Finite Group Databasebeta

← #148401

Subgroups — #148401

Order 512.

702 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#368C211yesmaximal, characteristic, complemented
2C22×#359C211yesmaximal, characteristic
3C2×#2138C211yesmaximal, characteristic
4C2×#1974C211yesmaximal, characteristic
5C2×#1382C211yesmaximal, characteristic, complemented
6C2×#1374C211yesmaximal, characteristic, complemented
7#10859C211yesmaximal, characteristic, complemented
8C22×#135C2211yescharacteristic
9C23×C16C2211yescharacteristic
10C23×C16C2211yescharacteristic
11C22×#108C2211yescharacteristic
12C22×#108C2211yescharacteristic
13C2×#427C2211yescharacteristic
14C22×#104—21no—
15C2×#408—21no—
16C22×C4×C8C2211yescharacteristic
17C2×#368—22no—
18C2×#368—22no—
19C2×#368—22no—
20C2×#359—22no—
21C2×#359—22no—
22C2×#359—22no—
23#2138—22no—
24#2138—22no—
25C2×#358C411yescharacteristic, complemented
26C2×#358C411yescharacteristic, complemented
27C2×#357C411yescharacteristic, complemented
28C2×#357C411yescharacteristic, complemented
29#1974—22no—
30#1974—22no—
31C2×#339—21no—
32#1729—21no—
33#1382—22no—
34#1374—22no—
35C22×#41—21no—
36C2×#135—21no—
37C2×#135—21no—
38C2×#135—21no—
39C2×#135—21no—
40C2×#135—21no—
41C2×#135—21no—
42C2×#135—21no—
43C2×#135—21no—
44C2×#135—21no—
45C2×#135—21no—
46C2×#135D412yes—
47C2×#135Q812yes—
48C22×Q16C2×C411yescharacteristic
49C22×SD16—21no—
50C22×D8C2×C411yescharacteristic
51C22×M16—21no—
52C22×M16—21no—
53C23×C8D411yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#117—21no—
56C22×C42D411yescharacteristic
57C22×C16—21no—
58C22×C16—21no—
59C22×C16—21no—
60C22×C16—21no—
61C22×C16—21no—
62C22×C16—21no—
63C22×C16—21no—
64C22×C16—21no—
65C22×C16—21no—
66C22×C16—21no—
67C22×C16C2×C411yescharacteristic
68C22×C16C2×C411yescharacteristic
69C22×C16D412yes—
70C22×C16D412yes—
71C22×C16D412yes—
72C2×#108—21no—
73C2×#108—21no—
74C2×#108—21no—
75C2×#108—21no—
76C2×#108—21no—
77C2×#108—21no—
78C2×#108—21no—
79C2×#108—21no—
80C2×#108—21no—
81C2×#108—21no—
82C2×#108D412yes—
83C2×#108D412yes—
84#427—21no—
85#427—21no—
86#427—21no—
87#427—21no—
88C2×#107C2×C411yescharacteristic
89C2×#107C2×C411yescharacteristic
90C2×#106—21no—
91C2×#104—24no—
92C2×#104—42no—
93C2×#104—42no—
94#408—24no—
95#408—42no—
96C2×C22⋊C8—42no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103#368—28no—
104#359—28no—
105#358—22no—
106#358—22no—
107#357—22no—
108#357—22no—
109#339—42no—
110#336—42no—
111C2×#41—22no—
112C2×#41—22no—
113C2×#41—41no—
114C2×#41—41no—
115C2×#41—41no—
116C2×#41—41no—
117C2×#41—41no—
118C2×#41—41no—
119C2×#41—41no—
120C2×#41—41no—
121C2×#41—41no—
122C2×#41—41no—
123C22×Q8—21no—
124C22×D4—21no—
125C22×D4—41no—
126C23×C4—21no—
127C23×C4—41no—
128C23×C4C2×D411yescharacteristic
129#135—22no—
130#135—22no—
131#135—22no—
132#135—22no—
133#135—22no—
134#135—22no—
135#135—22no—
136#135—22no—
137#135—41no—
138#135—41no—
139#135—41no—
140#135—41no—
141#135—41no—
142#135—41no—
143#135—41no—
144#135—41no—
145C2×Q16—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16C4⋊C412yes—
151C2×SD16—22no—
152C2×SD16—22no—
153C2×SD16—41no—
154C2×SD16—41no—
155C2×SD16—41no—
156C2×SD16—41no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8C4⋊C412yes—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—41no—
166C2×M16—41no—
167C2×M16—41no—
168C2×M16—41no—
169C2×M16—41no—
170C2×M16—41no—
171C2×M16—41no—
