Finite Group Databasebeta

← #220031

Subgroups — #220031

Order 512.

1197 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
1D4×C4⋊Q8C211yesmaximal, characteristic
2#22795C211yesmaximal, characteristic
3#22650C211yesmaximal, characteristic
4C2×#1548C212yesmaximal, complemented
5#9691C211yesmaximal, characteristic, complemented
6#9691C211yesmaximal, characteristic, complemented
7#9135C212yesmaximal, complemented
8#9128C212yesmaximal, complemented
9#8587C211yesmaximal, characteristic, complemented
10#8342C212yesmaximal, complemented
11C4×#333C211yesmaximal, characteristic, complemented
12C2×D4×Q8C412yes—
13C22×C4⋊Q8C2212yes—
14C2×#436—21no—
15C2×#431C2212yes—
16C2×#423C2212yes—
17#2868C2211yescharacteristic
18#2857C2212yes—
19#2551—21no—
20#2551—21no—
21#2546C412yes—
22#2440—22no—
23#2440—22no—
24#2434C414yes—
25D4×C4⋊C4—21no—
26D4×C4⋊C4—21no—
27C4×C4⋊Q8C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1939—21no—
31#1923—22no—
32#1851—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1797C2212yes—
36#1797C2212yes—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1548—28no—
41C2×#333C2211yescharacteristic
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#333C2212yes—
46C2×#333C2212yes—
47C2×#333C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2211yescharacteristic
50#1296C2212yes—
51C23×Q8C2×C414yes—
52C22×M16—22no—
53D4×Q8—48no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8C2×C414yes—
62C2×C4⋊Q8C2×C412yes—
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8C2×C414yes—
68C2×C4×Q8C2×C412yes—
69C2×C4×D4—22no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C42C2312yes—
76#436—44no—
77#431—24no—
78#431—42no—
79#423—24no—
80#423—42no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C4×C8—22no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98#398—24no—
99#395—24no—
100#395—24no—
101#391C2×C412yes—
102#391C2×C414yes—
103#389—22no—
104#389—22no—
105#389—21no—
106#389—21no—
107#389—22no—
108#389—22no—
109#385—24no—
110#385—24no—
111#384—21no—
112#384—21no—
113#384—21no—
114#384—21no—
115#384C2311yescharacteristic
116#380—22no—
117#380—22no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4C2×C412yes—
124C4×C22⋊C4D412yes—
125C4×C22⋊C4D412yes—
126C43C2311yescharacteristic
127#336—22no—
128#336—22no—
129#336—21no—
130#336—21no—
131#333—24no—
132#333—22no—
133#333—24no—
134#333—24no—
135#333—42no—
136#333—42no—
137#333D414yescomplemented
138C22×Q8—24no—
139C22×Q8—28no—
140C22×Q8—44no—
141C22×Q8—48no—
142C22×Q8—84no—
143C22×Q8C22×C414yes—
144C22×Q8C22×C412yes—
145C22×D4C22⋊C411yescharacteristic
146C23×C4—24no—
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D412yes—
150C2×M16—44no—
151C2×M16—44no—
152C2×M16—82no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C4⋊Q8—28no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170C4⋊Q8—44no—
171C4⋊Q8—44no—
172C4⋊Q8—48no—
173C4⋊Q8—82no—
174C4⋊Q8C2×D412yes—
175C4⋊Q8#4112yes—
176C22⋊Q8—416no—
177C22⋊Q8—48no—
178C4×Q8—44no—
179C4×D4—24no—
180C4×D4—22no—
181C4×D4—22no—
182C4×D4—48no—
183C4×D4SD1614yes—
184C4×D4Q1614yes—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4C22⋊C411yescharacteristic
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22×C412yes—
207C2×C22⋊C4—24no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C412yes—
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2411yescharacteristic
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—44no—
237C4⋊C8—42no—
238C4⋊C8—42no—
239C4⋊C8—81no—
240C4⋊C8—81no—
241C4⋊C8C2×D412yes—
242C4⋊C8C2×D412yes—
243C22⋊C8—44no—
244C22⋊C8—44no—
245C22⋊C8—82no—
246C4×C8—44no—
247C4×C8—42no—
248C4×C8—42no—
249#95—22no—
250#95—22no—
251#95—24no—
