Finite Group Databasebeta

← #415207

Subgroups — #415207

Order 512.

812 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#21322C211yesmaximal, characteristic
2#21312C211yesmaximal, characteristic
3#21302C211yesmaximal, characteristic
4C2×#1948C211yesmaximal, characteristic
5#11366C212yesmaximal
6#11303C212yesmaximal
7#11303C212yesmaximal
8#11271C212yesmaximal
9#10857C211yesmaximal, characteristic
10#10853C211yesmaximal, characteristic
11C4×#343C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C22×#106C2212yes—
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18C2×#384C2211yescharacteristic
19C2×C4×C4⋊C4C2211yescharacteristic
20#1992—22no—
21#1990—21no—
22#1990—21no—
23#1989—22no—
24#1988—21no—
25#1987—21no—
26#1985—22no—
27#1984—22no—
28#1948—22no—
29#1948—22no—
30C2×#343C2211yescharacteristic
31C2×#343C2211yescharacteristic
32C2×#343C2212yes—
33C2×#343C2211yescharacteristic
34C2×#343C2212yes—
35C2×#343C2211yescharacteristic
36C2×#341C2212yes—
37C2×#341C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41#1736C2212yes—
42C2×#340C2212yes—
43#1729C2212yes—
44#1729C2212yes—
45C22×M16C2312yes—
46C22×M16C2311yescharacteristic
47C22×M16C2311yescharacteristic
48C22×M16C2311yescharacteristic
49C22×M16C2311yescharacteristic
50C23×C8C2312yes—
51#533—22no—
52#533—22no—
53#533—22no—
54#533—22no—
55C2×C4⋊Q8D411yescharacteristic
56C2×C4⋊Q8D411yescharacteristic
57C2×#125D412yes—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—21no—
62C2×#117C2312yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70C2×#106—24no—
71C2×#106—24no—
72C2×#106—42no—
73#423—22no—
74#423—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8D412yes—
80C2×C4⋊C8D412yes—
81C4×M16—22no—
82C4×M16—22no—
83C4×M16—22no—
84C4×M16—22no—
85C2×#97—21no—
86C2×#97—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C2×#95—21no—
98#343—22no—
99#343—22no—
100#343—24no—
101#343—22no—
102#343—42no—
103#343Q814yescomplemented
104#341—24no—
105#341—42no—
106#340—24no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#336—24no—
116#336—24no—
117C22×Q8—21no—
118C22×Q8—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4#4111yescharacteristic
123C23×C4#4111yescharacteristic
124C23×C4C2411yescharacteristic
125C2×M16—21no—
126C2×M16—21no—
127C2×M16—21no—
128C2×M16—21no—
129C2×M16—21no—
130C2×M16—21no—
131C2×M16—21no—
132C2×M16—21no—
133C2×M16—21no—
134C2×M16—21no—
135C2×M16—21no—
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—22no—
150C2×M16—22no—
151C2×M16C2×Q812yes—
152C2×M16C2×Q812yes—
153C2×M16#4112yes—
154C2×M16#4112yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8C2×D414yes—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171#125—24no—
172#125—24no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175C22⋊Q8—42no—
176C22⋊Q8—42no—
177C4×Q8—42no—
178C4×Q8—42no—
179C4×Q8—42no—
180C4×Q8—42no—
181#117—22no—
182#117—22no—
183#117—22no—
184#117—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×Q812yes—
214C2×C4⋊C4#4112yes—
215C2×C22⋊C4—22no—
216C2×C42—22no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42—82no—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4112yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42#4112yes—
228#106—48no—
229C4⋊C8—24no—
230C4⋊C8—24no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233#97—42no—
234#97—42no—
235C4×C8—42no—
236C4×C8—42no—
237#95—41no—
238#95—41no—
239C24C2×#4111yescharacteristic
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—81no—
245C2×Q8—81no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—81no—
262C22×C4—81no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4#12111yescharacteristic
268C22×C4#12111yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12612yes—
272C22×C4#12612yes—
