic
113C2×#95C2×C412yes—
114C2×#95C2×C412yes—
115C2×#95C2×C412yes—
116C2×#95C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123#324—24no—
124#324—24no—
125#324—24no—
126#324—41no—
127#324—41no—
128#324—42no—
129#324—42no—
130C22×Q8—24no—
131C22×Q8—41no—
132C22×Q8—41no—
133C22×Q8—42no—
134C22×Q8C22⋊C412yes—
135C22×Q8C22⋊C411yescharacteristic
136C22×Q8C22⋊C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C412yes—
141C23×C4C22×C412yes—
142C23×C4C22×C412yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152#125—48no—
153#125—44no—
154#125—44no—
155C4×Q8—28no—
156C4×Q8—44no—
157C4×Q8—44no—
158C4×Q8—44no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4—41no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22⋊C412yes—
193C2×C4⋊C4C22⋊C412yes—
194C2×C4⋊C4C22⋊C412yes—
195C2×C4⋊C4C22⋊C412yes—
196C2×C4⋊C4C22⋊C412yes—
197C2×C4⋊C4C22⋊C411yescharacteristic
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—44no—
203C2×C42—42no—
204C2×C42—41no—
205C2×C42—41no—
206C2×C42—42no—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C4⋊C8—44no—
214C4⋊C8—44no—
215C4⋊C8—44no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8#4114yes—
224#97—44no—
225#97—44no—
226C4×C8—44no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241C24C23×C411yescharacteristic
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—41no—
250C2×Q8—48no—
251C2×Q8—48no—
252C2×Q8#10114yes—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4C2×C22⋊C412yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4#13112yes—
312C22×C4C22×D412yes—
313C22×C4C2×#4112yes—
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C42−1+411yescharacteristic
316C2×C8—24no—
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—
327C2×C8—42no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—44no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—44no—
349C4⋊C4—44no—
350C4⋊C4—42no—
351C4⋊C4#9914yes—
352C42—24no—
353C42—24no—
354C42—24no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—42no—
359C42—42no—
360C42—42no—
361C42—42no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23C22×C22⋊C411yescharacteristic
367C23C4×#4111yescharacteristic
368C23#52011yescharacteristic
369C23C2×#13111yescharacteristic
370C23#56811yescharacteristic
371C23#56811yescharacteristic
372C23#56811yescharacteristic
373Q8—44no—
374Q8—48no—
375Q8—44no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—42no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4C2×#9912yes—
441C2×C4C2×#9914yes—
442C2×C4C2×#10112yes—
443C2×C4C2×#10114yes—
444C2×C4#54812yes—
445C2×C4#55012yes—
446C8—44no—
447C8—82no—
448C8—82no—
449C8—82no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22#234111yescharacteristic
463C22#287911yescharacteristic
464C22#288011yescharacteristic
465C22C22×#9911yescharacteristic
466C22#288411yescharacteristic
467C22C22×#10111yescharacteristic
468C22#288911yescharacteristic
469C22#296711yescharacteristic
470C22#296811yescharacteristic
471C22#299211yescharacteristic
472C22#299511yescharacteristic
473C4—22no—
474C4—22no—
475C4—24no—
476C4—22no—
477C4—22no—
478C4—24no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—81no—
484C4—82no—
485C4—81no—
486C4—82no—
487C4—81no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2#1739711yescharacteristic
493C2#1741011yescharacteristic
494C2#1741611yescharacteristic
495C2#1743011yescharacteristic
496C2#2253311yescharacteristic
497C2#2253411yescharacteristic
498C2#2253811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutator, prime radicalC24C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#24532
#225102
#96542
#96502
#85622
#85602
#81542
#81532