Finite Group Databasebeta

← #320829

Subgroups — #320829

Order 512.

758 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30228C211yesmaximal, characteristic, complemented
2#29207C211yesmaximal, characteristic, complemented
3#29190C211yesmaximal, characteristic
4#28687C211yesmaximal, characteristic, complemented
5#28438C211yesmaximal, characteristic
6#22268C211yesmaximal, characteristic
7#10686C211yesmaximal, characteristic, complemented
8#10684C211yesmaximal, characteristic, complemented
9#10315C211yesmaximal, characteristic
10#10315C211yesmaximal, characteristic
11#9842C211yesmaximal, characteristic
12#9828C211yesmaximal, characteristic, complemented
13#9820C211yesmaximal, characteristic
14#9819C211yesmaximal, characteristic, complemented
15#8264C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#498C2211yescharacteristic
18C2×#492C2211yescharacteristic
19C2×#492C2211yescharacteristic
20C2×#492C2211yescharacteristic
21C2×#461—22no—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#127C2211yescharacteristic
26#2099—22no—
27#2099—22no—
28#2084—22no—
29#2065—22no—
30#2059—22no—
31#2013—22no—
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1990C2211yescharacteristic
40#1957C2211yescharacteristic
41#1957C2211yescharacteristic
42#1957C2211yescharacteristic
43#1956C2211yescharacteristic
44#1956C2211yescharacteristic
45#1956C2211yescharacteristic
46#1949—22no—
47#1948—22no—
48#1880—22no—
49#1877—22no—
50#1864—22no—
51#1845C2211yescharacteristic
52#1844C2211yescharacteristic
53#1844C2211yescharacteristic
54#1834C2211yescharacteristic
55#1834C2211yescharacteristic
56C4×C4⋊C8C2211yescharacteristic
57C4×C4⋊C8C2211yescharacteristic
58#1744C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1273C2211yescharacteristic
62#1273C2211yescharacteristic
63C22×SD16—22no—
64C22×SD16—22no—
65C22×SD16—22no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×#127C2311yescharacteristic
72C2×#125—21no—
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—21no—
77#498—22no—
78#498—22no—
79#498—41no—
80#498—41no—
81#492—22no—
82#492—22no—
83#492—22no—
84#492—22no—
85#492—41no—
86#492—41no—
87#461—48no—
88C2×#107—22no—
89C2×#106—22no—
90C2×#106—22no—
91C2×#106—22no—
92C2×#106—22no—
93C2×C4⋊C8D412yes—
94C2×C4⋊C8D412yes—
95C2×#103—22no—
96C2×#103—22no—
97C2×#103—22no—
98C2×#102—22no—
99C2×#102—22no—
100C2×#102—22no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#390—21no—
109#390—21no—
110#390—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384—21no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#382—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C43C2311yescharacteristic
129#340—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#340—22no—
139#336D412yes—
140#336D412yes—
141#336D412yes—
142#336D412yes—
143#322—41no—
144C82—41no—
145C22×Q8—21no—
146C22×Q8—21no—
147C22×Q8—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C23×C4—41no—
152C2×SD16—28no—
153C2×SD16—44no—
154C2×SD16—44no—
155C2×SD16—44no—
156C2×SD16—44no—
157C2×SD16—82no—
158C2×SD16—82no—
159C2×SD16—82no—
160C2×SD16—82no—
161C22×C8C2×D412yes—
162C22×C8C2×D412yes—
163C22×C8C2×D412yes—
164C22×C8C2×D412yes—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
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—44no—
179C4⋊Q8—82no—
180C4⋊Q8#4112yes—
181#127—21no—
182#127—21no—
183#127—21no—
184#127#4112yes—
185#125—82no—
186C4×Q8—82no—
187C4×D4—44no—
188C4×D4—82no—
189C4×D4—82no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
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—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—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—82no—
227#106—44no—
228#106—44no—
229#106—82no—
230#106—82no—
231C4⋊C8—24no—
232C4⋊C8SD1618yescomplemented
233#103—44no—
234#103—44no—
235#103—82no—
236#102—44no—
237#102—82no—
238#102—82no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255C24—41no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4#12711yescharacteristic
291C22×C4#12711yescharacteristic
292C22×C4#12711yescharacteristic
293C22×C4#12711yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297SD16—88no—
298SD16—88no—
299SD16—88no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
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×C8C2×SD1614yes—
316C2×C8C2×SD1614yes—
317C2×C8#13714yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—82no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C22⋊C4—82no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—82no—
358C42C2×SD1614yes—
359C42C2×#4111yescharacteristic
360C42C2×#4111yescharacteristic
361C42H(5,𝔽2)11yescharacteristic
362C422−1+411yescharacteristic
363C23—42no—
364C23—81no—
365C23—81no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23C2×#12711yescharacteristic
371Q8—44no—
372Q8—82no—
373Q8—82no—
374Q8—82no—
375Q8—82no—
376Q8—82no—
377D4—44no—
378D4—44no—
379D4—44no—
380D4—82no—
381D4—82no—
382D4—82no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
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—164no—
415C2×C4#49212yes—
416C2×C4#49212yes—
417C2×C4#52511yescharacteristic
418C2×C4#53711yescharacteristic
419C2×C4#54611yescharacteristic
420C2×C4#54611yescharacteristic
421C2×C4#54811yescharacteristic
422C2×C4#54811yescharacteristic
423C2×C4C22×SD1612yes—
424C2×C4C2×#13712yes—
425C2×C4#57612yes—
426C2×C4#57712yes—
427C2×C4#57712yes—
428C2×C4#57712yes—
429C8—24no—
430C8—24no—
431C8—24no—
432C8—42no—
433C8—42no—
434C8—42no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22#254011yescharacteristic
443C22C2×#49211yescharacteristic
444C22#315211yescharacteristic
445C22#315411yescharacteristic
446C22#315411yescharacteristic
447C22#315411yescharacteristic
448C22#315511yescharacteristic
449C4—22no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—82no—
454C4—161no—
455C4—161no—
456C4—161no—
457C4—161no—
458C4—161no—
459C4—161no—
460C4#312312yes—
461C4#326012yes—
462C4#329512yes—
463C4#332012yes—
464C4#336212yes—
465C4#337212yes—
466C2—162no—
467C2#1962211yescharacteristic
468C2#1965111yescharacteristic
469C2#1972611yescharacteristic
470C2#1974211yescharacteristic
471C2#2838811yescharacteristic
472C2#2840411yescharacteristic
473C2#2843311yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×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. #30228C2semidirect
2. #29207C2semidirect
4. #28687C2semidirect
7. #10686C2semidirect
8. #10684C2semidirect
12. #9828C2semidirect
14. #9819C2semidirect
15. #8264C2semidirect
232. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#302282
#292072
#291902
#286872
#284382
#222682
#106862
#106842
#103152
#98422
#98282
#98202
#98192
#82642