Finite Group Databasebeta

← #213112

Subgroups — #213112

Order 512.

890 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22246C211yesmaximal, characteristic, complemented
2#22219C211yesmaximal, characteristic
3#21726C211yesmaximal, characteristic, complemented
4#21692C211yesmaximal, characteristic
5C4×#492C211yesmaximal, characteristic
6C4×#487C211yesmaximal, characteristic, complemented
7C2×#1714C211yesmaximal, characteristic
8#9829C211yesmaximal, characteristic
9#9826C211yesmaximal, characteristic, complemented
10#9823C211yesmaximal, characteristic, complemented
11#9822C211yesmaximal, characteristic
12#9820C211yesmaximal, characteristic
13#9819C211yesmaximal, characteristic, complemented
14#8819C211yesmaximal, characteristic, complemented
15C8×#97C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#498C2211yescharacteristic
18C2×#492C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#487C2211yescharacteristic
21C2×#486C2211yescharacteristic
22C2×#440—22no—
23C2×C4×SD16—22no—
24C4×C4⋊Q8C2211yescharacteristic
25C4×C4⋊Q8C2211yescharacteristic
26C4×C4⋊Q8C2211yescharacteristic
27C4×#127C2211yescharacteristic
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1973—22no—
35#1968—22no—
36#1957C2211yescharacteristic
37#1957C2211yescharacteristic
38#1957C2211yescharacteristic
39#1956C2211yescharacteristic
40#1956C2211yescharacteristic
41#1956C2211yescharacteristic
42#1943—22no—
43#1933—22no—
44#1926—22no—
45#1920—22no—
46#1919—22no—
47#1864—22no—
48#1845C2211yescharacteristic
49#1844C2211yescharacteristic
50#1844C2211yescharacteristic
51#1844C2211yescharacteristic
52#1834C2211yescharacteristic
53#1834C2211yescharacteristic
54#1776—22no—
55C4×#103—22no—
56#1744C2211yescharacteristic
57C4×#97C2211yescharacteristic
58C42×C8C2211yescharacteristic
59#1714C418yescomplemented
60C2×#322C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63C2×C82C2211yescharacteristic
64C22×SD16—22no—
65C22×SD16—22no—
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×#127C2311yescharacteristic
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76#500—22no—
77#500C2×C414yescomplemented
78#498—22no—
79#498C2×C414yescomplemented
80#492—22no—
81#492—22no—
82#492C2×C414yes—
83#492C2×C414yes—
84#487—22no—
85#487C2×C414yescomplemented
86#486—22no—
87#486C2×C414yes—
88#440—48no—
89C4×SD16—48no—
90C2×#107—22no—
91C2×#107—22no—
92C2×#106—22no—
93C2×#106—22no—
94C2×#106—22no—
95C2×#106—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#103—22no—
99C2×#103—22no—
100C2×#102—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×#102—22no—
104C2×#97D412yes—
105C2×#97D412yes—
106C2×C4×C8D412yes—
107C2×C4×C8D412yes—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#391—21no—
116#391—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#389—21no—
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C43C2311yescharacteristic
137#340—22no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#340—22no—
143#340—22no—
144#340—22no—
145#336D412yes—
146#336D412yes—
147#322—22no—
148#322C2×C414yescomplemented
149C82—22no—
150C82—22no—
151C22×Q8—21no—
152C22×Q8—21no—
153C22×Q8—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—21no—
157C23×C4—41no—
158C2×SD16—28no—
159C2×SD16—28no—
160C2×SD16—44no—
161C2×SD16—44no—
162C2×SD16—44no—
163C2×SD16—44no—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8C22×C412yes—
180C4⋊Q8C22×C412yes—
181C4⋊Q8C22×C412yes—
182#127—21no—
183#127—21no—
184#127—21no—
185#127C22×C412yes—
186C4×Q8—44no—
187C4×Q8—44no—
