Finite Group Databasebeta

← #193314

Subgroups — #193314

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34393C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic, complemented
3#21959C211yesmaximal, characteristic, complemented
4#21410C211yesmaximal, characteristic
5C2×#1540C211yesmaximal, characteristic
6#8925C212yesmaximal, complemented
7#8924C212yesmaximal, complemented
8#8923C212yesmaximal, complemented
9#8921C212yesmaximal, complemented
10#7896C211yesmaximal, characteristic, complemented
11#7895C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2377—22no—
17#2366C2212yes—
18#2365C2212yes—
19#2357—21no—
20#2349—21no—
21#2341—21no—
22#2339—21no—
23#2326—22no—
24#2324—22no—
25#2318—22no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2283C2211yescharacteristic
29#1990C2212yes—
30#1957C2212yes—
31#1948—22no—
32#1948C2212yes—
33#1948C2212yes—
34#1948C2212yes—
35#1947C2212yes—
36#1895—22no—
37#1854—22no—
38#1846—21no—
39#1835—21no—
40#1807—21no—
41#1807—21no—
42#1762—22no—
43#1540C418yescomplemented
44C2×#332C2211yescharacteristic
45C2×#332C2211yescharacteristic
46C2×#331C2211yescharacteristic
47C2×#331C2211yescharacteristic
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×#41—21no—
53C22×M16—21no—
54C23×C8—21no—
55#532—22no—
56#532—22no—
57#532C2×C414yes—
58#532Q814yes—
59C2×C4⋊Q8C2311yescharacteristic
60C2×#127C2311yescharacteristic
61C2×#124—21no—
62C2×#121—21no—
63C2×#121D412yes—
64C2×C4×Q8—22no—
65C2×C4×D4—22no—
66C2×C4×D4—22no—
67C2×C4×D4—21no—
68C2×C4×D4D412yes—
69C2×#117—21no—
70C2×#117—21no—
71C22×C42C2311yescharacteristic
72#431—22no—
73#431C2×C414yescomplemented
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—22no—
77C2×#106—22no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8C2×C414yescomplemented
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×#102—24no—
86C2×C22⋊C8—21no—
87C4×M16—22no—
88C4×M16—22no—
89C2×#97—21no—
90C2×C4×C8—21no—
91C2×C4×C8C2312yes—
92#391—22no—
93#390—24no—
94#390—22no—
95#390—22no—
96#389—22no—
97#388—24no—
98#388—24no—
99#387—24no—
100#386—24no—
101#386—24no—
102#385—24no—
103#385—24no—
104#384D412yes—
105#384C2312yes—
106#383—21no—
107#382—21no—
108#382—21no—
109#382D411yescharacteristic
110#382D411yescharacteristic
111#381—22no—
112#381—21no—
113#380—22no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380D411yescharacteristic
118#380D411yescharacteristic
119#380D412yes—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4C2312yes—
125C4×C4⋊C4C2312yes—
126C4×C22⋊C4—22no—
127C4×C22⋊C4—21no—
128#376—21no—
129C2×#95—22no—
130#340—22no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—24no—
135#332C2×C414yescomplemented
136#332C2×C414yescomplemented
137#331C2×C414yes—
138#331C2×C414yes—
139C2×#41—24no—
140C2×#41—24no—
141C2×#41—82no—
142C2×#41—82no—
143C22×Q8—21no—
144C22×D4—22no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4C2×D412yes—
148C23×C4—22no—
149C23×C4—24no—
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C2×M16—44no—
154C2×M16—42no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—24no—
161C22×C8—82no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8C22×C412yes—
165C4⋊Q8C2×Q812yes—
166#127—21no—
167#127—21no—
168#127C22×C412yes—
169#127C2×Q812yes—
170#124—24no—
171#124—24no—
172#121—24no—
173#121—24no—
174#121—24no—
175#121C4⋊C418yes—
176C4×Q8—82no—
177C4×Q8—82no—
178C4×D4—24no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—82no—
182C4×D4—82no—
183C4×D4—82no—
184C4×D4—82no—
185C4×D4C4⋊C418yes—
186#117—81no—
187#117—81no—
188#117—81no—
189#117—81no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×D412yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—41no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—41no—
215C2×C42C22×C412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42C2411yescharacteristic
