Finite Group Databasebeta

← #190278

Subgroups — #190278

Order 512.

971 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34123C211yesmaximal, characteristic
2#20637C211yesmaximal, characteristic
3#17366C211yesmaximal, characteristic
4#8769C212yesmaximal, complemented
5#8768C212yesmaximal, complemented
6#8765C212yesmaximal, complemented
7#8764C212yesmaximal, complemented
8#8097C212yesmaximal, complemented
9#7960C212yesmaximal, complemented
10C2×C4×#41C2211yescharacteristic
11#2920C2212yes—
12C2×#423C2212yes—
13C2×#413—21no—
14#2356C2211yescharacteristic
15#2356C2211yescharacteristic
16#2353C412yes—
17#2353C412yes—
18#2349C2211yescharacteristic
19#2349C2211yescharacteristic
20#2346—21no—
21#2343—22no—
22#2339—21no—
23#2295—21no—
24#2290—22no—
25#2283C412yes—
26#2283C2211yescharacteristic
27C42×D4—21no—
28C4×#117C2211yescharacteristic
29C2×#376C412yes—
30#1801—21no—
31#1801C2212yes—
32#1800—21no—
33#1800C2212yes—
34#1742—21no—
35#1465—22no—
36#1464—22no—
37#1461C2212yes—
38#1460C2212yes—
39#1303C2214yes—
40#1303C2214yes—
41#1286C2214yes—
42#1286C2214yes—
43C22×#41D411yescharacteristic
44C22×M16—22no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41—22no—
48C4×#41—22no—
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8C2311yescharacteristic
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4D412yes—
55C2×C4×D4D411yescharacteristic
56C2×C4×D4C2311yescharacteristic
57C2×#117—21no—
58C2×#117D411yescharacteristic
59C2×#117D412yes—
60C22×C42—22no—
61C22×C42C2×C412yes—
62C22×C42C2311yescharacteristic
63#432—22no—
64#432—22no—
65#431—22no—
66#431C2314yes—
67#423—22no—
68#423—22no—
69#423—22no—
70#423—22no—
71#423C2314yes—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74#413—44no—
75C2×#101—22no—
76C2×#100—22no—
77#407—22no—
78C4×M16—22no—
79#391—21no—
80#391—21no—
81#389—21no—
82#389—21no—
83#388—22no—
84#388—22no—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#386—22no—
90#386—22no—
91#385—22no—
92#385—22no—
93#383—21no—
94#383C2×C412yes—
95#383C2×C412yes—
96#382—22no—
97#382C2311yescharacteristic
98#382C2311yescharacteristic
99#381—22no—
100#381C2×C412yes—
101#380—21no—
102#380C2×C412yes—
103#380C2×C412yes—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2311yescharacteristic
110C4×C4⋊C4C2311yescharacteristic
111C4×C22⋊C4—22no—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4C2×C411yescharacteristic
117C4×C22⋊C4C2×C411yescharacteristic
118#376—22no—
119#376—22no—
120#376C2×C412yes—
121#376C2×C412yes—
122#376C2×C411yescharacteristic
123#376C2×C412yes—
124#376C2×C411yescharacteristic
125C2×#95—22no—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C43—21no—
129#343—22no—
130#343—22no—
131#324—24no—
132#323—24no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—42no—
138C22×Q8C2×D411yescharacteristic
139C22×D4C22⋊C412yes—
140C22×D4C2×D411yescharacteristic
141C23×C4—22no—
142C23×C4—22no—
143C23×C4C22⋊C412yes—
144C23×C4C22⋊C412yes—
145C23×C4C22×C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—24no—
150C2×M16—42no—
151C2×M16—42no—
152C2×M16—44no—
153C2×M16—44no—
154C22×C8—42no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—44no—
168C4×D4—42no—
169C4×D4—42no—
170#117—22no—
171#117—22no—
172#117—24no—
173#117—24no—
174#117—44no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C22⋊C4—21no—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22⋊C412yes—
192C2×C22⋊C4C2×D412yes—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—48no—
207C2×C42C22⋊C412yes—
208C2×C42C22⋊C412yes—
209C2×C42C22×C412yes—