172C2×M16—41no—
173C2×M16—41no—
174C2×M16—41no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—41no—
182C22×C8—41no—
183C22×C8D812yes—
184C22×C8C22×C411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×Q811yescharacteristic
192C22×C8#4111yescharacteristic
193C22×C8#4111yescharacteristic
194C22×C8#4111yescharacteristic
195C22×C8#4111yescharacteristic
196C22×C8#4111yescharacteristic
197C22×C8#4111yescharacteristic
198#117—41no—
199#117—41no—
200#117—41no—
201#117—41no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C22⋊C4—41no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—41no—
211C2×C42D812yes—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—22no—
217C2×C16—22no—
218C2×C16—22no—
219C2×C16—22no—
220C2×C16—22no—
221C2×C16—22no—
222C2×C16—22no—
223C2×C16—41no—
224C2×C16—41no—
225C2×C16—41no—
226C2×C16—41no—
227C2×C16C22⋊C412yes—
228C2×C16C22⋊C412yes—
229#108—22no—
230#108—22no—
231#108—22no—
232#108—22no—
233#108—41no—
234#108—41no—
235#108—41no—
236#108—41no—
237#107—21no—
238#107—21no—
239#107—21no—
240#107—21no—
241#106—41no—
242#106—41no—
243#104—48no—
244C22⋊C8—48no—
245C4×C8—41no—
246C4×C8—41no—
247C4×C8—41no—
248C4×C8—41no—
249C24—41no—
250C24#13511yescharacteristic
251#41—42no—
252#41—42no—
253#41—42no—
254#41—42no—
255#41—42no—
256#41—42no—
257#41—42no—
258#41—42no—
259#41—81no—
260#41—81no—
261#41—81no—
262#41—81no—
263#41—81no—
264#41—81no—
265#41—81no—
266#41—81no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×D4—22no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—81no—
282C2×D4—81no—
283C2×D4—81no—
284C2×D4—81no—
285C22×C4—24no—
286C22×C4—22no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—81no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4C4×D411yescharacteristic
300C22×C4#12011yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4C22⋊Q811yescharacteristic
304C22×C4#12311yescharacteristic
305C22×C4#12711yescharacteristic
306C22×C4C2×D811yescharacteristic
307C22×C4C2×D811yescharacteristic
308C22×C4#13511yescharacteristic
309C22×C4AGL(1,ℤ/8)11yescharacteristic
310C22×C4AGL(1,ℤ/8)11yescharacteristic
311C22×C4#13711yescharacteristic
312C22×C4#13711yescharacteristic
313Q16—22no—
314Q16—22no—
315Q16—41no—
316Q16—41no—
317SD16—42no—
318SD16—42no—
319SD16—42no—
320SD16—42no—
321D8—22no—
322D8—22no—
323D8—41no—
324D8—41no—
325M16—42no—
326M16—42no—
327M16—42no—
328M16—42no—
329M16—81no—
330M16—81no—
331M16—81no—
332M16—81no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—81no—
343C2×C8—81no—
344C2×C8—81no—
345C2×C8—81no—
346C2×C8C2×C22⋊C411yescharacteristic
347C2×C8C2×C4⋊C411yescharacteristic
348C2×C8#11711yescharacteristic
349C2×C8C4×D411yescharacteristic
350C2×C8C4×D411yescharacteristic
351C2×C8C4×D411yescharacteristic
352C2×C8#12011yescharacteristic
353C2×C8#12011yescharacteristic
354C2×C8#12111yescharacteristic
355C2×C8#12111yescharacteristic
356C2×C8#12111yescharacteristic
357C2×C8#12111yescharacteristic
358C2×C8#12111yescharacteristic
359C2×C8#12111yescharacteristic
360C2×C8C22⋊Q811yescharacteristic
361C2×C8C22⋊Q811yescharacteristic
362C2×C8C22⋊Q811yescharacteristic
363C2×C8C22⋊Q811yescharacteristic
364C2×C8C22⋊Q811yescharacteristic
365C2×C8#12311yescharacteristic
366C2×C8#12311yescharacteristic
367C2×C8#12311yescharacteristic
368C2×C8#12311yescharacteristic
369C2×C8#12411yescharacteristic
370C2×C8#12411yescharacteristic
