Finite Group Databasebeta

← #117930

Subgroups — #117930

Order 512.

637 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20826C211yesmaximal, characteristic, complemented
2#17546C211yesmaximal, characteristic
3#17473C211yesmaximal, characteristic
4#17466C211yesmaximal, characteristic
5#7950C211yesmaximal, characteristic, complemented
6#7935C211yesmaximal, characteristic, complemented
7C2×#1279C211yesmaximal, characteristic, complemented
8#3444C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C22×C4×C8C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1848—21no—
16#1837—21no—
17#1810—22no—
18C4×#108—22no—
19#1509—22no—
20#1504—22no—
21#1503—21no—
22#1502—21no—
23#1494—21no—
24#1494—21no—
25#1493C411yescharacteristic, complemented
26#1493C411yescharacteristic, complemented
27#1492C411yescharacteristic, complemented
28#1492C411yescharacteristic, complemented
29#1483—22no—
30#1481—21no—
31#1480—21no—
32#1477—22no—
33#1476—21no—
34#1476—21no—
35#1285—22no—
36#1283—22no—
37#1279—21no—
38#1279—21no—
39C23×C8—21no—
40C23×C8—21no—
41#533C2×C411yescharacteristic
42#533D411yescharacteristic
43#533Q811yescharacteristic
44#532C2×C411yescharacteristic
45#532D411yescharacteristic
46#532Q811yescharacteristic
47#529—21no—
48#529—21no—
49#529—21no—
50#529—21no—
51#529—21no—
52#529—21no—
53#525—21no—
54#525—21no—
55#525—21no—
56C4×#41—21no—
57C4×#41—21no—
58C4×#41—21no—
59C22×C4⋊C4—21no—
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74C2×#108—24no—
75#423—21no—
76#423—21no—
77#423C2×C411yescharacteristic
78#423C2×C411yescharacteristic
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8D412yes—
85#407—21no—
86#407—21no—
87#407D412yes—
88C4×M16—21no—
89C4×M16—21no—
90C4×M16D412yes—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8C2×C411yescharacteristic
95C2×C4×C8C2×C411yescharacteristic
96C2×C4×C8D411yescharacteristic
97C2×C4×C8D411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8D411yescharacteristic
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4D412yes—
104C2×#95—21no—
105#341—24no—
106#328—24no—
107#327—24no—
108Q8⋊C8—24no—
109Q8⋊C8—21no—
110Q8⋊C8—21no—
111Q8⋊C8—21no—
112Q8⋊C8—21no—
113D4⋊C8—24no—
114D4⋊C8—21no—
115D4⋊C8—21no—
116D4⋊C8—21no—
117D4⋊C8—21no—
118C2×#41—21no—
119C2×#41—21no—
120C2×#41—21no—
121C23×C4—41no—
122C23×C4C2×D411yescharacteristic
123C23×C4#4111yescharacteristic
124C23×C4#4111yescharacteristic
125C2×M16—21no—
126C2×M16—21no—
127C2×M16—22no—
128C2×M16—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—44no—
137C22×C8—41no—
138C22×C8—41no—
139C22×C8—81no—
140C22×C8—81no—
141C4⋊Q8C4⋊C411yescharacteristic
142C4⋊Q8C4⋊C411yescharacteristic
143C4⋊Q8C4⋊C411yescharacteristic
144#127C4⋊C411yescharacteristic
145#126—41no—
146#126—41no—
147#125—21no—
148#125—21no—
149#125—21no—
150#124—21no—
151#124—21no—
152#124—21no—
153#123—41no—
154#123—41no—
155#123—41no—
156C22⋊Q8—21no—
157C22⋊Q8—21no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160C22⋊Q8—21no—
161C22⋊Q8—21no—
162#121—21no—
163#121—21no—
164#121—21no—
165#121—21no—
166#121—21no—
167#121—21no—
168#120—41no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×Q8—21no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187#117—21no—
188#117—21no—
189#117—21no—
190#117—41no—
191#117—41no—
192#117—41no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—41no—
197C2×C42—22no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—81no—
201C2×C42—81no—
202C2×C42C22×C411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×Q811yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42#4111yescharacteristic
214#108—48no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—41no—
222C4⋊C8—41no—
223C4⋊C8—41no—
224C4⋊C8—41no—
225C4⋊C8—41no—
226C4⋊C8—41no—
227C22⋊C8—22no—
228C22⋊C8—22no—
229C22⋊C8—41no—
230C22⋊C8—41no—
231C22⋊C8—41no—
232C22⋊C8—41no—
233#97—22no—
234#97—41no—
235#97—41no—
236C4×C8—21no—
237C4×C8—21no—
238C4×C8—22no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—41no—
