Finite Group Databasebeta

← #471727

Subgroups — #471727

Order 512.

891 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23630C211yesmaximal, characteristic
2#23452C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#21046C211yesmaximal, characteristic
5#11382C211yesmaximal, characteristic, complemented
6#11354C211yesmaximal, characteristic, complemented
7#11303C212yesmaximal, complemented
8#11242C212yesmaximal, complemented
9#11192C212yesmaximal
10#11105C212yesmaximal
11#10864C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#108C2212yes—
14C2×#424C2211yescharacteristic
15C2×#417C2212yes—
16C2×#417C2212yes—
17C2×#408C2211yescharacteristic
18C2×#408C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20#2297C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2072—21no—
23#2072—21no—
24#2071—21no—
25#2071—21no—
26#2024—21no—
27#2024—21no—
28#2023—21no—
29#2023—21no—
30#1990—22no—
31#1990—22no—
32#1923—22no—
33#1923—22no—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#341C2212yes—
39#1736C2212yes—
40#1736C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1729C2211yescharacteristic
44#1728C2211yescharacteristic
45#1728C2211yescharacteristic
46#1728C2211yescharacteristic
47C22×#41C2311yescharacteristic
48C23×Q8—21no—
49C22×M16C2311yescharacteristic
50C22×M16C2311yescharacteristic
51C22×M16C2312yes—
52C22×M16C2312yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—42no—
61C2×#117C2311yescharacteristic
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C42C2311yescharacteristic
68C2×#108—24no—
69C2×#108—42no—
70C2×#108—42no—
71#424—24no—
72C2×C4⋊C8—22no—
73#417—24no—
74#417—42no—
75#417—42no—
76#417—42no—
77C2×#103—22no—
78C2×#103—22no—
79C2×#102—22no—
80C2×#102—22no—
81#408—24no—
82#408—42no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C4×C8—42no—
86C2×C4×C8D412yes—
87C2×C4×C8D412yes—
88#391D411yescharacteristic
89#391D411yescharacteristic
90#390D411yescharacteristic
91#390D411yescharacteristic
92#386—21no—
93#386—21no—
94#384—21no—
95#384—21no—
96#384—21no—
97#384—21no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100#343—24no—
101#343—42no—
102#343—42no—
103#343—42no—
104#341—42no—
105#340—24no—
106#340—24no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—22no—
111#336—22no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—22no—
121C2×#41—21no—
122C2×#41—21no—
123C2×#41—22no—
124C2×#41—21no—
125C2×#41—21no—
126C22×Q8—44no—
127C22×Q8—41no—
128C22×Q8—41no—
129C22×Q8—82no—
130C22×Q8—82no—
131C22×Q8C2×D411yescharacteristic
132C22×D4—21no—
133C22×D4C2×D411yescharacteristic
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—22no—
137C23×C4#4111yescharacteristic
138C23×C4#4111yescharacteristic
139C23×C4C2411yescharacteristic
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—22no—
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—22no—
159C2×M16—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—21no—
171C22×C8C2×D412yes—
172C22×C8C2×D412yes—
173C22×C8#4114yes—
174C4⋊Q8—24no—
175C4⋊Q8—24no—
176C4⋊Q8—44no—
177C4⋊Q8—44no—
178C4⋊Q8—42no—
179C4⋊Q8—42no—
180#117—22no—
181#117—22no—
182C2×C4⋊C4—22no—
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—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4#4114yes—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—21no—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217#108—48no—
218C4⋊C8—44no—
219#103—44no—
220#103—44no—
221#102—44no—
222#102—44no—
223C22⋊C8—44no—
224C22⋊C8—44no—
225C4×C8—42no—
226C4×C8—42no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231C24—21no—
232C24C2×#4111yescharacteristic
233#41—42no—
234#41—42no—
235#41—42no—
236#41—42no—
237#41—42no—
238#41—42no—
239#41—42no—
240#41—42no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—816no—
248C2×Q8—82no—
249C2×Q8—82no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×D4—21no—
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4112yes—
292C22×C4C2×#4112yes—
293C22×C4H(5,𝔽2)12yes—
294C22×C4H(5,𝔽2)11yescharacteristic
295C22×C4H(5,𝔽2)11yescharacteristic
296C22×C42−1+411yescharacteristic
297C22×C42−1+411yescharacteristic
298C22×C42−1+412yes—
299M16—42no—
300M16—42no—
301M16—42no—
302M16—42no—
303M16—42no—
304M16—42no—
305M16—42no—
306M16—42no—
307M16—42no—
308M16—42no—
309M16—42no—
310M16—42no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8#12414yes—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—42no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—22no—
359C42—22no—
360C42—41no—
361C42—41no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23C2×#12111yescharacteristic
374C23#54211yescharacteristic
375C23#54211yescharacteristic
376C23#55112yes—
377C23#55612yes—
378Q8—42no—
379Q8—42no—
380Q8—88no—
381Q8—88no—
382D4—42no—
383D4—42no—
384D4—82no—
385D4—82no—
386C2×C4—28no—
387C2×C4—28no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—44no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—84no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—84no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—162no—
418C2×C4—162no—
419C2×C4#47112yes—
420C2×C4#47112yes—
421C2×C4#47212yes—
422C2×C4#47312yes—
423C2×C4#52511yescharacteristic
424C2×C4C2×#12411yescharacteristic
425C2×C4#53411yescharacteristic
426C2×C4#53511yescharacteristic
427C2×C4#53611yescharacteristic
428C2×C4#53611yescharacteristic
429C2×C4#53711yescharacteristic
430C2×C4#53711yescharacteristic
431C2×C4#53812yes—
432C2×C4#53912yes—
433C2×C4#54112yes—
434C2×C4#54212yes—
435C2×C4#54612yes—
436C2×C4#54712yes—
437C2×C4#55612yes—
438C2×C4#55912yes—
439C2×C4#56212yes—
440C2×C4#56312yes—
441C8—24no—
442C8—42no—
443C8—81no—
444C8—81no—
445C8—82no—
446C8—82no—
447C22—22no—
448C22—22no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—22no—
454C22—22no—
455C22—81no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22#253411yescharacteristic
463C22#254711yescharacteristic
464C22#262011yescharacteristic
465C22#275912yes—
466C22#279812yes—
467C22C2×#47111yescharacteristic
468C22C2×#47111yescharacteristic
469C22C2×#47211yescharacteristic
470C22C2×#47311yescharacteristic
471C4—22no—
472C4—22no—
473C4—22no—
474C4—22no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—84no—
479C4—162no—
480C4#259211yescharacteristic
481C4#263711yescharacteristic
482C4#263911yescharacteristic
483C4#268912yes—
484C4#271012yes—
485C4#281411yescharacteristic
486C2—22no—
487C2—22no—
488C2—161no—
489C2#1680511yescharacteristic
490C2#2813411yescharacteristic
491C2#2813511yescharacteristic
492C2#2873111yescharacteristic
493C2#2873211yescharacteristic
494C2#2985811yescharacteristic
495C2#2985911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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
3. C2×#1990C2semidirect
5. #11382C2semidirect
6. #11354C2semidirect
7. #11303C2semidirect
8. #11242C2semidirect
11. #10864C2semidirect

Maximal subgroups

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

subgroupindex
#236302
#234522
C2×#19902
#210462
#113822
#113542
#113032
#112422
#111922
#111052
#108642