Finite Group Databasebeta

← #412237

Subgroups — #412237

Order 512.

998 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3164C211yesmaximal, characteristic
2#23561C212yesmaximal
3#21181C211yesmaximal, characteristic
4#21156C211yesmaximal, characteristic
5#20444C211yesmaximal, characteristic
6#11388C212yesmaximal
7#11333C212yesmaximal
8#11253C212yesmaximal
9#10872C211yesmaximal, characteristic
10#10871C211yesmaximal, characteristic
11#10870C211yesmaximal, characteristic
12C2×#544C2212yes—
13#3164—22no—
14#3164—22no—
15#3164—22no—
16#3164—22no—
17C2×#501—22no—
18C2×#500—21no—
19C2×#499—21no—
20C2×#430C2211yescharacteristic
21C22×#108C2211yescharacteristic
22#2911C2211yescharacteristic
23#2911C2212yes—
24C2×#428C2212yes—
25C2×#427C2211yescharacteristic
26C2×#427C2211yescharacteristic
27C22×#107C2211yescharacteristic
28C2×#405C2211yescharacteristic
29C2×#405C2211yescharacteristic
30C2×#405C2211yescharacteristic
31#2317C2212yes—
32#2061—24no—
33#2060—24no—
34#1952—21no—
35#1952—21no—
36#1951—22no—
37#1944—22no—
38#1942—22no—
39#1773—22no—
40#1772—22no—
41C2×#344C2212yes—
42#1742C2212yes—
43C2×#343C2212yes—
44#1740C2212yes—
45#1736C2212yes—
46#1736C2212yes—
47#1733C2212yes—
48#1733C2212yes—
49#1733C2211yescharacteristic
50#1733C2211yescharacteristic
51C22×M16C2311yescharacteristic
52C22×M16C2312yes—
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C23×C8C2311yescharacteristic
56#544—24no—
57#544—24no—
58#544—24no—
59#544—42no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×#125—22no—
63C2×#125—22no—
64C2×C22⋊Q8—42no—
65C2×#117C2312yes—
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2312yes—
69C2×#117C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71#501—28no—
72#501—28no—
73#500—24no—
74#500—24no—
75#499—24no—
76#499—24no—
77#430—22no—
78#430Q814yescomplemented
79C2×#108—22no—
80C2×#108—21no—
81C2×#108—21no—
82C2×#108—22no—
83C2×#108Q814yescomplemented
84#428—24no—
85#428—24no—
86#428—21no—
87#428—21no—
88#428—22no—
89#428—22no—
90#428—22no—
91#428—22no—
92#428D412yes—
93#428D412yes—
94#427—21no—
95#427—21no—
96#427—21no—
97#427—21no—
98#427—22no—
99#427—22no—
100#427—42no—
101#427Q814yescomplemented
102C2×#107—21no—
103C2×#107—21no—
104C2×#107—21no—
105C2×#107—21no—
106C2×#107—21no—
107C2×#107—22no—
108C2×#107D411yescharacteristic
109C2×#107D411yescharacteristic
110C2×#107D412yes—
111C2×#107D412yes—
112C2×#106—21no—
113C2×#106—22no—
114C2×#106D412yes—
115#405—21no—
116#405—21no—
117#405—22no—
118#405—21no—
119#405—21no—
120#405—21no—
121#405—21no—
122#405D412yes—
123#405D412yes—
124#405Q814yescomplemented
125C2×#97—21no—
126C2×#97—21no—
127C2×#97—21no—
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131#389—24no—
132#385—42no—
133#384—22no—
134#382—22no—
135#344—24no—
136#343—24no—
137#343—22no—
138#343—22no—
139#342—42no—
140#340—24no—
141#340—42no—
142#339—24no—
143#339—22no—
144#339—24no—
145#339—22no—
146C22×Q8—42no—
147C23×C4—22no—
148C23×C4C2411yescharacteristic
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—21no—
154C2×M16—21no—
155C2×M16—22no—
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—21no—
163C2×M16—22no—
164C2×M16—21no—
165C2×M16C2×D412yes—
166C2×M16C2×D412yes—
167C2×M16C2×Q814yes—
168C2×M16C2×Q812yes—
169C2×M16C2×Q812yes—
170C2×M16C2×Q812yes—
171C2×M16C2×Q812yes—
172C2×M16#4112yes—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—22no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×Q812yes—
184C22×C8C2×Q812yes—
185C22×C8C2×Q812yes—
186C22×C8#4112yes—
187C4⋊Q8—28no—
188C4⋊Q8—44no—
189C4⋊Q8—44no—
190C4⋊Q8—44no—
191#125—28no—
192#125—44no—
193#125—44no—
194#125—44no—
195C22⋊Q8—48no—
196C22⋊Q8—48no—
197#117—21no—
198#117—21no—
199#117—21no—
200#117—22no—
201#117—22no—
202#117—21no—
203#117—22no—
204#117—22no—
205#117—22no—
206#117—21no—
207#117—21no—
208#117—22no—
209#117—21no—
210#117—22no—
211#117—21no—
212#117—21no—
213#117C2×D412yes—
214#117C2×D412yes—
215#117C2×Q812yes—
216#117C2×Q814yes—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4C2×D414yes—
229C2×C4⋊C4C2×D412yes—
230C2×C4⋊C4C2×D412yes—
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4#4111yescharacteristic
234C2×C4⋊C4#4111yescharacteristic
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C42—21no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245#108—28no—
246#108—24no—
247#108—24no—
248#107—28no—
249#107—22no—
250#107—22no—
251#107—22no—
252#107—22no—
253#107—21no—
254#107—21no—
255#107—21no—
256#107—21no—
