Finite Group Databasebeta

← #190280

Subgroups — #190280

Order 512.

1107 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34126C211yesmaximal, characteristic
2C2×#1800C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#8771C212yesmaximal, complemented
5#8770C212yesmaximal, complemented
6#8765C212yesmaximal, complemented
7#8764C212yesmaximal, complemented
8#8097C212yesmaximal, complemented
9#7962C212yesmaximal, complemented
10C22×C4×D4C2211yescharacteristic
11#2920C2212yes—
12C2×#423C2212yes—
13C22×#100—21no—
14#2357C2211yescharacteristic
15#2357C2211yescharacteristic
16#2353C412yes—
17#2353C412yes—
18#2349C2211yescharacteristic
19#2349C2211yescharacteristic
20#2345—21no—
21#2342—22no—
22D4×C22⋊C4—21no—
23#2293—21no—
24#2290—22no—
25#2283C2211yescharacteristic
26#2280C412yes—
27C42×D4—21no—
28C4×#117C2211yescharacteristic
29C2×C4×C22⋊C4C412yes—
30#1800—21no—
31#1800—21no—
32#1800C2212yes—
33#1800C2212yes—
34C2×#343—21no—
35#1466—22no—
36#1462—22no—
37#1460C2212yes—
38#1460C2212yes—
39#1303C2214yescomplemented
40#1303C2214yescomplemented
41#1286C2214yescomplemented
42#1286C2214yescomplemented
43C23×D4D411yescharacteristic
44C24×C4C2×C412yes—
45C22×M16—22no—
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—22no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D411yescharacteristic
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D412yes—
56C2×C4×D4C2311yescharacteristic
57C2×C4×D4C2311yescharacteristic
58C2×#117—21no—
59C2×#117—22no—
60C22×C4⋊C4D411yescharacteristic
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65#432—22no—
66#432—22no—
67#431—22no—
68#431C2314yes—
69#423—22no—
70#423—22no—
71#423—22no—
72#423—22no—
73#423C2314yes—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×#100—22no—
77C2×#100—22no—
78C2×#100—44no—
79#407—22no—
80C4×M16—22no—
81#390—21no—
82#390—21no—
83#389—21no—
84#389—21no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#387—22no—
90#387—22no—
91#386—22no—
92#386—22no—
93#385—22no—
94#385—22no—
95#383C2×C412yes—
96#382C2311yescharacteristic
97#382C2311yescharacteristic
98#381—21no—
99#381C2×C412yes—
100#381C2×C412yes—
101#380—22no—
102#380—21no—
103#380C2×C412yes—
104#380C2×C412yes—
105#379—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2311yescharacteristic
111C4×C4⋊C4C2311yescharacteristic
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4C2×C412yes—
120C4×C22⋊C4C2×C411yescharacteristic
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4C2×C411yescharacteristic
123#376—21no—
124#376C2×C411yescharacteristic
125#376C2×C411yescharacteristic
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C43—21no—
129#343—22no—
130#343—22no—
131#323—24no—
132#323—24no—
133C25C22⋊C412yes—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—42no—
139C22×D4C22⋊C412yes—
140C22×D4C2×D411yescharacteristic
141C22×D4C2×D411yescharacteristic
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—84no—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C412yes—
151C23×C4C22⋊C412yes—
152C23×C4C22×C412yes—
153C23×C4C22×C412yes—
154C23×C4C22×C411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C2×M16—24no—
158C2×M16—42no—
159C2×M16—42no—
160C2×M16—44no—
161C2×M16—44no—
162C22×C8—42no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171C4×D4—22no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—44no—
176C4×D4—42no—
177C4×D4—42no—
178#117—44no—
179#117—84no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C22⋊C4—21no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—48no—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C2×D412yes—
206C2×C22⋊C4C2×D412yes—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—84no—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4112yes—
221C2×C42#4112yes—
222C2×C42#4112yes—
223C2×C42#4112yes—
224C2×C42C2411yescharacteristic
225C4⋊C8—24no—
