Finite Group Databasebeta

← #98224

Subgroups — #98224

Order 512.

945 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic
2#9898C211yesmaximal, characteristic, complemented
3#9692C211yesmaximal, characteristic, complemented
4#8586C211yesmaximal, characteristic, complemented
5#8563C211yesmaximal, characteristic
6#8563C211yesmaximal, characteristic
7#7570C211yesmaximal, characteristic, complemented
8C2×D4×Q8—21no—
9C22×C4⋊Q8—21no—
10#2868C2211yescharacteristic
11#2857—21no—
12#2551—21no—
13#2551C412yes—
14#2546—21no—
15#2440—22no—
16#2440—22no—
17#2434—22no—
18D4×C4⋊C4—21no—
19D4×C4⋊C4C412yes—
20C4×C4⋊Q8C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#1842C2211yescharacteristic
23#1842C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26C2×#332—21no—
27C2×#332—22no—
28C2×D4⋊C8—21no—
29C2×D4⋊C8—21no—
30#1296—21no—
31#1296—22no—
32#1279—21no—
33#1279—21no—
34#1271—22no—
35C23×Q8—22no—
36D4×Q8—42no—
37D4×Q8—42no—
38D4×Q8—42no—
39D4×Q8—42no—
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—42no—
47C2×C4⋊Q8—42no—
48C2×C4⋊Q8—42no—
49C2×C4⋊Q8—42no—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D412yes—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—24no—
54C2×C4×Q8—21no—
55C2×C4×D4—22no—
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C42—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8—21no—
72#398—24no—
73#395—24no—
74#395—24no—
75#391—22no—
76#391—21no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—21no—
82#389D412yes—
83#385—24no—
84#385—24no—
85#384—21no—
86#384—21no—
87#384C2×C412yes—
88#384D411yescharacteristic
89#384D412yes—
90#380—22no—
91#380—22no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4C2×C411yescharacteristic
95C4×C4⋊C4D411yescharacteristic
96C4×C4⋊C4D411yescharacteristic
97C4×C22⋊C4—21no—
98C4×C22⋊C4—21no—
99C4×C22⋊C4—21no—
100C43C2311yescharacteristic
101#336—21no—
102#336—21no—
103#332—44no—
104#332—42no—
105D4⋊C8—42no—
106D4⋊C8—42no—
107C22×Q8—24no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—44no—
111C22×Q8—44no—
112C22×Q8—44no—
113C22×Q8—44no—
114C22×Q8—82no—
115C22×Q8—82no—
116C22×D4C22⋊C411yescharacteristic
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—24no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—41no—
124C22×C8—41no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8—42no—
128C4⋊Q8—42no—
129C4⋊Q8—44no—
130C4⋊Q8—44no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—44no—
134C4⋊Q8—44no—
135C4⋊Q8—44no—
136C4⋊Q8—44no—
137C4⋊Q8—41no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—41no—
141C4⋊Q8—41no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144C4⋊Q8—41no—
145C4⋊Q8D812yes—
146C4⋊Q8SD1612yes—
147C22⋊Q8—48no—
148C22⋊Q8—48no—
149C22⋊Q8—44no—
150C22⋊Q8—44no—
151C4×Q8—41no—
152C4×Q8—41no—
153C4×Q8—41no—
154C4×Q8—41no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—21no—
158C4×D4—21no—
159C4×D4—21no—
160C4×D4—21no—
161C4×D4—44no—
162C4×D4—44no—
163C4×D4D812yes—
164C4×D4SD1612yes—
165C4×D4SD1612yes—
166C4×D4Q1612yes—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C22⋊C414yes—
188C2×C4⋊C4C22⋊C412yes—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—24no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—41no—
211C4⋊C8—41no—
212C4⋊C8—42no—
213C4⋊C8—42no—
214C4⋊C8—41no—
215C4⋊C8—41no—
216C4⋊C8—41no—
217C4⋊C8—41no—
218C4⋊C8—41no—
219C4⋊C8—41no—
220C4⋊C8—41no—
221C4⋊C8—41no—
222C4×C8—41no—
223C4×C8—41no—
224C4×C8—41no—
225C4×C8—41no—
226#95—24no—
227#95—24no—
228#95—21no—
229#95—22no—
230#95—21no—
231#95—21no—
232#95—22no—
233C24—21no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—41no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—41no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×Q8—84no—
249C2×Q8—88no—
250C2×Q8—88no—
251C2×Q8—88no—
