Finite Group Databasebeta

← #191027

Subgroups — #191027

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 471 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3#13757C211yesmaximal, characteristic
4#8717C212yesmaximal
5#8574C214yesmaximal
6#8333C212yesmaximal
7#8219C214yesmaximal
8C2×#533C2211yescharacteristic
9C4×C4⋊Q8C411yescharacteristic, complemented
10C4×C4⋊Q8C411yescharacteristic, complemented
11C4×C4⋊Q8C411yescharacteristic, complemented
12C4×C4⋊Q8C411yescharacteristic, complemented
13C4×#125—21no—
14C4×#125—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8C2212yes—
18C4×C22⋊Q8C2212yes—
19#2296—21no—
20#2295—21no—
21#2295C412yescomplemented
22#2285C2211yescharacteristic
23C42×Q8—21no—
24C42×Q8C412yescomplemented
25C4×#117—21no—
26C2×#376—21no—
27C2×C43C2211yescharacteristic
28#1839—21no—
29#1831—21no—
30#1817C2212yes—
31#1797C2212yes—
32#1767—21no—
33#1749C2212yes—
34#1745—21no—
35C2×#328C2218yes—
36#1434—21no—
37#1434—21no—
38#1434C2212yes—
39#1434C2212yes—
40C4×#95—21no—
41C4×#95C2212yes—
42#1294C2218yes—
43#533—28no—
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×#125—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8C2312yes—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×#117—21no—
56C2×#117—21no—
57C22×C42—22no—
58C22×C42D411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×C4⋊C8—22no—
66C2×C22⋊C8C2314yes—
67C2×#97—22no—
68#391—21no—
69#391—21no—
70#391C2×C412yes—
71#391C2×C412yes—
72#389—21no—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389C2×C412yes—
79#389C2×C412yes—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#385—22no—
85#385—22no—
86#384C2×C411yescharacteristic
87#384C2×C411yescharacteristic
88#382—21no—
89#380—21no—
90#380C2312yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4C2×C412yes—
106C4×C4⋊C4C2×C412yes—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109#376—28no—
110#376—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95C2314yes—
117C43—22no—
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#328—216no—
131C22×Q8—21no—
132C22×Q8C22×C412yes—
133C23×C4—24no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—24no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144#125—28no—
145C22⋊Q8—28no—
146C22⋊Q8—24no—
147C22⋊Q8#4118yes—
148C4×Q8—28no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—44no—
152C4×Q8—44no—
153C4×Q8—44no—
154C4×Q8—44no—
155#117—28no—
156#117—44no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—28no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—44no—
190C2×C42—44no—
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22⋊C411yescharacteristic
198C2×C42C22⋊C411yescharacteristic
199C2×C42C22⋊C411yescharacteristic
200C2×C42C22⋊C411yescharacteristic
201C2×C42C22⋊C411yescharacteristic
202C2×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C4⋊C8—44no—
220C22⋊C8—28no—
221#97—44no—
222#95—28no—
223#95—28no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—28no—
235#95—44no—
236#95—44no—
237#95—44no—
238C24C22×D411yescharacteristic
239C2×Q8—24no—
240C2×Q8—22no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8#11714yes—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—28no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4#13714yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C2×#4114yes—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—28no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—28no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4#11714yes—
312C22⋊C4—28no—
313C22⋊C4—44no—
314C42—28no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—416no—
343C42—44no—
344C42—44no—
345C42—44no—
346C42—44no—
347C42—44no—
348C42#10414yes—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23C22×C22⋊C411yescharacteristic
354C23#52512yes—
355C23C2×#13712yes—
356C23#57512yes—
357Q8—44no—
358Q8—48no—
359Q8—44no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—48no—
406C2×C4—48no—
407C2×C4—48no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4#41012yes—
419C2×C4#41012yes—
420C2×C4#41912yes—
421C2×C4#41912yes—
422C2×C4C2×#10412yes—
423C2×C4C2×#10412yes—
424C2×C4C2×#10412yes—
425C2×C4#42112yes—
426C2×C4C2×#11712yes—
427C2×C4#52512yes—
428C8—84no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22#228011yescharacteristic
436C22C2×#41011yescharacteristic
437C22C2×#41011yescharacteristic
438C22C2×#41911yescharacteristic
439C22#289611yescharacteristic
440C22C22×#10411yescharacteristic
441C22C2×#42111yescharacteristic
442C22#311714yes—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—22no—
453C4—41no—
454C4—44no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—44no—
459C4—88no—
460C4—82no—
461C4—82no—
462C2—24no—
463C2#1754111yescharacteristic
464C2#1754511yescharacteristic
465C2#1758411yescharacteristic
466C2#1758611yescharacteristic
467C2#2072311yescharacteristic
468C2#2074111yescharacteristic
469C2#2074711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. C4×C4⋊Q8C4semidirect
10. C4×C4⋊Q8C4semidirect
11. C4×C4⋊Q8C4semidirect
12. C4×C4⋊Q8C4semidirect
21. #2295C4semidirect
24. C42×Q8C4semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
#207782
#137572
#87172
#85742
#83332
#82192