Finite Group Databasebeta

← #194170

Subgroups — #194170

Order 512.

1260 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
1C22×#477C211yesmaximal, characteristic
2Q8×#117C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#8975C214yesmaximal, complemented
7#8974C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#477—24no—
13C2×#477—21no—
14C2×#477—21no—
15C2×#477C418yescomplemented
16C2×#477C2212yes—
17C2×#477C2212yes—
18C2×#431C2211yescharacteristic
19C22×#107—21no—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×#103—21no—
23#2355—22no—
24#2355C2212yes—
25Q8×C4⋊C4C2212yes—
26#2345—22no—
27#2345—21no—
28Q8×C22⋊C4—21no—
29#2295—22no—
30#2295—21no—
31#2285—21no—
32#2285C2211yescharacteristic
33C42×Q8—21no—
34C4×#117—21no—
35C4×#117C2211yescharacteristic
36#1988—22no—
37#1987C2214yes—
38#1981—22no—
39#1956C2214yes—
40#1787—22no—
41#1736—21no—
42#1736—21no—
43#1487—24no—
44C2×Q8⋊C8C2212yes—
45C2×Q8⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C23×Q8—21no—
49C23×Q8D411yescharacteristic
50C22×M16—21no—
51C23×C8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8C2312yes—
60C2×C4×Q8—22no—
61C2×C4×Q8—21no—
62C2×C4×Q8—21no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8C2×C414yes—
66C2×C4×Q8D414yes—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4D411yescharacteristic
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#477—28no—
81#477—28no—
82#477—24no—
83#477—24no—
84#477C2×C418yes—
85#477C2×C418yes—
86#431—22no—
87#431—22no—
88C2×#107—28no—
89C2×#107—22no—
90C2×#107—22no—
91#423—22no—
92#423—22no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2×C414yes—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100C2×#103—28no—
101C2×#103—22no—
102C2×#103—22no—
103C2×#103—28no—
104C2×C22⋊C8—21no—
105C2×C4×C8C2312yes—
106#391—24no—
107#391—22no—
108#389—22no—
109#389—24no—
110#387—24no—
111#387—24no—
112#387—24no—
113#384—22no—
114#384C2312yes—
115#382—22no—
116#382—21no—
117#380—22no—
118#380—22no—
119#380—21no—
120#380D412yes—
121C4×C4⋊C4—24no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D412yes—
129C4×C4⋊C4D412yes—
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—22no—
136#376—21no—
137C43—22no—
138C43—21no—
139#340—24no—
140#340—24no—
141Q8⋊C8—24no—
142Q8⋊C8—24no—
143C22×Q8—21no—
144C22×Q8—22no—
145C22×Q8—24no—
146C22×Q8—24no—
147C22×Q8—22no—
148C22×Q8—42no—
149C22×Q8—41no—
150C22×Q8—42no—
151C22×Q8—84no—
152C22×Q8C22⋊C414yes—
153C22×Q8C2×D411yescharacteristic
154C22×Q8C2×D411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—41no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C2×M16—42no—
162C2×M16—42no—
163C2×M16—42no—
164C22×C8—24no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—44no—
169C4⋊Q8—24no—
170C4⋊Q8—24no—
171C4⋊Q8—24no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8C22×C414yes—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×Q8—44no—
190C4×Q8—44no—
191C4×Q8C22⋊C414yes—
192C4×Q8C22⋊C414yes—
193C4×Q8C22×C412yes—
194C4×Q8C22×C412yes—
195C4×Q8C2×D412yes—
196C4×Q8C2×D412yes—
197#117—22no—
198#117—22no—
199#117—22no—
200#117—42no—
201#117—42no—
202#117Q1614yes—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—42no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4C22⋊C414yes—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—21no—
232C2×C42—24no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—24no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—24no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—22no—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42C22×C412yes—
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D411yescharacteristic
249C2×C42C2×D411yescharacteristic
250C2×C42#4112yes—
251C2×C42#4112yes—
252C2×C42C2411yescharacteristic
253#107—28no—
