Finite Group Databasebeta

← #188522

Subgroups — #188522

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34212C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic
3C2×#1520C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic
5#13965C211yesmaximal, characteristic
6#8578C214yesmaximal, complemented
7#8572C214yesmaximal, complemented
8#8023C211yesmaximal, characteristic
9C2×#1294C211yesmaximal, characteristic
10C22×C22⋊Q8C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2368—22no—
14#2360C414yescomplemented
15#2345—22no—
16#2345C412yescomplemented
17Q8×C22⋊C4C412yescomplemented
18#2329—22no—
19#2315—21no—
20C4×C22⋊Q8C2214yes—
21C2×#385—21no—
22C2×#385—21no—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25#2273—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27C22×#95C2211yescharacteristic
28#1850—22no—
29#1850—22no—
30#1797C2214yes—
31#1520—24no—
32C2×#328—24no—
33C2×#328C2214yes—
34C2×#328C2212yes—
35C2×#328C2212yes—
36#1444—22no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39#1294C2214yes—
40#1294C2212yes—
41#1294C2212yes—
42C23×Q8C2×C411yescharacteristic
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8—21no—
46C23×C8—21no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8—22no—
51C2×C22⋊Q8C2×C412yes—
52C2×C22⋊Q8D414yes—
53C2×C22⋊Q8Q814yes—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8C2×C414yescomplemented
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4D411yescharacteristic
66C22×C22⋊C4D411yescharacteristic
67C22×C42C2312yes—
68C2×C4⋊C8—24no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C2×C22⋊C8C2312yes—
76C2×C22⋊C8C2312yes—
77#391—24no—
78#391C2×C414yescomplemented
79#389—24no—
80#389C2×C414yescomplemented
81#387—24no—
82#387—24no—
83#387—24no—
84#385—24no—
85#385—24no—
86#385—44no—
87#385—44no—
88#380—21no—
89#380—21no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380C2×C412yes—
95#380C2312yes—
96C4×C4⋊C4—24no—
97C4×C4⋊C4C2×C414yescomplemented
98C4×C22⋊C4—24no—
99C4×C22⋊C4—22no—
100C4×C22⋊C4—22no—
101C4×C22⋊C4—24no—
102C4×C22⋊C4C2×C412yes—
103C4×C22⋊C4C2×C412yes—
104C4×C22⋊C4D412yes—
105C4×C22⋊C4D412yes—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95D412yes—
113C2×#95C2312yes—
114C2×#95C2311yescharacteristic
115C2×#95C2311yescharacteristic
116#336—24no—
117#328—28no—
118#328—24no—
119#328—24no—
120C25C2×D411yescharacteristic
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—82no—
124C22×Q8C4⋊C414yescomplemented
125C22×Q8C22×C412yes—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—82no—
137C23×C4—82no—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D412yes—
146C23×C4C2×Q812yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—42no—
153C22×C8—42no—
154C22×C8—82no—
155C22×C8—82no—
156C22⋊Q8—24no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—22no—
162C22⋊Q8—22no—
163C22⋊Q8C4⋊C418yescomplemented
164C22⋊Q8C2×D414yes—
165C22⋊Q8C2×Q814yes—
166C4×Q8—84no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4C4⋊C414yescomplemented
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C22×C414yes—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—42no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4SD1614yes—
206C2×C22⋊C4Q1614yes—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—82no—
210C2×C42C22×C414yes—
211C2×C42C2×D414yes—
212C4⋊C8—84no—
213C22⋊C8—22no—
214C22⋊C8—22no—
215C22⋊C8—22no—
216C22⋊C8—22no—
217C22⋊C8—42no—
218C22⋊C8—42no—
219#95—22no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—44no—
232#95—42no—
233#95—44no—
234#95C22⋊C414yes—
235#95C22⋊C414yes—
236C24—42no—
237C24—42no—
238C24C2×C22⋊C411yescharacteristic
239C24C2×C22⋊C411yescharacteristic
240C24#12112yes—