257#107—24no—
258#107—22no—
259#107—22no—
260#107—42no—
261#107—42no—
262#106—28no—
263#106—22no—
264#106—22no—
265#106—24no—
266#106—22no—
267#106—22no—
268#106—42no—
269#106—42no—
270#97—22no—
271#97—22no—
272#97—24no—
273#97—41no—
274#97—41no—
275C4×C8—22no—
276C4×C8—22no—
277C4×C8—24no—
278C4×C8—41no—
279C4×C8—41no—
280#95—44no—
281C24H(5,𝔽2)11yescharacteristic
282C2×Q8—48no—
283C2×Q8—164no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—162no—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C22×Q811yescharacteristic
302C22×C4C22×Q811yescharacteristic
303C22×C4C22×Q811yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4H(5,𝔽2)11yescharacteristic
308C22×C42−1+411yescharacteristic
309C22×C42−1+411yescharacteristic
310M16—22no—
311M16—22no—
312M16—22no—
313M16—22no—
314M16—22no—
315M16—22no—
316M16—22no—
317M16—24no—
318M16—22no—
319M16—24no—
320M16—42no—
321M16—41no—
322M16—41no—
323M16—42no—
324M16C22⋊Q814yes—
325M16C4⋊Q814yes—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—24no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—41no—
338C2×C8#12011yescharacteristic
339C2×C8#12011yescharacteristic
340C2×C8#12011yescharacteristic
341C2×C8#12011yescharacteristic
342C2×C8C22⋊Q812yes—
343C2×C8C22⋊Q812yes—
344C2×C8C22⋊Q812yes—
345C2×C8C22⋊Q812yes—
346C2×C8C22⋊Q812yes—
347C2×C8C22⋊Q812yes—
348C2×C8C4⋊Q812yes—
349C2×C8C4⋊Q812yes—
350C2×C8C4⋊Q812yes—
351C2×C8C4⋊Q812yes—
352C2×C8C4⋊Q812yes—
353C2×C8C4⋊Q812yes—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—48no—
369C4⋊C4—48no—
370C4⋊C4—48no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—164no—
376C4⋊C4#12112yes—
377C4⋊C4#12112yes—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C22⋊C4—24no—
382C22⋊C4—41no—
383C22⋊C4—42no—
384C22⋊C4—41no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—41no—
388C42—22no—
389C42—22no—
390C42—22no—
391C42—24no—
392C42—41no—
393C42—42no—
394C42—42no—
395C42—41no—
396C42—41no—
397C42—42no—
398C23—41no—
399C23—41no—
400C23#53511yescharacteristic
401C23#54211yescharacteristic
402C23#54411yescharacteristic
403C23D4211yescharacteristic
404C23#54611yescharacteristic
405C23#55411yescharacteristic
406C23Q8211yescharacteristic
407Q8—168no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—84no—
426C2×C4—84no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—162no—
430C2×C4C2×#12011yescharacteristic
431C2×C4C2×#12111yescharacteristic
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×C22⋊Q811yescharacteristic
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×C4⋊Q811yescharacteristic
437C2×C4C2×C4⋊Q811yescharacteristic
438C2×C4C2×C4⋊Q811yescharacteristic
439C2×C4C2×C4⋊Q811yescharacteristic
440C2×C4#53211yescharacteristic
441C2×C4#53311yescharacteristic
442C2×C4#53311yescharacteristic
443C2×C4#53611yescharacteristic
444C2×C4#54211yescharacteristic
445C2×C4#54411yescharacteristic
446C2×C4D4211yescharacteristic
447C2×C4#54611yescharacteristic
448C2×C4#54811yescharacteristic
449C2×C4#54811yescharacteristic
450C2×C4D4×Q811yescharacteristic
451C2×C4D4×Q811yescharacteristic
452C2×C4D4×Q811yescharacteristic
453C2×C4D4×Q811yescharacteristic
454C2×C4#55411yescharacteristic
455C2×C4#55711yescharacteristic
456C2×C4#55711yescharacteristic
457C2×C4Q8211yescharacteristic
458C8—24no—
459C8—24no—
460C8—42no—
461C8—44no—
462C8#39512yes—
463C8#39512yes—
464C8#39512yes—
465C8#39512yes—
466C22—21no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22#240811yescharacteristic
477C22#245111yescharacteristic
478C22#245211yescharacteristic
479C22#255011yescharacteristic
480C22#255111yescharacteristic
481C22#256711yescharacteristic
482C22#262811yescharacteristic
483C4—42no—
484C4—42no—
485C4—84no—
486C4—82no—
487C4—81no—
488C4—82no—
489C4—81no—
490C4—84no—
491C4—82no—
492C4C2×#39511yescharacteristic
493C4#240911yescharacteristic
494C4#245111yescharacteristic
495C4#245211yescharacteristic
496C4#254611yescharacteristic
497C4#255011yescharacteristic
498C4#255111yescharacteristic
499C4#285711yescharacteristic
500C2—21no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—21no—
506C2#1512811yescharacteristic
507C2#2429611yescharacteristic
508C2#2429711yescharacteristic
subgroupG/H
center, socleC22#2408
commutator, Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
78. #430Q8semidirect
83. C2×#108Q8semidirect
101. #427Q8semidirect
124. #405Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31642
#235612
#211812
#211562
#204442
#113882
#113332
#112532
#108722
#108712
#108702