Finite Group Databasebeta

← #147073

Subgroups — #147073

Order 512.

775 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20637C211yesmaximal, characteristic
2#20629C211yesmaximal, characteristic
3C4×#410C211yesmaximal, characteristic
4C2×#1395C214yesmaximal
5C2×C4×#41C2211yescharacteristic
6C2×#423—21no—
7C2×#423C411yescharacteristic, complemented
8C22×C4⋊C8C411yescharacteristic, complemented
9C2×#413C2212yes—
10C2×#410C2212yes—
11C2×#408C2212yes—
12C4×#117—21no—
13#1801C412yescomplemented
14#1801C412yescomplemented
15#1800—21no—
16#1800—21no—
17#1798—21no—
18#1798—21no—
19#1797—21no—
20#1797—21no—
21C4×#99—21no—
22C4×#99—21no—
23C4×#99C412yes—
24C4×#99C412yes—
25#1742—21no—
26#1740—21no—
27#1729C412yescomplemented
28#1395—28no—
29C22×#41C2311yescharacteristic
30C22×M16—22no—
31C22×M16C2×C412yes—
32C23×C8C2×C412yes—
33C4×#41—41no—
34C4×#41—41no—
35C4×#41—41no—
36C4×#41—41no—
37C2×C4×Q8C2×C411yescharacteristic
38C2×C4×Q8C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4C2×C411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4D411yescharacteristic
43C2×C4×D4Q811yescharacteristic
44C2×C4×D4Q811yescharacteristic
45C2×#117—22no—
46C2×#117C2×C411yescharacteristic
47C2×#117D411yescharacteristic
48C2×#117D411yescharacteristic
49C22×C42C2×C411yescharacteristic
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52#423—22no—
53#423—22no—
54#423—41no—
55#423—41no—
56#423—41no—
57#423—41no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—41no—
63C2×C4⋊C8—41no—
64C2×C4⋊C8—41no—
65C2×C4⋊C8—41no—
66#413—24no—
67#413—24no—
68C2×#101C2×C414yes—
69C2×#100—22no—
70#410—22no—
71#410—22no—
72#410D414yes—
73#410Q814yes—
74C2×#99—22no—
75C2×#99C2×C414yes—
76#408—24no—
77#408—24no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C4×C4⋊C4—41no—
81C4×C4⋊C4—41no—
82C4×C22⋊C4—22no—
83C4×C22⋊C4—22no—
84#376—41no—
85C43—41no—
86#343—22no—
87#343—22no—
88#342—22no—
89#342—22no—
90#336—22no—
91#336—22no—
92C2×#41—41no—
93C2×#41—41no—
94C2×#41—41no—
95C2×#41—41no—
96C2×#41C2×D412yes—
97C2×#41C2×Q812yes—
98C2×#41#4112yes—
99C2×#41#4112yes—
100C22×Q8C22×C411yescharacteristic
101C22×D4C22×C411yescharacteristic
102C22×D4C2×D411yescharacteristic
103C22×D4C2×Q811yescharacteristic
104C23×C4C22⋊C411yescharacteristic
105C23×C4C22⋊C411yescharacteristic
106C23×C4C4⋊C411yescharacteristic
107C23×C4C4⋊C411yescharacteristic
108C23×C4C4⋊C411yescharacteristic
109C23×C4C4⋊C411yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C2×M16—24no—
114C2×M16—24no—
115C2×M16—42no—
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—42no—
119C2×M16—42no—
120C2×M16—42no—
121C2×M16—42no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—24no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8—42no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×D4—41no—
133C4×D4—41no—
134C4×D4—41no—
135C4×D4—41no—
136C4×D4—41no—
137C4×D4—41no—
138C4×D4—41no—
139C4×D4—41no—
140C4×D4—41no—
141C4×D4—41no—
142C4×D4—41no—
143C4×D4—41no—
144#117—24no—
145#117—24no—
146#117—41no—
147#117—41no—
148#117—41no—
149#117—41no—
150#117—41no—
151#117—41no—
152#117—42no—
153#117—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4C4211yescharacteristic
156C2×C4⋊C4C4211yescharacteristic
157C2×C4⋊C4C22⋊C411yescharacteristic
158C2×C4⋊C4C22⋊C411yescharacteristic
159C2×C4⋊C4C4⋊C411yescharacteristic
160C2×C4⋊C4C4⋊C411yescharacteristic
161C2×C22⋊C4—24no—
162C2×C22⋊C4C22⋊C411yescharacteristic
163C2×C22⋊C4C22⋊C411yescharacteristic
164C2×C22⋊C4C4⋊C411yescharacteristic
165C2×C22⋊C4C4⋊C411yescharacteristic
166C2×C22⋊C4C4⋊C411yescharacteristic
167C2×C22⋊C4C4⋊C411yescharacteristic
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42C4211yescharacteristic
184C2×C42C4211yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C4⋊C411yescharacteristic
192C2×C42C4⋊C411yescharacteristic
193C4⋊C8—42no—
194C4⋊C8—42no—
195C4⋊C8—42no—
196C4⋊C8—42no—
197C4⋊C8—42no—
