Finite Group Databasebeta

← #101683

Subgroups — #101683

Order 512.

507 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29156C211yesmaximal, characteristic
2#23116C211yesmaximal, characteristic
3#9885C211yesmaximal, characteristic, complemented
4#8577C211yesmaximal, characteristic
5#7667C211yesmaximal, characteristic, complemented
6#7029C211yesmaximal, characteristic, complemented
7#7025C211yesmaximal, characteristic, complemented
8C2×C8×D4—21no—
9#2857C2211yescharacteristic
10C4×#121C2211yescharacteristic
11#2100C411yescharacteristic, complemented
12#2094C411yescharacteristic, complemented
13#2058C411yescharacteristic, complemented
14#2023C411yescharacteristic, complemented
15#1923—21no—
16#1920—21no—
17#1916—21no—
18#1893—21no—
19#1869—21no—
20#1869—21no—
21#1864—21no—
22#1807—21no—
23#1797C2211yescharacteristic
24C8×C22⋊C4—21no—
25C4×C22⋊C8C2211yescharacteristic
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28C2×#327—21no—
29C4⋊#95C2211yescharacteristic
30#1294—21no—
31#1294—21no—
32#1294—21no—
33#1272—21no—
34#1272—21no—
35C23×C8—21no—
36C23×C8—21no—
37C23×C8—21no—
38C2×C4⋊Q8C2×C411yescharacteristic
39C2×C22⋊Q8—21no—
40C2×#121D411yescharacteristic
41C2×C4×D4—21no—
42C2×C4×D4D411yescharacteristic
43C22×C4⋊C4—21no—
44C22×C42C2311yescharacteristic
45C8×D4—42no—
46C8×D4—42no—
47C2×#106—21no—
48C2×#106—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8—21no—
51C2×#102—21no—
52C2×#102—21no—
53C2×C22⋊C8—21no—
54C2×C22⋊C8—21no—
55C2×C22⋊C8—21no—
56C2×C22⋊C8—21no—
57C2×C22⋊C8D411yescharacteristic
58C2×C22⋊C8D411yescharacteristic
59C2×C4×C8—21no—
60C2×C4×C8—21no—
61#391—21no—
62#390C2×C411yescharacteristic
63#389—21no—
64#389—21no—
65#389C2×C411yescharacteristic
66#388—21no—
67#386—21no—
68#384C2×C411yescharacteristic
69#380—21no—
70#380—21no—
71#380D411yescharacteristic
72C4×C4⋊C4D411yescharacteristic
73C4×C22⋊C4—21no—
74C2×#95—21no—
75C2×#95—21no—
76#340—21no—
77#340—21no—
78#340—21no—
79#340—21no—
80#336—21no—
81#336—21no—
82#336—21no—
83#336—21no—
84#336—21no—
85#336—21no—
86#336C2×C411yescharacteristic
87#336C2×C411yescharacteristic
88#336D411yescharacteristic
89#336D411yescharacteristic
90#327—42no—
91C22×Q8—21no—
92C22×D4—21no—
93C22×D4C22⋊C411yescharacteristic
94C23×C4—21no—
95C23×C4—41no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4#4111yescharacteristic
99C22×C8—21no—
100C22×C8—21no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—42no—
106C22×C8—42no—
107C22×C8—42no—
108C22×C8—42no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—81no—
115C22×C8—81no—
116C22×C8—81no—
117C4⋊Q8—81no—
118C22⋊Q8—81no—
119C22⋊Q8—81no—
120#121—21no—
121#121—21no—
122#121SD1612yes—
123#121Q1612yes—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—81no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—81no—
139C2×C4⋊C4—81no—
140C2×C4⋊C4C22⋊C411yescharacteristic
141C2×C4⋊C4C22⋊C411yescharacteristic
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—41no—
149C2×C42—21no—
150C2×C42—81no—
151C2×C42C22×C411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155#106—42no—
156#106—81no—
157C4⋊C8—42no—
158C4⋊C8—81no—
159#102—42no—
160#102—81no—
161C22⋊C8—21no—
162C22⋊C8—21no—
163C22⋊C8—21no—
164C22⋊C8—42no—
165C22⋊C8—42no—
166C22⋊C8—41no—
167C22⋊C8—41no—
168C22⋊C8—81no—
169C22⋊C8SD1612yes—
170C4×C8—42no—
171C4×C8—81no—
172#95—21no—
173#95—21no—
174#95—21no—
175#95—41no—
176#95—41no—
