Finite Group Databasebeta

← #493728

Subgroups — #493728

Order 512.

1143 conjugacy classes of subgroups.

The subgroup lattice has 497 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2C2×#2133C212yesmaximal
3#20639C212yesmaximal, complemented
4#12804C212yesmaximal, complemented
5#12803C212yesmaximal, complemented
6#12802C212yesmaximal, complemented
7#12801C212yesmaximal, complemented
8C2×#1394C212yesmaximal, complemented
9C2×#536C2211yescharacteristic
10C2×#520C411yescharacteristic
11C2×#519C2211yescharacteristic
12#2905—22no—
13C22×#101C2214yes—
14#2362—22no—
15#2360—22no—
16#2359—22no—
17#2359—22no—
18#2327C411yescharacteristic
19#2327C2211yescharacteristic
20#2326C411yescharacteristic
21#2326C2211yescharacteristic
22#2317C411yescharacteristic
23C4×C4⋊Q8C411yescharacteristic
24C4×C4⋊Q8C2211yescharacteristic
25C4×#124C411yescharacteristic
26C4×#124C2211yescharacteristic
27#2297C411yescharacteristic
28C2×#391C2211yescharacteristic
29C4×#117C411yescharacteristic
30#2133—28no—
31C2×#356C2214yes—
32C2×#355C2214yes—
33#1801—22no—
34#1801—22no—
35#1801C2214yes—
36#1801C2214yes—
37#1742—22no—
38#1394C2214yes—
39#1394C2214yes—
40C22×#41C2×C411yescharacteristic
41C23×Q8C2311yescharacteristic
42C22×M16—24no—
43#536—24no—
44#536D414yes—
45#536Q814yes—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2311yescharacteristic
48C2×#124C2×C411yescharacteristic
49C2×#124C2311yescharacteristic
50C2×#123—22no—
51C2×C22⋊Q8—22no—
52#520—44no—
53#519—44no—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8C2×C411yescharacteristic
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8D412yes—
60C2×C4×Q8C2311yescharacteristic
61C2×C4×Q8C2311yescharacteristic
62C2×C4×D4—21no—
63C2×C4×D4—21no—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×#117—21no—
67C2×#117C2×C411yescharacteristic
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117D412yes—
71C2×#117D412yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4C2×C411yescharacteristic
74C22×C42C2×C411yescharacteristic
75#424—48no—
76#423—24no—
77#423—24no—
78C2×#101—28no—
79C2×#101—28no—
80C2×#101—24no—
81C2×#101C2318yes—
82#391—22no—
83#391C2×C411yescharacteristic
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#391C2311yescharacteristic
88#391C2311yescharacteristic
89#390C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#389—22no—
92#389—22no—
93#389—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#386—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#384C2×C411yescharacteristic
107#384C2×C411yescharacteristic
108#382C2×C411yescharacteristic
109#382C2×C411yescharacteristic
110C4×C4⋊C4—21no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119#376C2×C411yescharacteristic
120#376C2×C411yescharacteristic
121C2×#95—22no—
122C43C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124#356—28no—
125#355—28no—
126#343—24no—
127#343—24no—
128C2×#41—22no—
129C2×#41—42no—
130C2×#41—42no—
131C2×#41C4⋊C414yes—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8C22×C411yescharacteristic
135C22×Q8C22×C411yescharacteristic
136C22×Q8C2×D412yes—
137C22×Q8C2×D412yes—
138C22×Q8C2×Q812yes—
139C22×Q8#4114yes—
140C22×Q8C2411yescharacteristic
141C22×D4—21no—
142C22×D4C22×C411yescharacteristic
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D412yes—
150C2×M16—28no—
151C2×M16—44no—
152C2×M16—48no—
153C2×M16—48no—
154C22×C8—44no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8C4⋊C414yes—
158C4⋊Q8C2×D412yes—
159C4⋊Q8C2×Q812yes—
160#124—22no—
161#124—22no—
162#124C4⋊C414yes—
163#124C2×D412yes—
164#124C2×Q812yes—
165#123—28no—
166#123—28no—
167C22⋊Q8—28no—
168C22⋊Q8—28no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—44no—
174C4×Q8—44no—
175C4×Q8—42no—
176C4×Q8—42no—
177C4×D4—42no—
178C4×D4—42no—
179C4×D4—44no—
180C4×D4—44no—
181#117—22no—
182#117—24no—
183#117—28no—
184#117—42no—
185#117—44no—
186#117—44no—
187#117C4⋊C414yes—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4C22⋊C411yescharacteristic
