Finite Group Databasebeta

← #416312

Subgroups — #416312

Order 512.

802 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21171C211yesmaximal, characteristic
2C2×#1948C211yesmaximal, characteristic
3#20755C211yesmaximal, characteristic
4C4×#428C211yesmaximal, characteristic
5#11366C212yesmaximal
6#11330C212yesmaximal
7#11290C212yesmaximal
8#11290C212yesmaximal
9#10857C211yesmaximal, characteristic
10#10853C211yesmaximal, characteristic
11#10847C211yesmaximal, characteristic
12C2×#428C2211yescharacteristic
13C2×#428C2211yescharacteristic
14C2×#428C2212yes—
15C22×#106C2212yes—
16C2×#423C2211yescharacteristic
17C2×#423C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C4×#117C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1948—22no—
24#1948—22no—
25#1948—21no—
26#1948—21no—
27#1947—21no—
28#1947—21no—
29#1832—22no—
30#1831—22no—
31#1776—21no—
32#1776—21no—
33C4×#107—21no—
34C4×#106—21no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2212yes—
40C2×#341C2212yes—
41#1736C2211yescharacteristic
42#1736C2212yes—
43#1736C2211yescharacteristic
44C2×#340C2212yes—
45#1729C2212yes—
46#1729C2212yes—
47C22×M16C2312yes—
48C22×M16C2311yescharacteristic
49C22×M16C2312yes—
50C22×M16C2311yescharacteristic
51C23×C8C2312yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#428—22no—
62#428—22no—
63#428—22no—
64#428—22no—
65#428—42no—
66#428—42no—
67C2×#107—21no—
68C2×#107—21no—
69C2×#107—22no—
70C2×#106—21no—
71C2×#106—21no—
72C2×#106—24no—
73C2×#106—22no—
74C2×#106—24no—
75C2×#106—42no—
76#423—22no—
77#423—22no—
78#423—22no—
79#423Q814yes—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8D412yes—
87C2×C4⋊C8D412yes—
88C4×M16—22no—
89C4×M16Q814yes—
90C2×#97—21no—
91C2×C4×C8—21no—
92#384—21no—
93#384—21no—
94#384—21no—
95#384—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4D411yescharacteristic
101C4×C4⋊C4D411yescharacteristic
102C4×C4⋊C4D411yescharacteristic
103C4×C4⋊C4D411yescharacteristic
104C4×C22⋊C4—21no—
105C4×C22⋊C4—21no—
106#376—21no—
107C2×#95—21no—
108C43—21no—
109#343—24no—
110#343—24no—
111#343—42no—
112#343—42no—
113#341—24no—
114#341—42no—
115#340—24no—
116#340—22no—
117#340—22no—
118#340—21no—
119#340—21no—
120#340—21no—
121#340—21no—
122#336—24no—
123#336—24no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4#4111yescharacteristic
128C23×C4#4111yescharacteristic
129C23×C4C2411yescharacteristic
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—21no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—21no—
138C2×M16—22no—
139C2×M16—21no—
140C2×M16—22no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16C2×Q812yes—
150C2×M16C2×Q814yes—
151C2×M16C2×Q812yes—
152C2×M16#4112yes—
153C2×M16#4112yes—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8C2×D414yes—
165#117—21no—
166#117—21no—
167#117—21no—
168#117—21no—
169#117—21no—
170#117—21no—
171#117—21no—
172#117—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4#4112yes—
200C2×C4⋊C4#4112yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—41no—
210C2×C42—41no—
211C2×C42—82no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×Q812yes—
217C2×C42C2×Q812yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220#107—24no—
221#107—24no—
222#107—44no—
223#106—24no—
224#106—24no—
225#106—48no—
226#106—44no—
227C4⋊C8—24no—
228C4⋊C8—24no—
229C4⋊C8—24no—
230C4⋊C8—24no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233#97—24no—
234C4×C8—24no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—41no—
240#95—41no—
241C24C2×#4111yescharacteristic
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—82no—
264C22×C4—82no—
265C22×C4#12111yescharacteristic