252C2×Q8—88no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4#10112yes—
257C2×D4#10212yes—
258C2×D4#10312yes—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—42no—
271C22×C4—48no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—48no—
283C22×C4—84no—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—81no—
294C2×C8—81no—
295C2×C8—81no—
296C2×C8—81no—
297C2×C8—81no—
298C2×C8—81no—
299C2×C8—81no—
300C2×C8—81no—
301C2×C8—81no—
302C2×C8—81no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—48no—
312C4⋊C4—48no—
313C4⋊C4—48no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—82no—
329C4⋊C4#10112yes—
330C4⋊C4#10214yes—
331C4⋊C4#10212yes—
332C4⋊C4#10312yes—
333C22⋊C4—24no—
334C22⋊C4—22no—
335C22⋊C4—22no—
336C22⋊C4—22no—
337C22⋊C4—22no—
338C22⋊C4—48no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—41no—
346C42—41no—
347C42—42no—
348C42—41no—
349C42—42no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42C2×D811yescharacteristic
355C42C2×D811yescharacteristic
356C42C2×SD1611yescharacteristic
357C42C2×SD1611yescharacteristic
358C42C2×SD1611yescharacteristic
359C42C2×Q1611yescharacteristic
360C42AGL(1,ℤ/8)11yescharacteristic
361C42AGL(1,ℤ/8)11yescharacteristic
362C42AGL(1,ℤ/8)11yescharacteristic
363C42AGL(1,ℤ/8)11yescharacteristic
364C42#13711yescharacteristic
365C42#13711yescharacteristic
366C23—22no—
367C23—22no—
368C23—22no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23#37911yescharacteristic
374Q8—42no—
375Q8—42no—
376Q8—42no—
377Q8—42no—
378Q8—816no—
379Q8—82no—
380Q8—82no—
381Q8—82no—
382Q8—82no—
383D4—22no—
384D4—22no—
385D4—22no—
386D4#33214yescomplemented
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—816no—
425C2×C4C2×#10111yescharacteristic
426C2×C4C2×#10111yescharacteristic
427C2×C4C2×#10211yescharacteristic
428C2×C4C2×#10211yescharacteristic
429C2×C4C2×#10311yescharacteristic
430C2×C4#41811yescharacteristic
431C2×C4#41811yescharacteristic
432C2×C4#41911yescharacteristic
433C2×C4#44711yescharacteristic
434C2×C4#44711yescharacteristic
435C2×C4#44711yescharacteristic
436C2×C4#44711yescharacteristic
437C2×C4#44811yescharacteristic
438C2×C4#44811yescharacteristic
439C2×C4#45011yescharacteristic
440C2×C4#45011yescharacteristic
441C2×C4#45011yescharacteristic
442C2×C4#45011yescharacteristic
443C2×C4#45111yescharacteristic
444C2×C4#45111yescharacteristic
445C2×C4#45311yescharacteristic
446C2×C4#45411yescharacteristic
447C2×C4#45511yescharacteristic
448C2×C4#45611yescharacteristic
449C8—82no—
450C8—82no—
451C8—161no—
452C8—161no—
453C22—24no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22#178411yescharacteristic
461C22#178611yescharacteristic
462C22#178611yescharacteristic
463C22#178711yescharacteristic
464C22#179211yescharacteristic
465C22#179311yescharacteristic
466C22#179611yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—41no—
474C4—42no—
475C4—42no—
476C4—44no—
477C4—42no—
478C4—82no—
479C4—84no—
480C4C2×#33211yescharacteristic
481C4#154611yescharacteristic
482C4#154911yescharacteristic
483C4#155611yescharacteristic
484C4#161911yescharacteristic
485C4#162711yescharacteristic
486C4#162911yescharacteristic
487C4#163711yescharacteristic
488C4#163911yescharacteristic
489C4#163911yescharacteristic
490C4#164911yescharacteristic
491C4#165111yescharacteristic
492C2—44no—
493C2#834011yescharacteristic
494C2#838811yescharacteristic
495C2#842111yescharacteristic
496C2#842511yescharacteristic
497C2#844311yescharacteristic
498C2#844711yescharacteristic
499C2#847311yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
prime radicalC22×D4C22⋊C4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #9898C2semidirect
3. #9692C2semidirect
4. #8586C2semidirect
7. #7570C2semidirect
386. D4#332semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#98982
#96922
#85862
#85632
#75702