Finite Group Databasebeta

← #202696

Subgroups — #202696

Order 512.

876 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21975C211yesmaximal, characteristic, complemented
2#21786C211yesmaximal, characteristic
3#20733C211yesmaximal, characteristic
4C2×#1486C211yesmaximal, characteristic
5C2×#1454C211yesmaximal, characteristic, complemented
6#9560C212yesmaximal, complemented
7#9558C212yesmaximal, complemented
8#9556C212yesmaximal, complemented
9#9554C212yesmaximal, complemented
10#7894C211yesmaximal, characteristic, complemented
11#7880C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#423C2211yescharacteristic
18#2283C2211yescharacteristic
19#1954—21no—
20#1954—21no—
21#1954C2212yes—
22#1954C2212yes—
23#1939C2212yescomplemented
24#1939C2212yescomplemented
25#1851—21no—
26#1851—21no—
27#1843—21no—
28#1832—21no—
29#1824C412yes—
30#1804C412yes—
31#1804C412yes—
32#1786C412yes—
33#1779—21no—
34#1779—21no—
35C4×#102C2212yes—
36C4×#102C2212yes—
37#1486—24no—
38C2×D4⋊C8C2212yes—
39C2×D4⋊C8C2212yes—
40#1454—24no—
41C2×#322C2211yescharacteristic, complemented
42C2×#322C2211yescharacteristic, complemented
43C2×#322C2211yescharacteristic, complemented
44C2×C82C2211yescharacteristic, complemented
45#1279C2212yescomplemented
46#1279C2212yescomplemented
47#1277C2212yescomplemented
48#1277C2212yescomplemented
49C23×D4C2×C411yescharacteristic
50C24×C4—21no—
51C22×M16—21no—
52C2×C4×D4—21no—
53C2×C4×D4—22no—
54C2×C4×D4—21no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2312yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C22⋊C4—21no—
62C22×C42C2311yescharacteristic
63#431—22no—
64#431—22no—
65#431—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431—22no—
70#431D414yes—
71#423—22no—
72#423—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8C2×C412yes—
78C2×C4⋊C8D411yescharacteristic
79C2×C4⋊C8D411yescharacteristic
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×#102C2×C412yes—
83C2×#102C2×C412yes—
84C2×#102C2×C412yes—
85C2×#102C2×C412yes—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8C2×C412yes—
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D411yescharacteristic
95C2×C4×C8—21no—
96C2×C4×C8C2311yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101C2×C4×C8C2311yescharacteristic
102#383—21no—
103#382C2×C411yescharacteristic
104#382C2×C411yescharacteristic
105#381—21no—
106#380C2×C412yes—
107C4×C4⋊C4C2312yes—
108C4×C22⋊C4—21no—
109#376—21no—
110#340C2×C412yes—
111#340C2×C412yes—
112#340C2×C412yes—
113#340C2×C412yes—
114#336—22no—
115#336—22no—
116D4⋊C8—24no—
117D4⋊C8—24no—
118#322—22no—
119#322—22no—
120#322—22no—
121C82—22no—
122C25—21no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—42no—
128C22×D4C22×C412yes—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—82no—
135C23×C4—82no—
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4#4111yescharacteristic
139C2×M16—42no—
140C2×M16—82no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8C22×C412yes—
146C22×C8C22×C412yes—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—42no—
158C4×D4—42no—
159C4×D4—42no—
160C4×D4—42no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C22×C412yes—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C42—22no—
173C2×C42C22×C411yescharacteristic
174C2×C42C2×D412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42#4112yes—
177C2×C42#4112yes—
178C2×C42#4112yes—
179C2×C42#4111yescharacteristic
180C2×C42C2411yescharacteristic
181C4⋊C8—24no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—24no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—42no—
190C4⋊C8—82no—
191C4⋊C8—82no—
192#102—24no—
193#102—24no—
194#102—42no—
