Finite Group Databasebeta

← #190862

Subgroups — #190862

Order 512.

1286 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2293C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic, complemented
3#20777C211yesmaximal, characteristic, complemented
4#8707C212yesmaximal, complemented
5#8567C214yesmaximal, complemented
6#8310C212yesmaximal
7#8179C214yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9C22×#117—21no—
10C2×#432—21no—
11#2293—21no—
12#2293—21no—
13#2293—21no—
14#2293—21no—
15#2293C412yes—
16#2293C412yes—
17#2293C2212yes—
18#2293C2212yes—
19C2×#390C411yescharacteristic
20C2×#390C411yescharacteristic
21C2×#389C411yescharacteristic
22C2×#389C411yescharacteristic
23C2×#388—21no—
24C2×#388—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C22⋊C4—21no—
27C2×C43C2211yescharacteristic
28#1842C2212yescomplemented
29#1839—21no—
30#1831—21no—
31#1831—21no—
32#1817—21no—
33#1797—21no—
34#1797—21no—
35#1797C2212yes—
36#1767—21no—
37#1767C2212yescomplemented
38C4×C4⋊C8C2212yescomplemented
39C4×C22⋊C8C2212yes—
40#1744C2212yescomplemented
41C2×D4⋊C8C2218yes—
42#1279C2218yescomplemented
43C23×D4C2×C411yescharacteristic
44C24×C4—21no—
45C2×C4×D4—28no—
46C2×C4×D4—21no—
47C2×C4×D4—21no—
48C2×C4×D4C2×C412yes—
49C2×C4×D4C2312yes—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—48no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C42—21no—
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42C2311yescharacteristic
73#432—44no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8C2314yes—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—22no—
81C2×C4×C8C2314yes—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390C2×C412yes—
87#390C2×C412yes—
88#390C2×C412yes—
89#390C2×C412yes—
90#389—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
97#389C2×C412yes—
98#388—22no—
99#388—22no—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2312yes—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95—21no—
117C43C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43C2311yescharacteristic
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131D4⋊C8—216no—
132C25—21no—
133C22×D4—24no—
134C22×D4—21no—
135C22×D4—24no—
136C22×D4—42no—
137C22×D4C22×C412yes—
138C23×C4—28no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—44no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C22×C8—24no—
155C22×C8—24no—
156C4×D4—28no—
157C4×D4—28no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4#4118yes—
162#117—48no—
163#117—48no—
164#117—48no—
165#117—48no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—44no—
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2411yescharacteristic
235C4⋊C8—28no—
236C4⋊C8—44no—
237C4⋊C8—44no—
238C4⋊C8—44no—
239C22⋊C8—44no—
240C22⋊C8—44no—
241#97—44no—
242C4×C8—28no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249C24—24no—
250C24—22no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24C2×C22⋊C411yescharacteristic
258C2×D4—24no—
259C2×D4—22no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—416no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4#11714yes—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—28no—
272C22×C4—22no—
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—28no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—44no—
292C22×C4—416no—
293C22×C4—416no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4#13114yes—
318C22×C4AGL(1,ℤ/8)12yes—
319C22×C4#13712yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—44no—
328C2×C8—44no—
329C4⋊C4—24no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—44no—
339C4⋊C4#11714yes—
340C22⋊C4—28no—
341C22⋊C4—28no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—21no—
358C42—21no—
359C42—22no—
360C42—21no—
361C42—21no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—44no—
379C42—44no—
380C42AGL(1,ℤ/8)14yes—
381C42C2×#4114yes—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—24no—
387C23—48no—
388C23—48no—
389C23—48no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—84no—
397C23—84no—
398C23—84no—
399C23—84no—
400C23#40812yes—
401C23#41712yes—
402C23#42112yes—
403C23C22×C22⋊C411yescharacteristic
404D4—28no—
405D4—48no—
406D4—44no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—28no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—24no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—416no—
446C2×C4—48no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—48no—
451C2×C4—48no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4#40812yes—
461C2×C4#41712yes—
462C2×C4#41812yes—
463C2×C4#41812yes—
464C2×C4#42112yes—
465C2×C4C2×#11712yes—
466C2×C4#52512yes—
467C2×C4#52512yes—
468C2×C4C2×#13112yes—
469C2×C4#56812yes—
470C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
471C2×C4C2×AGL(1,ℤ/8)12yes—
472C2×C4C2×#13711yescharacteristic
473C8—84no—
474C8—84no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—816no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22#149514yes—
490C22#228011yescharacteristic
491C22C2×#40811yescharacteristic
492C22#287911yescharacteristic
493C22C2×#41811yescharacteristic
494C22C2×#41711yescharacteristic
495C22C2×#42111yescharacteristic
496C22C2×#42111yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—44no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—82no—
511C4—82no—
512C4#292414yes—
513C4#310912yes—
514C4#311012yes—
515C2—24no—
516C2—84no—
517C2C2×#149511yescharacteristic
518C2#1747811yescharacteristic
519C2#1749211yescharacteristic
520C2#1749511yescharacteristic
521C2#2071711yescharacteristic
522C2#2073811yescharacteristic
523C2#2074811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20810C2semidirect
3. #20777C2semidirect
4. #8707C2semidirect
5. #8567C2semidirect
7. #8179C2semidirect
28. #1842C22semidirect
37. #1767C22semidirect
38. C4×C4⋊C8C22semidirect
40. #1744C22semidirect
42. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
#208102
#207772
#87072
#85672
#83102
#81792