177#95—41no—
178#95—41no—
179#95—41no—
180#95—41no—
181C24—21no—
182C24#12111yescharacteristic
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—81no—
186C2×Q8—81no—
187C2×D4—22no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4#10312yes—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—81no—
219C22×C4—81no—
220C22×C4—81no—
221C22×C4—81no—
222C22×C4—81no—
223C22×C4—81no—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4C4×D411yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4C2×SD1611yescharacteristic
231C22×C4C2×SD1611yescharacteristic
232C22×C4C2×Q1611yescharacteristic
233C22×C4#13511yescharacteristic
234C22×C4#13511yescharacteristic
235C22×C4#13511yescharacteristic
236C22×C4AGL(1,ℤ/8)11yescharacteristic
237C22×C4#13711yescharacteristic
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—84no—
257C2×C8—82no—
258C2×C8—82no—
259C2×C8—82no—
260C2×C8—82no—
261C2×C8—81no—
262C2×C8—81no—
263C2×C8—81no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4—81no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4#10312yes—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—81no—
289C22⋊C4—81no—
290C22⋊C4—81no—
291C42—41no—
292C42—41no—
293C42—41no—
294C42—41no—
295C42—41no—
296C42—41no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—22no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23#38611yescharacteristic
309C23#46111yescharacteristic
310C23#46311yescharacteristic
311C23#46511yescharacteristic
312C23#46711yescharacteristic
313C23#47111yescharacteristic
314C23#47211yescharacteristic
315Q8—161no—
316Q8—161no—
317D4—42no—
318D4—42no—
319D4—42no—
320D4—42no—
321D4—42no—
322D4—81no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—161no—
371C2×C4—161no—
372C2×C4—161no—
373C2×C4—161no—
374C2×C4C2×#10311yescharacteristic
375C2×C4#41611yescharacteristic
376C2×C4C4×SD1611yescharacteristic
377C2×C4#44011yescharacteristic
378C2×C4#44311yescharacteristic
379C2×C4#44411yescharacteristic
380C2×C4#45011yescharacteristic
381C2×C4#45211yescharacteristic
382C2×C4#46511yescharacteristic
383C2×C4#46711yescharacteristic
384C8—42no—
385C8—42no—
386C8—42no—
387C8—81no—
388C8—81no—
389C8—81no—
390C8—161no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22#167111yescharacteristic
411C22#167511yescharacteristic
412C22#167711yescharacteristic
413C22#168311yescharacteristic
414C22#187611yescharacteristic
415C22#187711yescharacteristic
416C22#188411yescharacteristic
417C22#189211yescharacteristic
418C22#190611yescharacteristic
419C22#190711yescharacteristic
420C22#191311yescharacteristic
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—21no—
426C4—21no—
427C4—21no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—41no—
432C4—42no—
433C4—81no—
434C4—81no—
435C4—162no—
436C4—161no—
437C4—161no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—82no—
443C2#905511yescharacteristic
444C2#920711yescharacteristic
445C2#921511yescharacteristic
446C2#926411yescharacteristic
447C2#927511yescharacteristic
448C2#938311yescharacteristic
449C2#943211yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9885C2semidirect
5. #7667C2semidirect
6. #7029C2semidirect
7. #7025C2semidirect
11. #2100C4semidirect
12. #2094C4semidirect
13. #2058C4semidirect
14. #2023C4semidirect

Maximal subgroups

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

subgroupindex
#291562
#231162
#98852
#85772
#76672
#70292
#70252