188C4×Q8—44no—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—82no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2411yescharacteristic
226#107—28no—
227#107—44no—
228#106—28no—
229#106—44no—
230#106—44no—
231#106—82no—
232#103—28no—
233#103—44no—
234#103—44no—
235#103—44no—
236#102—28no—
237#102—44no—
238#102—44no—
239#102—82no—
240#97—24no—
241#97SD1618yescomplemented
242C4×C8—24no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—42no—
250C4×C8C22×C412yes—
251C4×C8C22×C412yes—
252C4×C8C22×C412yes—
253C4×C8#4112yes—
254C4×C8#4112yes—
255C4×C8#4112yes—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262C24—41no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4#12711yescharacteristic
298C22×C4#12711yescharacteristic
299C22×C4#12711yescharacteristic
300C22×C4#12711yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304SD16—416no—
305SD16—416no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8C4×D414yes—
318C2×C8C4×D414yes—
319C2×C8C4×D414yes—
320C2×C8C2×SD1614yes—
321C2×C8C2×SD1614yes—
322C2×C8#13514yes—
323C2×C8AGL(1,ℤ/8)14yes—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—82no—
360C42—82no—
361C42—82no—
362C42C2×SD1614yes—
363C42C23×C411yescharacteristic
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C42C2×#4111yescharacteristic
367C23—42no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23C2×#12711yescharacteristic
375Q8—44no—
376Q8—44no—
377Q8—44no—
378Q8—82no—
379Q8—82no—
380Q8—82no—
381D4—44no—
382D4—44no—
383D4—44no—
384D4—82no—
385D4—82no—
386D4—82no—
387C2×C4—21no—
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—21no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—164no—
419C2×C4#49212yes—
420C2×C4#49212yes—
421C2×C4C2×C4×D411yescharacteristic
422C2×C4C2×C4×D411yescharacteristic
423C2×C4C2×C4×D411yescharacteristic
424C2×C4#53211yescharacteristic
425C2×C4#53211yescharacteristic
426C2×C4#53211yescharacteristic
427C2×C4C22×SD1612yes—
428C2×C4C2×#13512yes—
429C2×C4C2×AGL(1,ℤ/8)12yes—
430C2×C4#57512yes—
431C2×C4#57712yes—
432C2×C4#57812yes—
433C8—24no—
434C8—24no—
435C8—24no—
436C8—24no—
437C8—42no—
438C8C4×SD1618yescomplemented
439C22—82no—
440C22—82no—
441C22—82no—
442C22—161no—
443C22—161no—
444C22—161no—
445C22—161no—
446C22C4×#12711yescharacteristic
447C22C2×#49211yescharacteristic
448C22#314711yescharacteristic
449C22#315111yescharacteristic
450C22#315311yescharacteristic
451C22#315411yescharacteristic
452C22#315611yescharacteristic
453C4—22no—
454C4—22no—
455C4—41no—
456C4—41no—
457C4—82no—
458C4—82no—
459C4—82no—
460C4—161no—
461C4—161no—
462C4—161no—
463C4—161no—
464C4C2×C4×SD1612yes—
465C4C4×AGL(1,ℤ/8)12yes—
466C4#294712yes—
467C4#316712yes—
468C4#317812yes—
469C4#318012yes—
470C2—162no—
471C2#1962111yescharacteristic
472C2#1962411yescharacteristic
473C2#1964511yescharacteristic
474C2#1964911yescharacteristic
475C2C4×#49211yescharacteristic
476C2#2170111yescharacteristic
477C2#2170311yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22246C2semidirect
3. #21726C2semidirect
6. C4×#487C2semidirect
9. #9826C2semidirect
10. #9823C2semidirect
13. #9819C2semidirect
14. #8819C2semidirect
15. C8×#97C2semidirect
59. #1714C4semidirect
77. #500C2×C4semidirect
79. #498C2×C4semidirect
85. #487C2×C4semidirect
148. #322C2×C4semidirect
241. #97SD16semidirect
438. C8C4×SD16semidirect

Maximal subgroups

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

subgroupindex
#222462
#222192
#217262
#216922
C4×#4922
C4×#4872
C2×#17142
#98292
#98262
#98232
#98222
#98202
#98192
#88192
C8×#972