226C4⋊C8—24no—
227C4⋊C8—24no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8—42no—
231#100—44no—
232#100—44no—
233C22⋊C8—24no—
234C22⋊C8—42no—
235#97—42no—
236C4×C8—24no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—21no—
241#95—21no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—21no—
246#95—21no—
247#95C22×C411yescharacteristic
248#95C22×C411yescharacteristic
249#95C22×C412yes—
250#95C22×C412yes—
251#95C22×C411yescharacteristic
252#95C22×C411yescharacteristic
253C24—22no—
254C24—22no—
255C24—24no—
256C24—42no—
257C24—44no—
258C24—82no—
259C24C2×C22⋊C411yescharacteristic
260C24C2×C22⋊C412yes—
261C24C2×C22⋊C412yes—
262C2×D4—24no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—42no—
275C2×D4AGL(1,ℤ/8)12yes—
276C2×D4#13712yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—84no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4—82no—
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4#11711yescharacteristic
319C22×C4#11711yescharacteristic
320C22×C4#11711yescharacteristic
321C22×C4#11711yescharacteristic
322C22×C4#12312yes—
323C22×C4#12312yes—
324C22×C4#12312yes—
325C22×C4#12312yes—
326C22×C4C23×C411yescharacteristic
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329M16—44no—
330M16—44no—
331M16—84no—
332M16—84no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—84no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—24no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—24no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—42no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C42—42no—
360C42—44no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—41no—
365C42—41no—
366C42—42no—
367C42—82no—
368C42—82no—
369C42#11712yes—
370C42#11712yes—
371C42#12312yes—
372C42#12312yes—
373C42#12312yes—
374C42#12312yes—
375C42C2×#4112yes—
376C42C2×#4112yes—
377C23—22no—
378C23—21no—
379C23—22no—
380C23—21no—
381C23—22no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—41no—
386C23—44no—
387C23—44no—
388C23—42no—
389C23—42no—
390C23—44no—
391C23—42no—
392C23—42no—
393C23—44no—
394C23—82no—
395C23—82no—
396C23—88no—
397C23—88no—
398C23#38111yescharacteristic
399C23#38111yescharacteristic
400C23#41714yes—
401C23C22×C22⋊C411yescharacteristic
402D4—24no—
403D4—24no—
404D4—48no—
405D4—42no—
406D4—42no—
407C2×C4—24no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—88no—
456C2×C4—88no—
457C2×C4#38112yes—
458C2×C4#38112yes—
459C2×C4#38111yescharacteristic
460C2×C4#38111yescharacteristic
461C2×C4C2×#11711yescharacteristic
462C2×C4C2×#11711yescharacteristic
463C2×C4C2×#12312yes—
464C2×C4C2×#12312yes—
465C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
466C2×C4C2×#13711yescharacteristic
467C2×C4#57512yes—
468C8—84no—
469C8—84no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—24no—
476C22—21no—
477C22—21no—
478C22—42no—
479C22—42no—
480C22—44no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—88no—
488C22—82no—
489C22—82no—
490C22—88no—
491C22#182712yes—
492C22C2×#38111yescharacteristic
493C22C2×#41711yescharacteristic
494C22#289611yescharacteristic
495C4—22no—
496C4—22no—
497C4—22no—
498C4—21no—
499C4—21no—
500C4—21no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—82no—
508C4—82no—
509C4#182912yes—
510C4#309412yes—
511C4#309512yes—
512C2—21no—
513C2—21no—
514C2—22no—
515C2—41no—
516C2—44no—
517C2—82no—
518C2#2070211yescharacteristic
519C2C2×#182711yescharacteristic
520C2C2×#182911yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8771C2semidirect
5. #8770C2semidirect
6. #8765C2semidirect
7. #8764C2semidirect
8. #8097C2semidirect
9. #7962C2semidirect
39. #1303C22semidirect
40. #1303C22semidirect
41. #1286C22semidirect
42. #1286C22semidirect

Maximal subgroups

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

subgroupindex
#341262
C2×#18002
#173642
#87712
#87702
#87652
#87642
#80972
#79622