Finite Group Databasebeta

← #193297

Subgroups — #193297

Order 512.

1248 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34364C211yesmaximal, characteristic
2#21985C211yesmaximal, characteristic, complemented
3#21963C211yesmaximal, characteristic, complemented
4#21803C211yesmaximal, characteristic
5C2×#1545C211yesmaximal, characteristic
6#8924C212yesmaximal, complemented
7#8921C212yesmaximal, complemented
8#8921C212yesmaximal, complemented
9#8920C212yesmaximal, complemented
10#7899C211yesmaximal, characteristic, complemented
11#7894C211yesmaximal, characteristic, complemented
12C22×#127C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2380—22no—
17#2366C2212yes—
18#2366C2212yes—
19D4×C22⋊C4—22no—
20D4×C22⋊C4—22no—
21#2328—21no—
22#2328—21no—
23#2328—22no—
24#2308—22no—
25#2285C2211yescharacteristic
26#2283C2211yescharacteristic
27#2283C2211yescharacteristic
28#1982—22no—
29#1957C2212yes—
30#1957C2212yes—
31#1949—22no—
32#1948C2212yescomplemented
33#1948C2212yescomplemented
34#1947C2212yescomplemented
35#1947C2212yescomplemented
36#1854—22no—
37#1846—21no—
38#1835—21no—
39#1809—22no—
40#1786—22no—
41#1545C418yescomplemented
42C2×#331C2212yes—
43C2×#331C2212yes—
44#1279C2211yescharacteristic, complemented
45#1279C2211yescharacteristic, complemented
46#1279C2211yescharacteristic, complemented
47#1279C2211yescharacteristic, complemented
48#1279C2212yescomplemented
49#1279C2212yescomplemented
50C23×D4—21no—
51C23×D4—21no—
52C23×D4D412yes—
53C22×M16—21no—
54C2×#127—21no—
55C2×#127—21no—
56C2×#127—22no—
57C2×#127—22no—
58C2×#127C2×C414yes—
59C2×#127D412yes—
60C2×#127Q814yes—
61C2×#127C2312yes—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4—22no—
65C2×C4×D4—22no—
66C2×#117—21no—
67C22×C22⋊C4—24no—
68C22×C42C2311yescharacteristic
69#431—22no—
70#431—22no—
71#431—22no—
72#431C2×C414yescomplemented
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—22no—
77#423—22no—
78#423C2×C414yescomplemented
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×#102—24no—
85C2×#102—24no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2311yescharacteristic
91#390—22no—
92#390—22no—
93#390—22no—
94#390—22no—
95#386—24no—
96#386—24no—
97#386—24no—
98#386—24no—
99#386—24no—
100#386—24no—
101#386—24no—
102#386—24no—
103#384C2312yes—
104#383—21no—
105#383—21no—
106#382—21no—
107#382D411yescharacteristic
108#382D411yescharacteristic
109#382D412yes—
110#381—21no—
111#381—21no—
112#380—21no—
113#380—21no—
114#380D412yes—
115#380D412yes—
116#379—24no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2312yes—
121C4×C4⋊C4C2311yescharacteristic
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124#376—21no—
125#376—21no—
126#340—22no—
127#340—22no—
128#340—22no—
129#340—22no—
130#331C2×C418yescomplemented
131#331C2×C418yescomplemented
132C25—22no—
133C22×D4—24no—
134C22×D4—24no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—24no—
139C22×D4—24no—
140C22×D4—24no—
141C22×D4—22no—
142C22×D4—22no—
143C22×D4—81no—
144C22×D4—81no—
145C22×D4—82no—
146C22×D4—81no—
147C22×D4—81no—
148C22×D4—82no—
149C22×D4—84no—
150C22×D4C4⋊C418yes—
151C22×D4C2×D414yes—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C2×M16—24no—
158C2×M16—82no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164#127—24no—
165#127—24no—
166#127—24no—
167#127—24no—
168#127—24no—
169#127—22no—
170#127—22no—
171#127C4⋊C418yes—
172#127C22×C414yes—
173#127C2×Q814yes—
174C4×D4—82no—
175C4×D4—82no—
176C4×D4—82no—
177C4×D4—82no—
178C4×D4—82no—
179C4×D4—82no—
180C4×D4—82no—
181C4×D4—82no—
182#117—81no—
183#117—81no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—88no—
201C2×C42—21no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—41no—
205C2×C42C22×C412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q812yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212C2×C42C2411yescharacteristic
213#107—82no—
214#107—82no—
215#106—82no—
216#106—82no—
217C4⋊C8—24no—
218C4⋊C8—22no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—42no—
222C4⋊C8C22×C414yes—
223C4⋊C8C22×C412yes—
224C4⋊C8C22×C412yes—
225#102—84no—
226#102—84no—
227C22⋊C8—24no—
228C22⋊C8—42no—