198C4⋊C8—42no—
199C4⋊C8—41no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203#101—28no—
204#100—44no—
205#99—24no—
206#99—24no—
207#99—24no—
208#99C4⋊C418yes—
209C22⋊C8—44no—
210C22⋊C8—42no—
211C22⋊C8—42no—
212#95—42no—
213#95—42no—
214C24C2×C22⋊C411yescharacteristic
215C24C2×C4⋊C411yescharacteristic
216C24C2×C4⋊C411yescharacteristic
217#41—42no—
218#41—42no—
219#41—42no—
220#41—42no—
221#41—42no—
222#41—42no—
223#41—42no—
224#41—42no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8C4×D412yes—
228C2×Q8C4×Q812yes—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4C2×C4⋊C412yes—
236C2×D4#11712yes—
237C2×D4#12112yes—
238C2×D4C22⋊Q812yes—
239C2×D4#12512yes—
240C2×D4C4⋊Q812yes—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—81no—
272C22×C4#9511yescharacteristic
273C22×C4#9511yescharacteristic
274C22×C4#9511yescharacteristic
275C22×C4#9511yescharacteristic
276C22×C4#9511yescharacteristic
277C22×C4#9511yescharacteristic
278C22×C4#9511yescharacteristic
279C22×C4#9511yescharacteristic
280C22×C4C2×C4211yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C4⋊C411yescharacteristic
284C22×C4C4×D412yes—
285C22×C4C4×Q812yes—
286C22×C4#12112yes—
287C22×C4C22⋊Q812yes—
288C22×C4#12411yescharacteristic
289C22×C4#12411yescharacteristic
290C22×C4#12711yescharacteristic
291C22×C4C4⋊Q811yescharacteristic
292M16—44no—
293M16—44no—
294M16—82no—
295M16—82no—
296M16—84no—
297C2×C8—44no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—82no—
305C2×C8—82no—
306C2×C8—84no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C22⋊C4—28no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—44no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—44no—
342C42—44no—
343C42—84no—
344C42—84no—
345C42#9911yescharacteristic
346C42#9911yescharacteristic
347C42#9911yescharacteristic
348C42#9911yescharacteristic
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23C2×#9511yescharacteristic
362C23#38012yes—
363C23#38211yescharacteristic
364C23#38411yescharacteristic
365C23#38912yes—
366Q8—42no—
367Q8—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—42no—
372D4—42no—
373D4—42no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—48no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4#34211yescharacteristic
426C2×C4#34211yescharacteristic
427C2×C4#34211yescharacteristic
428C2×C4#34211yescharacteristic
429C2×C4C4×C4⋊C412yes—
430C2×C4#38912yes—
431C2×C4#39011yescharacteristic
432C2×C4#39011yescharacteristic
433C2×C4#39111yescharacteristic
434C2×C4#39111yescharacteristic
435C2×C4C2×#9911yescharacteristic
436C2×C4C2×#9911yescharacteristic
437C2×C4#41011yescharacteristic
438C2×C4#41011yescharacteristic
439C8—84no—
440C8—82no—
441C8—82no—
442C22—21no—
443C22—21no—
444C22—22no—
445C22—21no—
446C22—21no—
447C22—22no—
448C22—42no—
449C22—41no—
450C22—41no—
451C22—42no—
452C22—42no—
453C22—41no—
454C22—41no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—81no—
460C22—81no—
461C22C4⋊#9511yescharacteristic
462C22C2×#34211yescharacteristic
463C22#174011yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—22no—
469C4—22no—
470C4—21no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—41no—
476C4—41no—
477C4—88no—
478C4#139512yescomplemented
479C4#196411yescharacteristic
480C4#196411yescharacteristic
481C4#199611yescharacteristic
482C4#199611yescharacteristic
483C2—21no—
484C2—21no—
485C2—42no—
486C2—42no—
487C2—41no—
488C2—41no—
489C2#1085411yescharacteristic
490C2C2×#139511yescharacteristic
491C2#1258911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, prime radical, cosocleC22×#41C23
socleC22C2×#342
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#423C4semidirect
8. C22×C4⋊C8C4semidirect
13. #1801C4semidirect
14. #1801C4semidirect
27. #1729C4semidirect
478. C4#1395semidirect

Maximal subgroups

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

subgroupindex
#206372
#206292
C4×#4102
C2×#13952