273C22×C4#12612yes—
274C22×C4#12612yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×Q811yescharacteristic
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C22×C4C2×#4111yescharacteristic
284C22×C4C2×#4111yescharacteristic
285C22×C4H(5,𝔽2)11yescharacteristic
286C22×C42−1+411yescharacteristic
287C22×C42−1+411yescharacteristic
288C22×C42−1+411yescharacteristic
289M16—22no—
290M16—22no—
291M16—22no—
292M16—22no—
293M16—22no—
294M16—22no—
295M16—22no—
296M16—22no—
297M16—41no—
298M16—41no—
299M16—41no—
300M16—41no—
301M16—41no—
302M16—41no—
303M16—41no—
304M16—41no—
305M16—42no—
306M16—42no—
307M16—42no—
308M16—42no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C22⋊C4—42no—
356C22⋊C4—42no—
357C42—42no—
358C42—42no—
359C42—82no—
360C42—82no—
361C42#12312yes—
362C42#12312yes—
363C42#12312yes—
364C42#12312yes—
365C42#12312yes—
366C42#12312yes—
367C42#12412yes—
368C42#12412yes—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23C2×#12111yescharacteristic
374C23C2×#12611yescharacteristic
375C23C2×#12611yescharacteristic
376C23#52911yescharacteristic
377C23#53311yescharacteristic
378C23#54111yescharacteristic
379C23#54211yescharacteristic
380Q8—84no—
381Q8—84no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—162no—
412C2×C4—162no—
413C2×C4#39712yes—
414C2×C4#39712yes—
415C2×C4#39712yes—
416C2×C4#39712yes—
417C2×C4#47112yes—
418C2×C4#47112yes—
419C2×C4C2×#12311yescharacteristic
420C2×C4C2×#12311yescharacteristic
421C2×C4C2×#12311yescharacteristic
422C2×C4C2×#12411yescharacteristic
423C2×C4#52911yescharacteristic
424C2×C4#52911yescharacteristic
425C2×C4#53311yescharacteristic
426C2×C4#53311yescharacteristic
427C2×C4#54111yescharacteristic
428C2×C4#54111yescharacteristic
429C2×C4#54211yescharacteristic
430C2×C4#54211yescharacteristic
431C2×C4#54611yescharacteristic
432C2×C4#54711yescharacteristic
433C2×C4#54811yescharacteristic
434C2×C4#54811yescharacteristic
435C2×C4#54811yescharacteristic
436C2×C4D4×Q811yescharacteristic
437C2×C4D4×Q811yescharacteristic
438C2×C4#55011yescharacteristic
439C2×C4#55211yescharacteristic
440C2×C4#55211yescharacteristic
441C2×C4#55411yescharacteristic
442C2×C4#55611yescharacteristic
443C2×C4#55611yescharacteristic
444C2×C4#55611yescharacteristic
445C2×C4#55711yescharacteristic
446C2×C4#55911yescharacteristic
447C8—42no—
448C8—42no—
449C8—44no—
450C8—42no—
451C8—42no—
452C8—82no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22C2×#39711yescharacteristic
466C22#242611yescharacteristic
467C22#242911yescharacteristic
468C22#251111yescharacteristic
469C22#251411yescharacteristic
470C22#264511yescharacteristic
471C22#286311yescharacteristic
472C22C2×#47111yescharacteristic
473C22C2×#47111yescharacteristic
474C22#306811yescharacteristic
475C22#306911yescharacteristic
476C4—22no—
477C4—22no—
478C4—22no—
479C4—82no—
480C4—82no—
481C4—82no—
482C4—82no—
483C4—162no—
484C4#207914yes—
485C4#242111yescharacteristic
486C4#242311yescharacteristic
487C4#244611yescharacteristic
488C4#244611yescharacteristic
489C4#244711yescharacteristic
490C4#244811yescharacteristic
491C4#248911yescharacteristic
492C4#249211yescharacteristic
493C2—21no—
494C2—21no—
495C2—21no—
496C2—21no—
497C2#1518611yescharacteristic
498C2C2×#207911yescharacteristic
499C2#2364711yescharacteristic
500C2#2531311yescharacteristic
501C2#2531511yescharacteristic
502C2#2549011yescharacteristic
503C2#2549211yescharacteristic
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

Complements

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

NH ≅ G/Ntype
103. #343Q8semidirect

Maximal subgroups

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

subgroupindex
#213222
#213122
#213022
C2×#19482
#113662
#113032
#112712
#108572
#108532
C4×#3432