229C4×C8—24no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—21no—
238#95—22no—
239#95—21no—
240#95—21no—
241#95—22no—
242#95—21no—
243#95—22no—
244#95—21no—
245#95—41no—
246#95—41no—
247C24—24no—
248C24—28no—
249C24—48no—
250C24—82no—
251C24—82no—
252C24—84no—
253C24#12011yescharacteristic
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—24no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—28no—
260C2×D4—28no—
261C2×D4—28no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—84no—
273C2×D4—84no—
274C2×D4—88no—
275C2×D4—84no—
276C2×D4—84no—
277C2×D4—84no—
278C2×D4—84no—
279C2×D4—84no—
280C2×D4—82no—
281C2×D4—82no—
282C2×D4—84no—
283C2×D4—88no—
284C2×D4—88no—
285C2×D4C2×C4⋊C418yes—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—84no—
299C22×C4—84no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4C2×C4⋊C412yes—
305C22×C4C4×D412yes—
306C22×C4C4×D412yes—
307C22×C4#12011yescharacteristic
308C22×C4#12011yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4C22⋊Q812yes—
311C22×C4C22⋊Q812yes—
312C22×C4#12312yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316M16—84no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—24no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—41no—
329C2×C8—41no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—44no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—84no—
346C22⋊C4—84no—
347C22⋊C4—84no—
348C22⋊C4—84no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C22⋊C4—82no—
352C42—41no—
353C42—41no—
354C42—42no—
355C42—41no—
356C42—41no—
357C42—42no—
358C42—82no—
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—28no—
374C23—416no—
375C23—44no—
376C23—44no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—88no—
382C23—88no—
383C23—168no—
384C23—168no—
385C23#38512yes—
386C23C2×#12011yescharacteristic
387D4—48no—
388D4—44no—
389D4—44no—
390D4—44no—
391D4—44no—
392D4—44no—
393D4—44no—
394D4—44no—
395D4—44no—
396D4—84no—
397D4—88no—
398D4—88no—
399D4—88no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
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—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—168no—
430C2×C4—168no—
431C2×C4#38512yes—
432C2×C4#38512yes—
433C2×C4#38512yes—
434C2×C4C22×C4⋊C411yescharacteristic
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×C4×D411yescharacteristic
437C2×C4C2×C22⋊Q811yescharacteristic
438C2×C4C2×C22⋊Q811yescharacteristic
439C2×C4C2×#12311yescharacteristic
440C2×C4#57611yescharacteristic
441C2×C4#57611yescharacteristic
442C2×C4#57612yes—
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57712yes—
446C8—42no—
447C8—42no—
448C8—44no—
449C8—81no—
450C8—81no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—42no—
455C22—42no—
456C22—48no—
457C22—48no—
458C22—48no—
459C22—84no—
460C22—84no—
461C22—84no—
462C22—84no—
463C22—1616no—
464C22C2×#38511yescharacteristic
465C22#300811yescharacteristic
466C22#300911yescharacteristic
467C22#301212yes—
468C22#301912yes—
469C4—21no—
470C4—21no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—82no—
475C4—82no—
476C4—82no—
477C4—82no—
478C4—162no—
479C4#294611yescharacteristic
480C4#294611yescharacteristic
481C4#294811yescharacteristic
482C4#294911yescharacteristic
483C4#308111yescharacteristic
484C4#308111yescharacteristic
485C4#308112yes—
486C2—42no—
487C2—88no—
488C2#1893012yes—
489C2#1893112yes—
490C2#2085611yescharacteristic
491C2#2087611yescharacteristic
492C2#2087911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#127C22
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. #21963C2semidirect
6. #8924C2semidirect
7. #8921C2semidirect
8. #8921C2semidirect
9. #8920C2semidirect
10. #7899C2semidirect
11. #7894C2semidirect
32. #1948C22semidirect
33. #1948C22semidirect
34. #1947C22semidirect
35. #1947C22semidirect
41. #1545C4semidirect
44. #1279C22semidirect
45. #1279C22semidirect
46. #1279C22semidirect
47. #1279C22semidirect
48. #1279C22semidirect
49. #1279C22semidirect
72. #431C2×C4semidirect
78. #423C2×C4semidirect
130. #331C2×C4semidirect
131. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343642
#219852
#219632
#218032
C2×#15452
#89242
#89212
#89202
#78992
#78942