Finite Group Databasebeta

← #465319

Subgroups — #465319

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2066C212yesmaximal
2C2×#1981C211yesmaximal, characteristic
3C2×#1920C211yesmaximal, characteristic
4#15188C211yesmaximal, characteristic
5#13918C211yesmaximal, characteristic
6#11322C211yesmaximal, characteristic
7#11228C211yesmaximal, characteristic
8#11190C212yesmaximal
9#11077C212yesmaximal
10#11067C212yesmaximal
11#11010C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×#382—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#2066—28no—
27#1981—24no—
28#1920—24no—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36#1442—21no—
37#1442—21no—
38#1442—22no—
39#1441—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42#1434—21no—
43C4×#95C2211yescharacteristic
44C23×Q8C2312yes—
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
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—22no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×#106—22no—
65C2×#106—22no—
66C2×#106D414yes—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—42no—
70C2×C4⋊C8—42no—
71C2×C4⋊C8—42no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—24no—
78C2×#103—42no—
79C2×#103—42no—
80C2×#103—42no—
81C2×#103—42no—
82C2×#103—42no—
83C2×#103—42no—
84#391—24no—
85#391—24no—
86#391—42no—
87#391—42no—
88#389—24no—
89#389—22no—
90#389—22no—
91#389—42no—
92#382—44no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4D414yes—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95D411yescharacteristic
107C2×#95D412yes—
108C2×#95D411yescharacteristic
109#340—24no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—42no—
114#336—22no—
115#336—42no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8#4114yes—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8C2×D412yes—
158C4⋊Q8—28no—
159C4⋊Q8—48no—
160C4⋊Q8—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4#4112yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C2×D412yes—
208C2×C42#4112yes—
209#106—28no—
210C4⋊C8—44no—
211C4⋊C8—48no—
212#103—48no—
213#103—48no—
214#103—48no—
215#103—48no—
216#95—24no—
217#95—22no—
218#95—42no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—42no—
226#95—44no—
227#95—42no—
228#95—81no—
229#95SD1614yes—
230C24C22×D411yescharacteristic
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4#12112yes—
315C22×C4#12112yes—
316C22×C4C2×SD1614yes—
317C22×C4C2×SD1612yes—
318C22×C4#13712yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4112yes—
323C22×C4C2×#4111yescharacteristic
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)12yes—
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C42−1+412yes—
328C22×C42−1+411yescharacteristic
329C2×C8—22no—
330C2×C8—24no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—48no—
354C4⋊C4—48no—
355C4⋊C4—41no—
356C4⋊C4—42no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—88no—
366C4⋊C4—84no—
367C4⋊C4#12114yes—
368C42—22no—
369C42—22no—
370C42—24no—
371C42—42no—
372C42—42no—
373C42—42no—
374C42—42no—
375C42—48no—
376C23C2×#12111yescharacteristic
377C23#53712yes—
378C23D4211yescharacteristic
379C23#54612yes—
380C23#54811yescharacteristic
381C23C22×SD1611yescharacteristic
382C23C2×#13711yescharacteristic
383C23#57512yes—
384C23#57711yescharacteristic
385C23#57811yescharacteristic
386C23#57812yes—
387Q8—44no—
388Q8—44no—
389Q8—44no—
390Q8—44no—
391Q8—82no—
392Q8—82no—
393Q8—82no—
394Q8—82no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#46114yes—
455C2×C4C2×#12111yescharacteristic
456C2×C4#52511yescharacteristic
457C2×C4#53511yescharacteristic
458C2×C4#53511yescharacteristic
459C2×C4#53512yes—
460C2×C4#53611yescharacteristic
461C2×C4#53711yescharacteristic
462C2×C4#53812yes—
463C2×C4#53912yes—
464C2×C4#54012yes—
465C2×C4#54212yes—
466C2×C4#54711yescharacteristic
467C2×C4#54812yes—
468C2×C4#55112yes—
469C2×C4#55212yes—
470C2×C4#55311yescharacteristic
471C2×C4#56212yes—
472C2×C4#56312yes—
473C8—42no—
474C8—82no—
475C8—81no—
476C22#241111yescharacteristic
477C22#241611yescharacteristic
478C22#262011yescharacteristic
479C22#267612yes—
480C22#270712yes—
481C22C2×#46111yescharacteristic
482C22#304011yescharacteristic
483C22#304411yescharacteristic
484C22#304611yescharacteristic
485C22#324612yes—
486C22#325212yes—
487C22#325812yes—
488C22#326112yes—
489C22#326912yes—
490C22#327611yescharacteristic
491C22D4×SD1611yescharacteristic
492C22#329812yes—
493C22#331212yes—
494C22#332112yes—
495C22#332712yes—
496C22#335911yescharacteristic
497C22#337211yescharacteristic
498C22#337712yes—
499C4—24no—
500C4—44no—
501C4—42no—
502C4—44no—
503C4—82no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4—161no—
508C4—162no—
509C4#253611yescharacteristic
510C4#261911yescharacteristic
511C4#263611yescharacteristic
512C4#272312yes—
513C4#276112yes—
514C4#282011yescharacteristic
515C2#1651911yescharacteristic
516C2#2448011yescharacteristic
517C2#2449911yescharacteristic
518C2#2473011yescharacteristic
519C2#2474311yescharacteristic
520C2#2982811yescharacteristic
521C2#2983711yescharacteristic
522C2#3090912yes—
523C2#3093212yes—
524C2#3196412yes—
525C2#3199112yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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×#20662
C2×#19812
C2×#19202
#151882
#139182
#113222
#112282
#111902
#110772
#110672
#110102