252#95—24no—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24C2×C22⋊C412yes—
257C2×Q8—22no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—48no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—44no—
264C2×Q8—44no—
265C2×Q8—48no—
266C2×Q8—84no—
267C2×Q8—816no—
268C2×Q8—816no—
269C2×Q8C4×D414yes—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4#10314yes—
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—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—42no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—84no—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C412yes—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×SD1614yes—
301C22×C4C2×Q1614yes—
302C22×C4C23×C411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305M16—88no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—41no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—44no—
343C4⋊C4—48no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4#10014yes—
347C4⋊C4#10314yes—
348C22⋊C4—24no—
349C22⋊C4—24no—
350C22⋊C4—48no—
351C22⋊C4#10318yes—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—21no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—22no—
363C42—41no—
364C42—44no—
365C42C2×SD1612yes—
366C42C2×Q1612yes—
367C42#13712yes—
368C42#13712yes—
369C42C22×D411yescharacteristic
370C42C2×#4111yescharacteristic
371C42H(5,𝔽2)11yescharacteristic
372C42H(5,𝔽2)11yescharacteristic
373C23—22no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23C2×#10314yes—
380C23C22×C22⋊C411yescharacteristic
381Q8—44no—
382Q8—44no—
383Q8—816no—
384Q8—84no—
385Q8—84no—
386D4—28no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—816no—
423C2×C4C2×#10014yes—
424C2×C4C2×#10012yes—
425C2×C4C2×#10312yes—
426C2×C4C2×#10314yes—
427C2×C4C2×#10312yes—
428C2×C4#41912yes—
429C2×C4#41912yes—
430C2×C4C2×C4×D411yescharacteristic
431C2×C4#51811yescharacteristic
432C2×C4D4211yescharacteristic
433C2×C4D4211yescharacteristic
434C2×C4#54611yescharacteristic
435C2×C4#54611yescharacteristic
436C2×C4C22×SD1611yescharacteristic
437C2×C4C22×Q1611yescharacteristic
438C2×C4C2×#13711yescharacteristic
439C2×C4C2×#13711yescharacteristic
440C2×C4#57711yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57811yescharacteristic
443C2×C4#57811yescharacteristic
444C8—42no—
445C8—42no—
446C8—81no—
447C8—82no—
448C8—82no—
449C8—81no—
450C22—24no—
451C22—24no—
452C22—24no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22D4×C22⋊C411yescharacteristic
458C22C22×#10011yescharacteristic
459C22#288811yescharacteristic
460C22C22×#10311yescharacteristic
461C22C2×#41911yescharacteristic
462C22#289811yescharacteristic
463C22#289811yescharacteristic
464C4—22no—
465C4—22no—
466C4—21no—
467C4—21no—
468C4—24no—
469C4—24no—
470C4—41no—
471C4—44no—
472C4—42no—
473C4—82no—
474C4—84no—
475C4#154814yescomplemented
476C4#327611yescharacteristic
477C4#327811yescharacteristic
478C4D4×SD1611yescharacteristic
479C4D4×Q1611yescharacteristic
480C4#329811yescharacteristic
481C4#329911yescharacteristic
482C4#332811yescharacteristic
483C4#332911yescharacteristic
484C2—44no—
485C2C2×#154811yescharacteristic
486C2#1772211yescharacteristic
487C2#1776011yescharacteristic
488C2#1777311yescharacteristic
489C2D4×#10011yescharacteristic
490C2D4×#10311yescharacteristic
491C2#2243011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1548C2semidirect
5. #9691C2semidirect
6. #9691C2semidirect
7. #9135C2semidirect
8. #9128C2semidirect
9. #8587C2semidirect
10. #8342C2semidirect
11. C4×#333C2semidirect
137. #333D4semidirect
475. C4#1548semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#227952
#226502
C2×#15482
#96912
#91352
#91282
#85872
#83422
C4×#3332