Finite Group Databasebeta

← #192391

Subgroups — #192391

Order 512.

1034 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34074C211yesmaximal, characteristic
2#21982C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#14277C211yesmaximal, characteristic
6#8868C214yesmaximal
7#8867C214yesmaximal
8#8026C211yesmaximal, characteristic
9#7906C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2331—21no—
14#2331—21no—
15#2330—21no—
16#2330—21no—
17#2329—21no—
18#2329—21no—
19#2329C2212yes—
20#2329C2212yes—
21#2296—21no—
22#2295—22no—
23#2295—22no—
24C2×#389—21no—
25#2285C2211yescharacteristic
26C2×#384C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C4×#117—21no—
29C2×#376C2211yescharacteristic
30#1990—21no—
31#1956—21no—
32#1947—22no—
33#1872C2214yes—
34#1852C2214yes—
35#1845—21no—
36#1834—21no—
37#1803—22no—
38#1518C418yescomplemented
39C2×#328C2214yes—
40#1444—21no—
41#1444C2212yes—
42#1442—21no—
43#1442C2212yes—
44C4⋊#95—21no—
45#1294C2214yes—
46#1280C2212yes—
47#1280C2212yes—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533Q814yes—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8D411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2312yes—
57C2×C4×Q8—21no—
58C2×C4×Q8D412yes—
59C2×#117—21no—
60C2×#117—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4C2312yes—
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69#431—22no—
70#431—22no—
71#431—22no—
72#431C2×C414yes—
73C2×#107—24no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×#103—24no—
77C2×C22⋊C8C2312yes—
78C2×C22⋊C8C2312yes—
79C2×C4×C8—21no—
80C2×C4×C8—21no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391—22no—
85#389—22no—
86#389—22no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—22no—
93#389—44no—
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#384—21no—
105#384—21no—
106#384—21no—
107#384—21no—
108#384D411yescharacteristic
109#384D411yescharacteristic
110#382—21no—
111#382—22no—
112#382C2×C414yes—
113#380—21no—
114#380C2312yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4D412yes—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125#376—22no—
126#376—21no—
127#376—21no—
128#376—21no—
129#376—21no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D411yescharacteristic
135C2×#95D411yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2312yes—
139C43—22no—
140#340—24no—
141#340—24no—
142#328C2×C4116yes—
143C22×Q8—21no—
144C22×Q8C2×D412yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8C4⋊C414yescomplemented
159C4⋊Q8C4⋊C414yescomplemented
160#125—24no—
161#125—24no—
162C22⋊Q8—24no—
163C22⋊Q8—24no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166C22⋊Q8C22×C414yes—
167C22⋊Q8C2×Q814yes—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8C4⋊C418yescomplemented
172#117—24no—
173#117—24no—
174#117—42no—
175#117—42no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4—82no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4—82no—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C42—22no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42Q1614yes—
224C2×C42C22×C412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×Q812yes—
230C2×C42#4112yes—
231C2×C42#4112yes—
232#107—84no—
233C4⋊C8—24no—
234C4⋊C8—42no—
235#103—84no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8C22×C414yes—
240C4×C8—24no—
241C4×C8—42no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—24no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—22no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—22no—
261#95—22no—
262#95—44no—
263#95—82no—
264#95—82no—
265#95C22×C414yes—
266C24C22×D411yescharacteristic
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—81no—
272C2×Q8—81no—
273C2×Q8C2×C4⋊C414yes—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4C4×D412yes—
315C22×C4C4×D412yes—
316C22×C4#12011yescharacteristic
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4C22⋊Q812yes—
321C22×C4C22⋊Q812yes—
322C22×C4C2×Q1612yes—
323C22×C4C2×Q1612yes—
324C22×C4C2×Q1612yes—
325C22×C4#13512yes—
326C22×C4AGL(1,ℤ/8)12yes—
327C22×C4#13712yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×Q811yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C2×C8—24no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—28no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4—48no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—88no—
363C4⋊C4—82no—
364C4⋊C4—82no—
365C4⋊C4C2×C4⋊C414yes—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C42—28no—
371C42—24no—
372C42—24no—
373C42—42no—
374C42—42no—
375C42—44no—
376C42—82no—
377C42—82no—
378C42—82no—
379C42—82no—
380C42C2×C4⋊C412yes—
381C42C2×C4⋊C412yes—
382C42C4×D412yes—
383C42C4×D412yes—
384C42C22⋊Q812yes—
385C42C22⋊Q812yes—
386C42#12312yes—
387C42#12312yes—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23C2×C4×D411yescharacteristic
393C23C2×#12011yescharacteristic
394C23C2×C22⋊Q811yescharacteristic
395C23C22×Q1611yescharacteristic
396C23C2×#13511yescharacteristic
397C23C2×AGL(1,ℤ/8)11yescharacteristic
398C23C2×#13711yescharacteristic
399Q8—44no—
400Q8—82no—
401Q8—84no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—42no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4#38512yes—
463C2×C4#38512yes—
464C2×C4#38512yes—
465C2×C4#38512yes—
466C2×C4C4×Q1614yes—
467C2×C4#44014yes—
468C2×C4#44912yes—
469C2×C4#44912yes—
470C2×C4#45112yes—
471C2×C4#45112yes—
472C2×C4#47714yes—
473C2×C4#47914yes—
474C2×C4C22×C4⋊C411yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C2×C22⋊Q811yescharacteristic
477C2×C4C2×#12311yescharacteristic
478C8—44no—
479C8—82no—
480C22—21no—
481C22—21no—
482C22—21no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—21no—
487C22—21no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22C2×#38511yescharacteristic
493C22C2×C4×Q1611yescharacteristic
494C22C2×#44011yescharacteristic
495C22C2×#45111yescharacteristic
496C22C2×#44911yescharacteristic
497C22#302111yescharacteristic
498C22#302211yescharacteristic
499C22#302711yescharacteristic
500C22#302811yescharacteristic
501C22C2×#47711yescharacteristic
502C22C2×#47911yescharacteristic
503C4—22no—
504C4—22no—
505C4—21no—
506C4—21no—
507C4—21no—
508C4—21no—
509C4—42no—
510C4—44no—
511C4—42no—
512C4—44no—
513C4—84no—
514C4—82no—
515C4—81no—
516C4—82no—
517C4—81no—
518C4—162no—
519C4#186314yes—
520C4#186314yes—
521C2—21no—
522C2—21no—
523C2—21no—
524C2—21no—
525C2#1830311yescharacteristic
526C2#1830411yescharacteristic
527C2#1831111yescharacteristic
528C2#1831211yescharacteristic
529C2#2086511yescharacteristic
530C2C2×#186311yescharacteristic
531C2#2089011yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
38. #1518C4semidirect
158. C4⋊Q8C4⋊C4semidirect
159. C4⋊Q8C4⋊C4semidirect
171. C4×Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#340742
#219822
#218602
C2×#15182
#142772
#88682
#88672
#80262
#79062