244C4×C8—41no—
245C4×C8C22⋊C411yescharacteristic
246C4×C8C22⋊C411yescharacteristic
247C4×C8C22⋊C411yescharacteristic
248C4×C8C22⋊C411yescharacteristic
249#95—22no—
250#95—41no—
251C24#12411yescharacteristic
252#41—42no—
253#41—42no—
254#41—41no—
255#41—41no—
256#41—41no—
257#41—41no—
258#41—82no—
259C2×Q8—21no—
260C2×Q8—21no—
261C2×Q8—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4#11711yescharacteristic
283C22×C4C4×D411yescharacteristic
284C22×C4C4×D411yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4C22⋊Q811yescharacteristic
292C22×C4C22⋊Q811yescharacteristic
293C22×C4C22⋊Q811yescharacteristic
294C22×C4#12311yescharacteristic
295C22×C4#12311yescharacteristic
296C22×C4#12311yescharacteristic
297C22×C4#12411yescharacteristic
298C22×C4#12411yescharacteristic
299C22×C4#12611yescharacteristic
300M16—44no—
301M16—44no—
302M16—42no—
303M16—42no—
304C2×C8—24no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—82no—
320C2×C8—81no—
321C2×C8—81no—
322C2×C8—82no—
323C4⋊C4—24no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—44no—
331C4⋊C4—42no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—82no—
351C42—81no—
352C42—81no—
353C42C2×C22⋊C411yescharacteristic
354C42C2×C4⋊C411yescharacteristic
355C42C4×D411yescharacteristic
356C42C4×D411yescharacteristic
357C42#12011yescharacteristic
358C42#12111yescharacteristic
359C42#12111yescharacteristic
360C42#12111yescharacteristic
361C42#12111yescharacteristic
362C42C22⋊Q811yescharacteristic
363C42C22⋊Q811yescharacteristic
364C42C22⋊Q811yescharacteristic
365C42#12311yescharacteristic
366C42#12311yescharacteristic
367C42#12611yescharacteristic
368C42#12711yescharacteristic
369C23—21no—
370C23—21no—
371C23—22no—
372C23—22no—
373C23—41no—
374C23#38811yescharacteristic
375C23#39311yescharacteristic
376C23#39411yescharacteristic
377Q8—41no—
378Q8—41no—
379Q8—42no—
380Q8—81no—
381D4—41no—
382D4—41no—
383D4—42no—
384D4—81no—
385D4—81no—
386D4—81no—
387D4—82no—
388D4—82no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—82no—
417C2×C4—88no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4#38011yescharacteristic
425C2×C4#38511yescharacteristic
426C2×C4#38511yescharacteristic
427C2×C4#38611yescharacteristic
428C2×C4#38811yescharacteristic
429C2×C4#39011yescharacteristic
430C2×C4#39211yescharacteristic
431C2×C4#39411yescharacteristic
432C2×C4#39611yescharacteristic
433C2×C4#39611yescharacteristic
434C2×C4#39711yescharacteristic
435C2×C4#39911yescharacteristic
436C2×C4#44312yes—
437C2×C4#44412yes—
438C2×C4#47111yescharacteristic
439C2×C4#47111yescharacteristic
440C2×C4#47211yescharacteristic
441C2×C4#47311yescharacteristic
442C8—22no—
443C8—22no—
444C8—41no—
445C8—41no—
446C8—82no—
447C8—82no—
448C8—82no—
449C8—81no—
450C8—81no—
451C22—24no—
452C22—24no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22#143911yescharacteristic
463C22#192912yes—
464C22#203112yes—
465C22#205111yescharacteristic
466C22#205211yescharacteristic
467C4—21no—
468C4—21no—
469C4—21no—
470C4—21no—
471C4—21no—
472C4—21no—
473C4—21no—
474C4—21no—
475C4—84no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4#181511yescharacteristic
480C4#181611yescharacteristic
481C4#186612yes—
482C4#197411yescharacteristic
483C4#197511yescharacteristic
484C4#205111yescharacteristic
485C4#205211yescharacteristic
486C2—21no—
487C2—21no—
488C2—22no—
489C2—22no—
490C2—161no—
491C2#1108111yescharacteristic
492C2#1110511yescharacteristic
493C2#1112511yescharacteristic
subgroupG/H
centerC2×C4#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radical#3444C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #20826C2semidirect
5. #7950C2semidirect
6. #7935C2semidirect
7. C2×#1279C2semidirect
25. #1493C4semidirect
26. #1493C4semidirect
27. #1492C4semidirect
28. #1492C4semidirect

Maximal subgroups

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

subgroupindex
#208262
#175462
#174732
#174662
#79502
#79352
C2×#12792