254#107—28no—
255C4⋊C8—24no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—22no—
259C4⋊C8—42no—
260C4⋊C8—42no—
261C4⋊C8C22×C414yes—
262#103—28no—
263#103—28no—
264#103—88no—
265C22⋊C8—42no—
266C4×C8—22no—
267C4×C8—22no—
268#95—24no—
269#95—24no—
270#95—24no—
271#95—24no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276C24#12011yescharacteristic
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—24no—
281C2×Q8—28no—
282C2×Q8—24no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—44no—
286C2×Q8—44no—
287C2×Q8—48no—
288C2×Q8—48no—
289C2×Q8—42no—
290C2×Q8—44no—
291C2×Q8—84no—
292C2×Q8—88no—
293C2×Q8—88no—
294C2×Q8C2×C22⋊C412yes—
295C2×Q8C2×C22⋊C412yes—
296C2×Q8#13714yes—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—24no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—44no—
312C22×C4—82no—
313C22×C4—84no—
314C22×C4C2×C22⋊C412yes—
315C22×C4C4×D412yes—
316C22×C4C4×D412yes—
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4#12312yes—
323C22×C4C2×Q1612yes—
324C22×C4#13512yes—
325C22×C4#13712yes—
326C22×C4#13712yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330M16—84no—
331C2×C8—24no—
332C2×C8—48no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—44no—
354C4⋊C4—48no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—84no—
364C4⋊C4C2×C22⋊C412yes—
365C4⋊C4C2×C22⋊C412yes—
366C4⋊C4C2×Q1614yes—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—44no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—24no—
375C42—24no—
376C42—24no—
377C42—48no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—44no—
382C42—82no—
383C42—84no—
384C42C2×C22⋊C412yes—
385C42C4×D412yes—
386C42C4×D412yes—
387C42#12112yes—
388C42#12112yes—
389C42#12312yes—
390C42C23×C411yescharacteristic
391C42C22×D411yescharacteristic
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C23—22no—
395C23—22no—
396C23#38612yes—
397C23#45112yes—
398C23#45212yes—
399C23C2×#12011yescharacteristic
400Q8—24no—
401Q8—22no—
402Q8—22no—
403Q8—22no—
404Q8—48no—
405Q8—48no—
406Q8—42no—
407Q8—84no—
408Q8—88no—
409Q8#41914yes—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—48no—
425C2×C4—42no—
426C2×C4—42no—
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—42no—
436C2×C4—44no—
437C2×C4—84no—
438C2×C4—88no—
439C2×C4—84no—
440C2×C4#38612yes—
441C2×C4#38612yes—
442C2×C4#38612yes—
443C2×C4C4×Q1614yes—
444C2×C4#44114yes—
445C2×C4#45112yes—
446C2×C4#45212yes—
447C2×C4#46214yes—
448C2×C4#46414yes—
449C2×C4C22×C22⋊C411yescharacteristic
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4C2×#12111yescharacteristic
454C2×C4C2×#12311yescharacteristic
455C2×C4C22×Q1611yescharacteristic
456C2×C4C2×#13511yescharacteristic
457C2×C4C2×#13712yes—
458C2×C4C2×#13711yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57512yes—
461C8—44no—
462C8—82no—
463C8—82no—
464C22—24no—
465C22—24no—
466C22—22no—
467C22—22no—
468C22#187814yes—
469C22C2×#38611yescharacteristic
470C22C2×#45111yescharacteristic
471C22C2×#45211yescharacteristic
472C22#300712yes—
473C22#302112yes—
474C4—22no—
475C4—24no—
476C4—21no—
477C4—21no—
478C4—22no—
479C4—44no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—82no—
484C4—84no—
485C4—84no—
486C4#187814yes—
487C4C2×#41911yescharacteristic
488C4#289611yescharacteristic
489C4C2×C4×Q1611yescharacteristic
490C4C2×#44111yescharacteristic
491C4C2×#46211yescharacteristic
492C4C2×#46411yescharacteristic
493C4#309512yes—
494C2—24no—
495C2#1878912yes—
496C2#1879112yes—
497C2#2091111yescharacteristic
498C2C2×#187811yescharacteristic
499C2#2094511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8975C2semidirect
7. #8974C2semidirect
15. C2×#477C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
Q8×#1172
#218042
#213132
C2×#14872
#89752
#89742
#80452
#78912