266C22×C4#12111yescharacteristic
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12512yes—
270C22×C4#12512yes—
271C22×C4#12512yes—
272C22×C4#12512yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×Q811yescharacteristic
276C22×C4C22×Q811yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C42−1+411yescharacteristic
286C22×C42−1+411yescharacteristic
287M16—22no—
288M16—22no—
289M16—22no—
290M16—22no—
291M16—22no—
292M16—24no—
293M16—24no—
294M16—41no—
295M16—41no—
296M16—42no—
297M16—42no—
298M16—42no—
299M16—42no—
300M16—42no—
301M16—42no—
302M16#12514yes—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—41no—
349C22⋊C4—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—42no—
355C42—42no—
356C42—81no—
357C42—81no—
358C42—82no—
359C42—82no—
360C42C22⋊Q812yes—
361C42C22⋊Q812yes—
362C42C22⋊Q812yes—
363C42C22⋊Q812yes—
364C42#12312yes—
365C42#12312yes—
366C42C4⋊Q812yes—
367C42C4⋊Q812yes—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23C2×#12111yescharacteristic
373C23C2×#12511yescharacteristic
374C23C2×#12511yescharacteristic
375C23#55211yescharacteristic
376C23#55311yescharacteristic
377C23#55511yescharacteristic
378C23#55611yescharacteristic
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—162no—
415C2×C4—162no—
416C2×C4#39812yes—
417C2×C4#39812yes—
418C2×C4#39812yes—
419C2×C4#39812yes—
420C2×C4#47112yes—
421C2×C4#47112yes—
422C2×C4C2×C22⋊Q811yescharacteristic
423C2×C4C2×C22⋊Q811yescharacteristic
424C2×C4C2×#12311yescharacteristic
425C2×C4#52511yescharacteristic
426C2×C4#52511yescharacteristic
427C2×C4C2×#12511yescharacteristic
428C2×C4#52911yescharacteristic
429C2×C4#52911yescharacteristic
430C2×C4C2×C4⋊Q811yescharacteristic
431C2×C4#53311yescharacteristic
432C2×C4#53511yescharacteristic
433C2×C4#53611yescharacteristic
434C2×C4#54211yescharacteristic
435C2×C4#54211yescharacteristic
436C2×C4#54411yescharacteristic
437C2×C4#54411yescharacteristic
438C2×C4#54611yescharacteristic
439C2×C4#54711yescharacteristic
440C2×C4#54811yescharacteristic
441C2×C4#55011yescharacteristic
442C2×C4#55411yescharacteristic
443C2×C4#55411yescharacteristic
444C2×C4#55611yescharacteristic
445C2×C4#55611yescharacteristic
446C2×C4#55711yescharacteristic
447C2×C4#55711yescharacteristic
448C2×C4#55911yescharacteristic
449C2×C4#55911yescharacteristic
450C8—24no—
451C8—44no—
452C8—42no—
453C8—44no—
454C8—82no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—21no—
467C22C2×#39811yescharacteristic
468C22#241411yescharacteristic
469C22#241511yescharacteristic
470C22#250811yescharacteristic
471C22#250911yescharacteristic
472C22#257011yescharacteristic
473C22#257111yescharacteristic
474C22C2×#47111yescharacteristic
475C22C2×#47111yescharacteristic
476C22#306811yescharacteristic
477C22#306911yescharacteristic
478C4—22no—
479C4—22no—
480C4—22no—
481C4—81no—
482C4—82no—
483C4—81no—
484C4—81no—
485C4—82no—
486C4—81no—
487C4—162no—
488C4#209014yes—
489C4#244711yescharacteristic
490C4#244811yescharacteristic
491C4#255011yescharacteristic
492C4#255211yescharacteristic
493C4#258011yescharacteristic
494C4#258111yescharacteristic
495C4#262811yescharacteristic
496C4#285111yescharacteristic
497C2—21no—
498C2—21no—
499C2—21no—
500C2—21no—
501C2#1522111yescharacteristic
502C2C2×#209011yescharacteristic
503C2#2367511yescharacteristic
504C2#2467111yescharacteristic
505C2#2467311yescharacteristic
506C2#2472111yescharacteristic
507C2#2472311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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
#211712
C2×#19482
#207552
C4×#4282
#113662
#113302
#112902
#108572
#108532
#108472