371C2×C8#12411yescharacteristic
372C2×C8#12611yescharacteristic
373C2×C8#12611yescharacteristic
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C22⋊C4—81no—
379C22⋊C4—81no—
380C42—22no—
381C42—22no—
382C42—41no—
383C42—41no—
384C42—81no—
385C42—81no—
386C16—22no—
387C16—22no—
388C16—42no—
389C23—22no—
390C23—22no—
391C23—42no—
392C23—82no—
393C23—82no—
394C23—81no—
395C23—81no—
396C23C4×D811yescharacteristic
397C23#45211yescharacteristic
398C23#46411yescharacteristic
399C23#46611yescharacteristic
400C23#47811yescharacteristic
401C23#48211yescharacteristic
402C23#49511yescharacteristic
403Q8—42no—
404Q8—42no—
405Q8—81no—
406Q8—81no—
407D4—42no—
408D4—42no—
409D4—82no—
410D4—82no—
411D4—82no—
412D4—82no—
413D4—81no—
414D4—81no—
415C2×C4—24no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4#38511yescharacteristic
432C2×C4#38811yescharacteristic
433C2×C4#39011yescharacteristic
434C2×C4#39211yescharacteristic
435C2×C4#39411yescharacteristic
436C2×C4#39611yescharacteristic
437C2×C4#39911yescharacteristic
438C2×C4C4×D811yescharacteristic
439C2×C4#44011yescharacteristic
440C2×C4#44011yescharacteristic
441C2×C4#44711yescharacteristic
442C2×C4#44711yescharacteristic
443C2×C4#45211yescharacteristic
444C2×C4#45911yescharacteristic
445C2×C4#45911yescharacteristic
446C2×C4#46411yescharacteristic
447C2×C4#46611yescharacteristic
448C2×C4#46811yescharacteristic
449C2×C4#46811yescharacteristic
450C2×C4#47411yescharacteristic
451C2×C4#47411yescharacteristic
452C2×C4#47811yescharacteristic
453C2×C4#48011yescharacteristic
454C2×C4#48011yescharacteristic
455C2×C4#48211yescharacteristic
456C2×C4#49311yescharacteristic
457C2×C4#49611yescharacteristic
458C2×C4#49711yescharacteristic
459C8—42no—
460C8—82no—
461C8—82no—
462C8#38011yescharacteristic
463C8#38511yescharacteristic
464C8#38611yescharacteristic
465C8#38811yescharacteristic
466C8#39311yescharacteristic
467C8#39411yescharacteristic
468C8#39611yescharacteristic
469C8#39711yescharacteristic
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—82no—
479C22—84no—
480C22—162no—
481C22—161no—
482C22—161no—
483C22#186211yescharacteristic
484C22#192511yescharacteristic
485C22#197011yescharacteristic
486C22#201511yescharacteristic
487C22#203311yescharacteristic
488C22#206711yescharacteristic
489C22#209211yescharacteristic
490C22#217812yes—
491C22#217912yes—
492C4—44no—
493C4—82no—
494C4—161no—
495C4—161no—
496C4#143911yescharacteristic
497C4#186211yescharacteristic
498C4#192511yescharacteristic
499C4#197311yescharacteristic
500C4#201111yescharacteristic
501C4#203311yescharacteristic
502C4#206711yescharacteristic
503C4#209311yescharacteristic
504C2—21no—
505C2—21no—
506C2—21no—
507C2—21no—
508C2—162no—
509C2#1112311yescharacteristic
510C2#1293912yes—
511C2#1313412yes—
512C2#1324111yescharacteristic
513C2#1324211yescharacteristic
subgroupG/H
centerC22×C4C2×D8
commutatorC22×C8C22×C4
Frattini, cosocleC23×C8C23
socleC23C4×D8
prime radicalC22×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C8 ◁ C22×C4×C8 ◁ G
LECSG ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#368C2semidirect
5. C2×#1382C2semidirect
6. C2×#1374C2semidirect
7. #10859C2semidirect
25. C2×#358C4semidirect
26. C2×#358C4semidirect
27. C2×#357C4semidirect
28. C2×#357C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3682
C22×#3592
C2×#21382
C2×#19742
C2×#13822
C2×#13742
#108592