223#107—82no—
224#106—82no—
225#106—82no—
226#106—82no—
227C4⋊C8—24no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8C22×C412yes—
233C4⋊C8C22×C412yes—
234C4⋊C8C22×C412yes—
235C4⋊C8C22×C412yes—
236#102—48no—
237C22⋊C8—24no—
238#97—42no—
239C4×C8—24no—
240C4×C8—22no—
241C4×C8—22no—
242#95—24no—
243#95—24no—
244#95—21no—
245#95—21no—
246#95—22no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—22no—
252#95—22no—
253#95—41no—
254#95—41no—
255C24—22no—
256C24#12011yescharacteristic
257#41—88no—
258#41—84no—
259#41—84no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×D4—28no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—84no—
278C2×D4C2×C4⋊C414yes—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—28no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—84no—
294C22×C4—84no—
295C22×C4—84no—
296C22×C4—84no—
297C22×C4C2×C4⋊C412yes—
298C22×C4C4×D412yes—
299C22×C4C4×D412yes—
300C22×C4#12011yescharacteristic
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4C22⋊Q812yes—
304C22×C4C22⋊Q812yes—
305C22×C4#12312yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—48no—
310C2×C8—24no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—84no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—84no—
343C4⋊C4C2×C4⋊C414yes—
344C22⋊C4—28no—
345C22⋊C4—28no—
346C22⋊C4—82no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C22⋊C4—84no—
352C22⋊C4—82no—
353C42—42no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—82no—
358C42—84no—
359C42C2×C4⋊C412yes—
360C42C4×D412yes—
361C42C4×D412yes—
362C42C22⋊Q812yes—
363C42C22⋊Q812yes—
364C42#12312yes—
365C42C23×C411yescharacteristic
366C42C22×Q811yescharacteristic
367C42C2×#4111yescharacteristic
368C42C2×#4111yescharacteristic
369C23—21no—
370C23—21no—
371C23—22no—
372C23—22no—
373C23—24no—
374C23—42no—
375C23—42no—
376C23—44no—
377C23—44no—
378C23#38512yes—
379C23C2×#12011yescharacteristic
380Q8—44no—
381Q8—42no—
382Q8—42no—
383D4—44no—
384D4—44no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—82no—
392D4—88no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—88no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—168no—
426C2×C4#38512yes—
427C2×C4#38512yes—
428C2×C4#38512yes—
429C2×C4C22×C4⋊C411yescharacteristic
430C2×C4C2×C4×D411yescharacteristic
431C2×C4C2×C4×D411yescharacteristic
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×#12311yescharacteristic
435C2×C4#57612yes—
436C2×C4#57712yes—
437C2×C4#57711yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57711yescharacteristic
441C8—44no—
442C8—42no—
443C8—42no—
444C8—42no—
445C8—42no—
446C22—22no—
447C22—22no—
448C22—22no—
449C22—42no—
450C22—42no—
451C22—44no—
452C22—44no—
453C22—44no—
454C22—84no—
455C22—84no—
456C22C2×#38511yescharacteristic
457C22#300911yescharacteristic
458C22#301011yescharacteristic
459C22#301212yes—
460C22#301812yes—
461C4—21no—
462C4—21no—
463C4—22no—
464C4—22no—
465C4—22no—
466C4—44no—
467C4—82no—
468C4—82no—
469C4—82no—
470C4—82no—
471C4—162no—
472C4#294811yescharacteristic
473C4#294911yescharacteristic
474C4#295211yescharacteristic
475C4#295211yescharacteristic
476C4#308112yes—
477C4#308411yescharacteristic
478C4#308411yescharacteristic
479C2—42no—
480C2—84no—
481C2#1873912yes—
482C2#1874212yes—
483C2#2085511yescharacteristic
484C2#2087911yescharacteristic
485C2#2088211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21985C2semidirect
3. #21959C2semidirect
6. #8925C2semidirect
7. #8924C2semidirect
8. #8923C2semidirect
9. #8921C2semidirect
10. #7896C2semidirect
11. #7895C2semidirect
43. #1540C4semidirect
73. #431C2×C4semidirect
80. C2×C4⋊C8C2×C4semidirect
135. #332C2×C4semidirect
136. #332C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343932
#219852
#219592
#214102
C2×#15402
#89252
#89242
#89232
#89212
#78962
#78952