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4112yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218C4⋊C8—24no—
219C4⋊C8—24no—
220C4⋊C8—24no—
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224#101—44no—
225#100—44no—
226C22⋊C8—24no—
227C22⋊C8—42no—
228#97—42no—
229C4×C8—24no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—21no—
243#95—21no—
244#95C22×C411yescharacteristic
245#95C22×C411yescharacteristic
246#95C22×C412yes—
247#95C22×C412yes—
248#95C22×C411yescharacteristic
249#95C22×C411yescharacteristic
250C24C2×C22⋊C411yescharacteristic
251C24C2×C22⋊C412yes—
252#41—44no—
253#41—44no—
254#41—44no—
255#41—44no—
256C2×Q8—22no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—42no—
265C2×D4AGL(1,ℤ/8)12yes—
266C2×D4#13712yes—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—24no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—81no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4C2×C22⋊C412yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4#11712yes—
301C22×C4#11712yes—
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4#12312yes—
305C22×C4#12312yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309M16—44no—
310M16—44no—
311M16—84no—
312M16—84no—
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—84no—
316C4⋊C4—22no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C42—24no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—22no—
342C42—44no—
343C42—44no—
344C42—44no—
345C42—42no—
346C42—41no—
347C42—41no—
348C42—42no—
349C42—81no—
350C42—81no—
351C42#11711yescharacteristic
352C42#11711yescharacteristic
353C42#11711yescharacteristic
354C42#11711yescharacteristic
355C42#12312yes—
356C42#12312yes—
357C42#12312yes—
358C42#12312yes—
359C42C2×#4112yes—
360C42C2×#4112yes—
361C23—22no—
362C23—22no—
363C23—21no—
364C23—21no—
365C23—41no—
366C23—41no—
367C23—42no—
368C23—42no—
369C23#38112yes—
370C23#41714yes—
371C23C22×C22⋊C411yescharacteristic
372Q8—42no—
373Q8—42no—
374D4—24no—
375D4—24no—
376D4—48no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—44no—
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—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—88no—
423C2×C4—88no—
424C2×C4#38111yescharacteristic
425C2×C4#38111yescharacteristic
426C2×C4#38111yescharacteristic
427C2×C4#38112yes—
428C2×C4#38111yescharacteristic
429C2×C4C2×#11711yescharacteristic
430C2×C4C2×#11711yescharacteristic
431C2×C4C2×#12312yes—
432C2×C4C2×#12312yes—
433C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57512yes—
436C8—84no—
437C8—84no—
438C22—21no—
439C22—21no—
440C22—22no—
441C22—24no—
442C22—22no—
443C22—22no—
444C22—44no—
445C22—44no—
446C22—42no—
447C22—42no—
448C22—81no—
449C22—81no—
450C22C2×#38111yescharacteristic
451C22C2×#41711yescharacteristic
452C22#289611yescharacteristic
453C4—21no—
454C4—21no—
455C4—22no—
456C4—21no—
457C4—21no—
458C4—21no—
459C4—22no—
460C4—42no—
461C4—42no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—81no—
466C4—81no—
467C4—82no—
468C4—82no—
469C4#182712yes—
470C4#309412yes—
471C4#309512yes—
472C2—21no—
473C2—21no—
474C2—42no—
475C2—44no—
476C2#2070211yescharacteristic
477C2C2×#182711yescharacteristic
478C2#2071311yescharacteristic
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
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. #8769C2semidirect
5. #8768C2semidirect
6. #8765C2semidirect
7. #8764C2semidirect
8. #8097C2semidirect
9. #7960C2semidirect

Maximal subgroups

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

subgroupindex
#341232
#206372
#173662
#87692
#87682
#87652
#87642
#80972
#79602