241C24C22⋊Q812yes—
242C24C2×SD1612yes—
243C24C2×Q1612yes—
244C24AGL(1,ℤ/8)12yes—
245C24#13712yes—
246C24C22×D411yescharacteristic
247C2×Q8—22no—
248C2×Q8—24no—
249C2×Q8—44no—
250C2×Q8—82no—
251C2×Q8—84no—
252C2×Q8—84no—
253C2×Q8C2×C4⋊C414yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—48no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4—88no—
299C22×C4—88no—
300C22×C4—84no—
301C22×C4—84no—
302C22×C4C2×C22⋊C414yes—
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C414yes—
306C22×C4C2×C4⋊C412yes—
307C22×C4C2×C4⋊C412yes—
308C22×C4#11712yes—
309C22×C4#11712yes—
310C22×C4#12112yes—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4C22⋊Q812yes—
314C22×C4C22⋊Q812yes—
315C22×C4C22⋊Q812yes—
316C22×C4C2×SD1614yes—
317C22×C4C2×Q1614yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q811yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—84no—
329C2×C8—84no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—28no—
334C4⋊C4—24no—
335C4⋊C4—44no—
336C4⋊C4—84no—
337C4⋊C4—82no—
338C4⋊C4—84no—
339C4⋊C4—88no—
340C4⋊C4—88no—
341C4⋊C4—84no—
342C4⋊C4C2×C4⋊C414yes—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—28no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—44no—
351C22⋊C4—84no—
352C22⋊C4#10318yes—
353C22⋊C4#10318yes—
354C42—44no—
355C42—44no—
356C42—84no—
357C42—84no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—44no—
368C23—44no—
369C23—42no—
370C23—44no—
371C23—44no—
372C23#38012yes—
373C23#38012yes—
374C23C2×#10312yes—
375C23C2×#10312yes—
376C23#41712yes—
377C23#41712yes—
378C23#46014yes—
379C23#46214yes—
380C23#47514yes—
381C23#47714yes—
382C23C22×C22⋊C411yescharacteristic
383C23C2×#12111yescharacteristic
384C23C2×C22⋊Q811yescharacteristic
385C23C22×SD1611yescharacteristic
386C23C22×Q1611yescharacteristic
387C23C2×AGL(1,ℤ/8)11yescharacteristic
388C23C2×#13711yescharacteristic
389Q8—44no—
390Q8—44no—
391Q8—84no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—84no—
430C2×C4—84no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—816no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—164no—
447C2×C4#38012yes—
448C2×C4#38012yes—
449C2×C4C2×#10318yes—
450C2×C4C22×C4⋊C411yescharacteristic
451C2×C4C2×#11711yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×C22⋊Q811yescharacteristic
454C8—82no—
455C8—82no—
456C22—21no—
457C22—22no—
458C22—21no—
459C22—22no—
460C22—21no—
461C22—22no—
462C22—21no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—24no—
467C22—24no—
468C22—22no—
469C22—22no—
470C22—24no—
471C22—24no—
472C22—44no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22#180314yes—
477C22#180314yes—
478C22C2×#38011yescharacteristic
479C22#288312yes—
480C22C22×#10311yescharacteristic
481C22C2×#41711yescharacteristic
482C22#290112yes—
483C22C2×#46011yescharacteristic
484C22C2×#46211yescharacteristic
485C22C2×#47511yescharacteristic
486C22C2×#47711yescharacteristic
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—22no—
492C4—22no—
493C4—28no—
494C4—44no—
495C4—44no—
496C4—44no—
497C4—84no—
498C4—84no—
499C4—161no—
500C4—161no—
501C2—21no—
502C2—21no—
503C2—21no—
504C2—24no—
505C2—24no—
506C2—21no—
507C2#1761412yes—
508C2#1762212yes—
509C2#2063311yescharacteristic
510C2C2×#180311yescharacteristic
511C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8578C2semidirect
7. #8572C2semidirect
14. #2360C4semidirect
16. #2345C4semidirect
17. Q8×C22⋊C4C4semidirect
55. C2×C4×Q8C2×C4semidirect
78. #391C2×C4semidirect
80. #389C2×C4semidirect
97. C4×C4⋊C4C2×C4semidirect
124. C22×Q8C4⋊C4semidirect
163. C22⋊Q8C4⋊C4semidirect
185. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342122
#219302
C2×#15202
C22×#3282
#139652
#85782
#85722
#80232
C2×#12942