195#102—42no—
196C22⋊C8—24no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199C22⋊C8—24no—
200C22⋊C8—42no—
201C22⋊C8—82no—
202C22⋊C8—82no—
203C22⋊C8—82no—
204C22⋊C8—82no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—21no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—21no—
211C4×C8—21no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—21no—
215C4×C8—42no—
216C4×C8C2×D412yes—
217C4×C8C2×D412yes—
218#95—22no—
219#95—21no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—22no—
224C24—24no—
225C24—22no—
226C24—42no—
227C24—42no—
228C24—82no—
229C24—82no—
230C24C4×D411yescharacteristic
231C2×D4—24no—
232C2×D4—22no—
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—48no—
238C2×D4—48no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4#13114yes—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—88no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4—82no—
264C22×C4#11712yes—
265C22×C4C4×D412yes—
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4#12112yes—
270C22×C4#12312yes—
271C22×C4#12412yes—
272C22×C4#12612yes—
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4111yescharacteristic
276M16—84no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—44no—
311C22⋊C4—82no—
312C42—42no—
313C42—42no—
314C42#11712yes—
315C42C4×D412yes—
316C42#12112yes—
317C42#12312yes—
318C42#12412yes—
319C42#12612yes—
320C42C22×D411yescharacteristic
321C42C2×#4111yescharacteristic
322C42C2×#4111yescharacteristic
323C42C2×#4111yescharacteristic
324C23—21no—
325C23—21no—
326C23—22no—
327C23—22no—
328C23—24no—
329C23—48no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—84no—
337C23—84no—
338C23—88no—
339C23—88no—
340C23#38812yes—
341C23C2×C4×D411yescharacteristic
342D4—24no—
343D4—24no—
344D4—48no—
345D4—42no—
346D4—42no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—88no—
369C2×C4—84no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4#38812yes—
375C2×C4#38812yes—
376C2×C4#38812yes—
377C2×C4C2×#11711yescharacteristic
378C2×C4C2×C4×D411yescharacteristic
379C2×C4C2×#12111yescharacteristic
380C2×C4C2×#12311yescharacteristic
381C2×C4C2×#12411yescharacteristic
382C2×C4C2×#12611yescharacteristic
383C2×C4C2×#13112yes—
384C2×C4#56812yes—
385C2×C4#57612yes—
386C2×C4#57712yes—
387C8—42no—
388C8—42no—
389C8—42no—
390C8—42no—
391C8—41no—
392C8—41no—
393C8—41no—
394C8—41no—
395C8—82no—
396C8—82no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—44no—
404C22—44no—
405C22—816no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—82no—
410C22C2×#38811yescharacteristic
411C22#292911yescharacteristic
412C22#293011yescharacteristic
413C22#294612yes—
414C22#295712yes—
415C4—22no—
416C4—22no—
417C4—21no—
418C4—21no—
419C4—22no—
420C4—82no—
421C4—82no—
422C4—82no—
423C4#292412yes—
424C4#293312yes—
425C4#305711yescharacteristic
426C4#305811yescharacteristic
427C4#313612yes—
428C2—42no—
429C2—84no—
430C2#1791812yes—
431C2#1794612yes—
432C2#2105311yescharacteristic
433C2#2109111yescharacteristic
434C2#2109811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21975C2semidirect
5. C2×#1454C2semidirect
6. #9560C2semidirect
7. #9558C2semidirect
8. #9556C2semidirect
9. #9554C2semidirect
10. #7894C2semidirect
11. #7880C2semidirect
23. #1939C22semidirect
24. #1939C22semidirect
41. C2×#322C22semidirect
42. C2×#322C22semidirect
43. C2×#322C22semidirect
44. C2×C82C22semidirect
45. #1279C22semidirect
46. #1279C22semidirect
47. #1277C22semidirect
48. #1277C22semidirect

Maximal subgroups

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

subgroupindex
#219752
#217862
#207332
C2×#14862
C2×#14542
#95602
#95582
#95562
#95542
#78942
#78802