199C2×C4⋊C4C22⋊C411yescharacteristic
200C2×C4⋊C4C22⋊C412yes—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4C22⋊C411yescharacteristic
216C2×C22⋊C4C22⋊C412yes—
217C2×C22⋊C4C22⋊C411yescharacteristic
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—42no—
226C2×C42C22⋊C412yes—
227C2×C42C22⋊C412yes—
228C2×C42C22⋊C412yes—
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C22×C411yescharacteristic
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C4⋊C8—48no—
238C4⋊C8—48no—
239#101—28no—
240#101—28no—
241#101—48no—
242#95—22no—
243#95—22no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254C24—21no—
255C24C2×C22⋊C411yescharacteristic
256#41—48no—
257#41—48no—
258C2×Q8—21no—
259C2×Q8—41no—
260C2×Q8—42no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—44no—
264C2×Q8—42no—
265C2×Q8—44no—
266C2×Q8—48no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8C2×C4⋊C412yes—
271C2×Q8C2×C4⋊C412yes—
272C2×Q8#11712yes—
273C2×Q8#12114yes—
274C2×Q8C22⋊Q814yes—
275C2×Q8C22×D411yescharacteristic
276C2×Q8C22×Q811yescharacteristic
277C2×Q8C2×#4112yes—
278C2×D4—24no—
279C2×D4—21no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4C2×C4⋊C412yes—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—24no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—81no—
305C22×C4—82no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C412yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C4⋊C412yes—
317C22×C4#11712yes—
318C22×C4#12114yes—
319C22×C4C22⋊Q814yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D412yes—
322M16—48no—
323M16—48no—
324M16—88no—
325C2×C8—48no—
326C2×C8—88no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—28no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—22no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—28no—
352C22⋊C4—44no—
353C22⋊C4—42no—
354C22⋊C4—42no—
355C22⋊C4—82no—
356C22⋊C4#9914yes—
357C42—21no—
358C42—21no—
359C42—24no—
360C42—24no—
361C42—22no—
362C42—22no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—41no—
367C42—41no—
368C42—42no—
369C42—42no—
370C42—42no—
371C42—42no—
372C42—42no—
373C42—81no—
374C42—81no—
375C42C2×C4⋊C412yes—
376C42C2×C4⋊C412yes—
377C23—22no—
378C23—41no—
379C23—41no—
380C23—42no—
381C23—42no—
382C23—41no—
383C23—41no—
384C23#38012yes—
385C23C2×#9912yes—
386C23#41012yes—
387C23C22×C22⋊C411yescharacteristic
388Q8—42no—
389Q8—42no—
390Q8—44no—
391Q8—44no—
392Q8—41no—
393Q8—41no—
394Q8—42no—
395Q8—42no—
396Q8—42no—
397D4—42no—
398D4—42no—
399D4—48no—
400C2×C4—28no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—28no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—48no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—84no—
443C2×C4—162no—
444C2×C4—162no—
445C2×C4—162no—
446C2×C4#38014yes—
447C2×C4#38012yes—
448C2×C4C2×#9912yes—
449C2×C4C2×#9912yes—
450C2×C4C22×C4⋊C411yescharacteristic
451C2×C4C2×#11711yescharacteristic
452C2×C4C2×#12112yes—
453C2×C4C2×C22⋊Q812yes—
454C8—88no—
455C22—21no—
456C22—22no—
457C22—21no—
458C22—22no—
459C22—42no—
460C22—42no—
461C22—44no—
462C22—42no—
463C22—82no—
464C22—81no—
465C22—81no—
466C22#179914yes—
467C22C2×#38011yescharacteristic
468C22C22×#9911yescharacteristic
469C22C2×#41011yescharacteristic
470C4—22no—
471C4—22no—
472C4—22no—
473C4—22no—
474C4—21no—
475C4—21no—
476C4—22no—
477C4—24no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—41no—
483C4—41no—
484C4—42no—
485C4—42no—
486C4—81no—
487C4—81no—
488C4—81no—
489C2—21no—
490C2—21no—
491C2—24no—
492C2—82no—
493C2C2×#179911yescharacteristic
494C2#3337011yescharacteristic
495C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#519 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20639C2semidirect
4. #12804C2semidirect
5. #12803C2semidirect
6. #12802C2semidirect
7. #12801C2semidirect
8. C2×#1394C2semidirect

Maximal subgroups

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

subgroupindex
#342762
C2×#21332
#206392
#128042
#128032
#128022
#128012
C2×#13942